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Chapter 28 Mathematics in Physics free study material by TEACHING CARE online tuition and coaching classes

Chapter 28 Mathematics in Physics free study material by TEACHING CARE online tuition and coaching classes

 

Introduction.

Mathematics is the language of physics. It becomes very easier to describe, understand and apply the physical principles, if we have a good knowledge of mathematics.

For example : Newton’s law of gravitation states that every body in this universe attracts every other body with a force which is directly proportional to the product of their masses and is inversely proportional to the square of the distance between them.

 

 

This law can be expressed by a single mathematical relationship

F µ m1m2

r 2

or F = Gm1m2

r 2

 

 

The techniques of mathematics such as algebra, trigonometry, calculus, graph and logarithm can be used to make predictions from the basic equation.

If we are poor at grammar and vocabulary, it would be difficult for us to communicate our feelings, similarly for better understanding and expressing of physics the basic knowledge of mathematics is must.

In this introductory chapter we will learn some fundamental mathematics.

Algebra.

  • Quadratic equation : An equation of second degree is called a quadratic Standard quadratic equation ax 2 + bx + c = 0

Here a is called the coefficient of x2, b is called the coefficient of x and c is a constant term, x is the variable whose value (roots of the equation) are to be determined

 

Roots of the equation are : x =                2a

 

This formula can be written as

x = – Coefficient of x ±

 

(Coefficient of x)2 – 4(Coefficient of x 2 ) ´ (Constant term) 2(Coefficient of x 2 )

 

Note : @ If a and b be the roots of the quadratic equation then

Sum of roots a + b = – b and product of roots = c

a                                                     a

Problem 1.         Solve the equation 10x 2 – 27x + 5 = 0

Solution :          By comparing the given equation with standard equation a = 10, b = – 27, and c = 5

 

 

x =               2a

= – (-27) ±

(-27)2 – 4 ´ 10 ´ 5

2 ´ 10

= 27 ± 23

20

 

x1 = 27 + 235 and x 2 = 27 231

20        2                     20        5

 

 

\ Roots of the equation are

5 and 1 .

2         5

 

 

  • Binomial theorem : If n is any number positive, negative or fraction and x is any real number, such that

x < 1 i.e. x lies between – 1 and + 1 then according to binomial theorem

 

 

(1 + x)n

= 1 + nx n(n – 1) x 2 + n(n – 1)(n – 2) x 3 + …..

 

 

2!                  3 !

 

Here 2 ! (Factorial 2) = 2 ´ 1, 3 ! (Factorial 3) = 3 ´ 2 ´ 1 and 4 ! (Factorial 4) = 4 ´ 3 ´ 2 ´ 1

Note : @If |x| << 1 then only the first two terms are significant. It is so because the values of second and the higher order terms being very very small, can be neglected. So the expression can be written as

(1 + x)n = 1 + nx

(1 + x)n = 1 – nx

(1 – x)n = 1 – nx

(1 – x)n = 1 + nx Problem 2.         Evaluate (1001)1/3 upto six places of decimal. Solution :          (1001)1/3 = (1000 + 1)1/3 = 10(1 + 0.001)1/3

By comparing the given equation with standard equation (1 + x)n = 1 + nx + n(n  1) x 2 + ……

2!

 

x = 0.001 and n = 1/3

é

1 æ 1 – 1ö ´ (.001)2           ù

 

ê                                 ç           ÷                           ú

 

\ 10(1 + 0.001)1/ 3 = 10ê1 + 1 (0.001) + 3 è 3             ø

 

+ ….ú = 10 é1 + 0.00033 – 1 (0.000001) + ….ù

 

 

ê       3                          2!

ê

ë

ú            êë                         9                       úû

ú

û

 

= 10[1.0003301] = 10.003301(Approx.)

Problem 3.         The value of acceleration due to gravity (g) at a height h above the surface of earth is given by

 

 

g‘ =

gR 2

(R + h)2

. If h << R then

 

 

 

 

(a)

g‘ = gæ1 – h ö

(b)

g‘ = gæ1 – 2h ö

(c)

g‘ = gæ1 + h ö

(d)

g‘ = gæ1 + 2h ö

 

R
R
R
R

ç            ÷                                     ç             ÷                                ç           ÷                                      ç             ÷

è            ø                                     è             ø                                è           ø                                      è             ø

 

 

æ   R   ö2

æ       1     ö2     æ

h ö-2        é

h     (-2)(-3) æ h ö2            ù

 

Solution : (b)

g‘ = gç R h ÷

= gç 1 + h / R ÷

=ç1 + R ÷

= g ê1 + (-2)      +

R

2!     ç R ÷

+ …….ú

 

è            ø          è

ø       è           ø            êë

è     ø               úû

 

 

g‘ = gæ1 – 2h ö ( if h << R then by neglecting higher power of h .)

ç             ÷

R
R

è             ø

  • Arithmetic progression : It is a sequence of numbers which are arranged in increasing order and having a constant difference between

Example : 1, 3, 5, 7, 9, 11, 13, ……   or       2, 4, 6, 8, 10, 12, …..

 

 

In general arithmetic progression can be written as a0, a1, a2, a3, a4, a5 …….

 

  • nth term of arithmetic progression

an = a0 + (n – 1)d

 

a0 = First term, n = Number of terms, d = Common difference = (a1a0) or (a2a1) or (a3a2)

 

 

  • Sum of arithmetic progression Sn

= n [2a

2

+ (n – 1)d] = n [a

0                             2   0

  • an ]

 

Problem 4.         Find the sum of series 7 + 10 + 13 + 16 + 19 + 22 + 25

 

 

Solution :

Sn = n [a0

2

  • an

] = 7 [7 + 25] = 112           [As n = 7; a

2                                               0

= 7; an

= a7

= 25]

 

  • Geometric progression : It is a sequence of numbers in which every term is obtained by multiplying the previous term by a constant This constant quantity is called the common ratio.

Example :    4, 8, 16, 32, 64, 128 ……          or       5, 10, 20, 40, 80, …….

In general geometric progression can be written as a, ar, ar2, ar3, ar4, …. Here a = first term, r = common ratio

 

  • Sum of ‘n’ terms of P.

 

 

 

  • Sum of infinite terms of P.

S a(1 – r n )

n          1 – r

S a (r n – 1)

n           r – 1

S     =  a

if r < 1

if r > 1 if r < 1

 

¥     1 – r

S     =  a

¥     r – 1

if r > 1

 

Problem 5.         Find the sum of series Q = 2q + q + q + q

 

 

+ ……

 

3    9    27

Solution :          Above equation can be written as Q = q + éq + q + q + q +……….. ù

ëê      3    9    27         ûú

é           ù

By using the formula of sum of infinite terms of G.P. Q = q + ê q   ú = q + 3 q = 5 q

 

 

 

(5)  Some common formulae of algebra

(i) (a + b)2 = a2 + b2 + 2ab

  • (a b)2 = a2 + b2 – 2ab
  • (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
  • (a + b) (a b) = a2b2

(v) (a + b)3 = a3 + b3 + 3ab(a + b)

(vi) (a b)3 = a3b3 – 3ab(a b)

(vii) (a + b)2 – (a b)2 = 4ab

ê

ê

ê1 –

ë

ú

1

ú             2       2

3 úû

 

 

 

(viii) (a + b)2 + (a b)2 = 2(a2 + b2)

(ix) a 3b 3 =(a b)(a 2 + b 2 + ab)

(x) a 3 + b 3 = (a + b)(a 2 + b 2ab)

 

(6)   Componendo and dividendo method : If

 

 

ac

 

 

 

 

then

 

 

a + bc + d

 

 

 Trigonometry.

b      d               a b

c d

 

 

  • Trigonometric ratio : In right angled triangle ABC, the largest side AC, which is opposite to the right angle is called hypotenuse, and if angle considered is q , then side opposite to q, AB, will be termed as perpendicular and BC is called the base of the

 

sinq = Perpendicular = AB

Hypotenuse    AC

cosecq

=  Hypotenuse  = AC

Perpendicular    AB

 

cosq =     Base    = BC                                      sec q Hypotenuse = AC

Hypotenuse   AC                                                            Base        BC

 

 

tanq

= Perpendicular = AB

Base         BC

cotq =

Base Perpendicular

BC

AB

 

(2)  Value of trigonometric ratio of standard angles

 

Angle 0o 30o 45o 60o 90o 120o 135o 150o 180o 270o 360o
sinq 0 1/2 1/Ö2 Ö3/2 1 Ö3/2 1/ Ö2 1/2 0 – 1 0
cosq 1 Ö3/2 1/Ö2 1/2 0 – 1/2 – 1/Ö2 – Ö3/2 – 1 0 1
tanq 0 1/Ö3 1 Ö3 ¥ – Ö3 – 1 – 1/Ö3 0 – ¥ 0
  • Important points :
    • Value of sinq or cosq lies between – 1 and +1, however tanq and cotq can have any real
    • Value of secq and cosecq can’t be numerically less than
    • (90oq) will lie in first quadrant (90o + q) will lie in second quadrant (180oq) will lie in second quadrant (180o + q) will lie in third quadrant

(270o + q) and (0oq) will lie in fourth quadrant.

 

 

(4)  Fundamental trigonometrical relation

 

 

(i)

 

tanq

= sinq

cosq

(ii) cosecq

= 1 sinq

(iii)

 

sec q

= 1 cosq

  • cot q

= 1 tanq

 

  • sin 2 q + cos 2 q = 1

sec 2 q – tan 2 q = 1

cosec 2q – cot 2 q = 1

 

  • T-Ratios of allied angles : The angles whose sum or difference with angle q is zero or a multiple of 90° are called angle allied to q.

 

(i)         sin(-q ) = – sinq  cos(-q ) = cosq  tan(-q ) = – tanq
(ii)                  sin(90 oq ) = cosq

(iii)                 sin(90 o + q ) = cosq

(iv)                 sin(180oq ) = sinq

(v)        sin(180o + q ) = – sinq

(vi)                 sin(270oq ) = – cosq

(vii)               sin(270o + q ) = – cosq

(viii)              sin(360oq ) = – sinq

(ix)                 sin(360o + q ) = sinq

 cos(90 oq ) = sinq

cos(90 o + q ) = – sinq

cos(180oq ) = – cosq

cos(180o + q ) = – cosq

cos(270oq ) = – sinq

cos(270o + q ) = sinq

cos(360oq ) = cosq

cos(360o + q ) = cosq

 tan(90oq ) = cot q

tan(90 o + q ) = – cot q

tan(180oq ) = – tanq

tan(180o + q ) = tanq

tan(270oq ) = cot q

tan(270o + q ) = – cot q

tan(360oq ) = – tanq

tan(360o + q ) = tanq

 

Note : @Angle ( 2np + q )

lies in first quadrant, if q in an acute angle. Similarly

(2npq )

will lie in fourth

 

quadrant. Where n = 0, 1, 2, 3, 4

@    Angle (-q ) is presumed always lie in fourth quadrant, whatever the value of q .

 

@    If parent angle is 90° or 270° then to cos ecq .

sinq change to

cosq ,

tanq

change to

cot q

and

sec q

change

 

@    If parent angle is 180° or 360° then no change in trigonometric function

 

Problem 6.         Find the values of (i) cos(-60o )

Solution :          (i) cos(-60 o ) = cos 60 o = 1

2

  • tan 210o
  • sin 300o
  • cos 120o

(v) sin(-1485o )

 

 

(ii) tan(210 o ) = tan(180 o + 30 o ) = tan 30 o =

 

(iii) sin(300 o ) = sin(360 o – 60 o ) = – sin 60 o =

 

(iv) cos(120 o ) = cos(90 o + 30 o ) = – sin 30 o = 1

2

 

(v) sin(-1485 o ) = – sin(3 ´ 360 o + 45 o ) = – sin 45 o = –

 

(6)  Addition formulae

  • sin(A + B) = sin A cos B + cos A sin B
  • cos(A + B) = cos A cos B – sin A sin B

 

 

 

 

tan(A + B) =

tan A + tan B

 

1 – tan A tan B

 

Putting B = A in these formulae, we get

  • sin 2A = 2 sin A cos A
  • cos 2A = cos 2 A – sin 2 A = 1 – 2 sin 2 A = 2 cos 2 A – 1

 

 

tan 2A =

2 tan A

 

1 – tan 2 A

 

Problem 7.         If

A = 60 o

then value of

sin 2A will be

 

 

 

 

(a)

3                            (b)

2

1                           (c)

2

(d)

 

 

 

 

Solution : (a)      sin 2A = 2 sin A cos A = 2 sin 60 cos 60 = 2 ´

3 ´ 1 =      3

 

 

(7)  Difference formulae

2     2      2

 

  • sin(A B) = sin A cos B – cos A sin B
  • cos(A B) = cos A cos B + sin A sin B

 

 

tan(A – B) = tan A tan B  

1 + tan A tan B

 

(8)  Transformation formulae

sin(A + B) + sin(A B) = 2 sin A cos B cos(A B) – cos(A + B) = 2 sin A sin B sin(A + B) – sin(A B) = 2 cos A sin B cos(A B) + cos(A + B) = 2 cos A cos B

If we put (A + B) = C and (A B) = D then on adding and subtracting, we get

 

A C + D

2

and

B C D

2

 

Putting these values in the above equation we get

  • sin C + sin D = 2 sin C +D cos C  D

2           2

  • cos C + cos D = 2 cos C +D cos C  D

2           2

  • sin C – sin D = 2 cos C +D sin C  D

2          2

  • cos C – cos D = -2 sin C +D sin C  D

2           2

  • The sine and cosine formulae for a triangle : In a triangle ABC of sides a, b, c and angles A, B and

C, the following formulae hold good.

 

 

 

(i)

  • =

sin A

  • =

sin B

c

 

sin C

 

  • a 2 = b 2 + c 2 – 2bc cos A
  • b 2 = c 2 + a 2 – 2ca cos B
  • c 2 = a 2 + b 2 – 2ab cos C
  • Area of a triangle ABC = ;   where, S = (a + b + c) / 3

Logarithm.

Logarithm of a number with respect to a given base is the power to which the base must be raised to represent that number.

If a x = N then log a N = x

Here x is called the logarithm of N to the base a.

There are two system of logarithm : Logarithm to the base 10 are called common logarithms where as logarithms to the base e are called natural logarithm. They are written as ln.

 

Conversion of natural log into common log :

Important formulae of logarithm :

  • log a (mn) = log am + log a n (Product formula)

log e x = 2.3026 log10 x

 

 

  • log

æ m ö = log

a ç     ÷

n               a

m – log a n

(Quotient formula)

 

è    ø

 

  • log amn = n log a m

(Power formula)

 

  • log am = log b m log a b

(Base change formula)

 

Note : @ Antilogarithm is the reverse process of logarithm i.e., the number whose logarithm is x is called

 

 

Graphs.

antilogarithm of x. If log n = x

then n = antilog of x

 

 

A graph is a line, straight or curved which shows the variation of one quantity w.r.t. other, which are interrelated with each other.

In a relation of two quantities, the quantity which is made to alter at will, is called the independent variable and the other quantity which varies as a result of this change is called the dependent variable. Conventionally, in any graph, the independent variable (i.e. cause) is represented along x-axis and dependent variable (i.e. effect) is represented along y-axis.

For example, we want to depict V = IR graphically, in which R is a constant called resistance, V is the applied

voltage (cause) and I (effect) is the resulting current. We will represent voltage on x-axis and current on y-axis.

 

 

 

 

Some important graphs for various equations

Y                                                                                                                                                               Y

q

 

q                                                                                        C

 

 

 

 

y = mx + c

 

O                                 X                                                                                                                             O                                X

 

 

 

y = mx

m = tanq = slope of line with x-axis

c = Positive intercept on y-axis and positive slope

 

 

Y                                                                                                                                                               Y

 

 

 

C            O  q

y = mx c

X

C                          q                          y = – mx + c

O                                 X

 

 

 

 

 

Negative intercept and positive slope

Y

Positive intercept and Negative slope

 

Y

 

 

 

X       y2 = kx

O

y2 = – kx

O                                X

 

 

 

Symmetric parabola about positive X-axis                           Symmetric parabola about negative X-axis

 

Y                                                                                                                                                              Y

 

 

X    x2 = ky

O

O                                                          x2 = – ky

X

 

 

 

 

Symmetric parabola about positive Y-axis

Y

Symmetric parabola about negative Y-axis

 

Y

 

 

 

X             y = ax + bx2

O

 

O

y = ax bx2

X

 

Asymmetric parabola                                                          Asymmetric parabola

 

 

 

 

Differential Calculus.

The differential coefficient or derivative of variable y with respect to variable x is defined as the instantaneous rate of change of y w.r.t. x. It is denoted by dy

dx

Geometrically the differential coefficient of y = f(x) with respect to x at any point

is equal to the slope of the tangent to the curve representing y = f(x) at that point

 

 

i.e.

dy = tanq .

dx

 

Note : @ Actually

dy is a rate measurer.

dx

 

 

@If

dy is positive, it means y is increasing with increasing of x and vice-versa.

dx

 

 

@For small change Dx

we use Dy = dy . Dx

dx

 

Example: (1) Instantaneous speed v = ds

dt

 

 

 

 

  • Instantaneous acceleration

a = dv =

dt

d 2 x

 

dt 2

 

  • Force

F dp

dt

 

  • Angular velocity w = dq

dt

  • Angular acceleration a = dw

dt

  • Power P = dW

dt

  • Torque t = dL

dt

(1)

Fundamental formulae of differentiation :

 

 

 

 

Problem 8.         Differentiate the following w.r.t x

 

  • x 3
  • ax 2 + bx + c
  • 2x 3e x
  • 6 log e x – – 7

 

Solution :          (i) d (x 3 ) = 3x 2

dx

 

 

(ii)

d (x)1 / 2 =

dx

1     1 -1

2 (x)2       =

1 (x)1 / 2 =        1

2

 

(iii) d (ax 2 + bx + c) = a d (x2) + b d (x) + d (c) = 2ax + b

dx                                    dx                 dx            dx

 

(iv)

d (2x 3e x ) = 2 d (x 3 ) – d (e x ) = 6 x2ex

 

 

dx                               dx              dx

(v) d (6 loge x –           – 7) = 6 d (loge x) – d (x1 / 2) – d (7) = 6 – 1  

 

dx                                              dx

dx                 dx             x      2

 

Problem 9.         Differentiate the following w.r.t. x

  • sin x + cos x
  • sin x + e x

 

 

Solution :          (i)

d (sin x + cos x) =

dx

d (sin x) +

dx

d (cos x) = cos x – sin x dx

 

 

 

(ii)

d (sin x + e x ) =

dx

d (sin x) +

dx

d (e x ) = cos x + e x

dx

 

Problem 10.      Differentiate the following w.r.t. t

 

  • sin t 2
  • e sin t
  • sin(wt + q )

 

 

Solution :          (i)

d (sin t 2 ) = cos t 2 d (t 2 ) = 2t cos t 2

 

 

 

(ii)

dt

d (e sin t ) = e sin t

dt

dt

d (sin t) = e sin t .cos t dt

 

(iii) d [sin(wt + q )] = cos(wt + q ). d (wt + q ) = cos(wt + q ).w

 

dt

Problem 11.      Differentiate

x 2 + e x

 

log x + 20

dt

w.r.t. x

 

 

Solution :          Let y =

x 2 + e x . log x + 20

 

dy        d  æ  x 2 + e x  ö

 

=                            

Then

ç                     ÷

dx       dx ç log x + 20 ÷

 

è                     ø

(log x + 20) d (x 2 + e x ) – (x 2 + e x ) d (log x + 20)

=                        dx                                           dx                      

(log x + 20)2

(log x + 20)(2x + e x ) – (x 2 + e x )æ 1 + 0 ö

ç x         ÷

=                                                               è            ø

(log x + 20)2

  • Maxima and minima : If a quantity y depends on another quantity x in a manner shown in It becomes maximum at x1 and minimum at x2.

At these points the tangent to the curve is parallel to X-axis and hence its

 

slope is tanq = 0. But the slope of the curve equals the rate of change

dy

dy . Thus,

dx

 

at a maximum or minimum

dx = 0

 

Just before the maximum the slope is positive, at the maximum it is zero and just after the maximum it is

 

negative. Thus

dy decreases at a maximum and hence the rate of change of

dx

dy is negative at a maximum. i.e.,

dx

 

d æ dy ö < 0 at a maximum.

ç     ÷

 

dx è dx ø

 

Hence the condition of maxima :

dy = 0

dx

 

and

d 2y < 0

dx 2

 

(Second derivative test)

 

 

Similarly, at a minimum the slope changes from negative to positive. The slope increases at such a point and

 

hence

d  æ dy ö > 0

 

 

 

ç       ÷

 

dx è dx ø

 

Hence the condition of minima :

dy = 0

dx

and

d2y > 0 . (Second derivative test)

dx2

 

Problem 12.       The height reached in time t by a particle thrown upward with a speed u is given by h = ut 1 gt 2 .

2

Find the

 

time taken in reaching the maximum height.

 

 

Solution :          For maximum height

dh = 0

d [ut – 1 gt 2 ] = u – 2gt = 0

 

\t = u

 

dt            dt          2                 2                 g

Problem 13.      A metal ring is being heated so that at any instant of time t in second, its area is given by

A = 3t 2 + t + 2 m2.

3

What will be the rate of increase of area at t = 10 sec .

 

 

Solution :          Rate of increase of area

dA = d (3t 2 + t + 2) = 6t + 1

 

 

dt      dt             3                 3

 

æ dA ö

= 6 ´ 10 + 1 = 181 m2 .

 

 

ç dt ÷

3      3 sec

 

è       øt =10 sec

 

 

Problem 14.      The radius of an air bubble is increasing at the rate of when the radius is 1 cm.

Solution :          Volume of the spherical bubble V = 4 pR3

3

1 cm / sec . Determine the rate of increase in its volume

2

 

Differentiating both sides w.r.t. time

dV = d æ 4 pR3 ö = 4 p .3R 2. dR = 4pR 2 dR

è
3
ø

dt      dt ç 3       ÷                        dt                  dt

 

 

 

at R = 1cm ,

dV = 4p ´ (1)2 ´ 1 = 2p

cm3 / sec .

[Given

dR = 1 cm / sec ]

 

dt                            2                                                               dt       2

Problem 15.      Find the angle of tangent drawn to the curve y = 3x 2 – 7x + 5 at the point (1, 1) with the x– axis.

 

 

Solution :

y = 3x 2 – 7x + 5

Slope of tangent = dy = 6x – 7

dx

 

 

at (1, 1)

 

Integral Calculus.

dy = -1

dx

\ tanq = -1 Þ q = 135o.

 

 

The process of integration is just the reverse of differentiation. The symbol ∫ is used to denote integration.

 

 

 

If f(x) is the differential coefficient of function

F(x) with respect to x, then by integrating

f(x) we can get

 

F(x) again.

(1)  Fundamental formulae of integration :

 

ò xndx = x        , provided n ¹ – 1                               ò sec 2 x dx = tan x

n+1

n + 1

  0+1

ò dx = ò x 0 dx = x          = x

0 + 1

  ò cos ec 2 x dx = – cot x  
 
ò(u + v) dx = òu dx + òv dx                                                     ò sec x tan x dx = sec x

ò cu dx = còu dx                                                                        ò cosec x cot x dx = -cosec x

where c is a constant and u is a function of x.

n+1                                                                                                                          n+1                            n+1

ò cxndx = c x                                                                               ò(ax + b)n dx =         (ax b)           = (ax b)

n + 1                                                                            (n + 1) d (ax + b)          a(n + 1)

dx

a log (ax + b)

ò x 1dx = ò dx = log e x                                                           ò    a      dx =            e                  = log e (ax + b)

x                                                                               (ax + b)            d (ax b)

dx

ò e x dx e x                                                                                                                            ò e ax +b dx =       e ax +b            e ax +b

 d (ax + b)        a

dx

x                                                                                                                                                                            cx +d                                    cx +d

ò a x dx =     a                                                                               ò acx+d dx =                a                     = a 

log e a                                                                                                 log e a d (cx d)  c log e a

dx

ò sin x dx = – cos x                                                                    ò sec 2 (ax b) dx tan(ax + b)tan(ax + b)

d (ax + b)           a

dx

ò sin nx dx cos nx                                                                ò cosec 2 (ax b) dx  cot (ax + b) =  cot (ax + v)

n                                                                                                                  d (ax + b)             a

dx

ò cos x dx = sin x                                                                       ò sec (ax + b) tan(ax + b) dx = sec (ax + b) = sec (ax + b)

d (ax + b)           a

dx

ò cos nx dx = sin nx                                                                   ò cosec (ax + b) cot (ax + b) dx = cosec (ax + b) = cosec (ax + b)

n                                                                                                                                        d (ax + b)                 a

dx

 

  • Method of integration : Sometimes, we come across some functions which cannot be integrated directly by using the standard In such cases, the integral of a function can be obtained by using one or more of the following methods.

 

 

  • Integration by substitution : Those functions which cannot be integrated directly can be reduced to standard integrand by making a suitable substitution and then can be integrated by using the standard integrals. To understand the method, we take the few
  • Integration by parts : This method of integration is based on the following rule :

Integral of a product of two functions = first function ´ integral of second function – integral of (differential coefficient of first function ´ integral of second function).

 

 

Thus, if u and v are the functions of x, then

 

Problem 16.      Integrate the following w.r.t. x

òuvdx = uò

vdx

é du ´

ò ê

ë dx

vdxù dx

ú
ò

û

 

 

  • x1/2 (ii) cot 2 x

(iii)

1

 

  • – sin x

 

1 / 2+1

Solution :          (i) ò x1/ 2dx = x            = 2 (x 3 / 2 )

1 + 1     3

2

 

 

(ii)    ò

cot 2 x dx = ò (cosec 2 x – 1)dx = ò

cosec 2 x dx – ò

dx = – cot x x

 

 

     1     dx =

æ     1     ´ 1 + sin x ö. dx =

1 + sin x

dx =

1     +   sin x   dx

 

ò 1 – sin x

ò ç 1 – sin x       1 + sin x ÷

ò 1 – sin2 x

ò cos 2 x      cos 2 x

 

è
ø

= ò(sec 2 x + tan x sec x)dx = tan x + sec x.

  • Definite integrals : When a function is integrated between definite limits, the integral is called definite For example,

b                                                                                                                                                                                                                                                b

f(x)dx is definite integral of f(x) between the limits a and b and is written as           f(x)dx =| F(x)|b = F(b) – F(a)

òa                                                                                                                                                                                                                                             òa                                                  a

Here a is called the lower limit and b is called the upper limit of integration.

b

Geometrically òa   f(x) dx equals to area of curve  F(x)  between the limits a and b.

 

ò

Problem 17.      Evaluate 6 (2x 2 + 3x + 5)dx

0

 

 

ò
ò

6                                         6                     6

6              é 2x 3 ù 6

é 3x 2 ù 6

 

0

Solution :

(2x 2 + 3x + 5)dx =

0                                         0

2x 2dx +

ò0 3x dx +

5 dx = ê

0               êë

ú   + ê

3 úû 0         êë

ú

2 úû 0

+ [5x]6 = 144 + 54 + 30 = 228.

 

ò

Problem 18.      Integrate the following

 

 

(i)

2 1 dx

(ii)

p / 2

cos x dx

(iii)

r2 Kq1q2 .dr

 

1

p / 4 tan 2 x dx

 

ò0

2   1             2

ò0

é x 1 / 2 ù 2                        2

òr            r 2                                  ò0

 

Solution :          (i) ò

dx  = ò

x -1/ 2 dx = ê

ú    = [2x1/ 2 ]0  = 2

 

0                      0                      êë 1 / 2 úû 0

 

 

 

p / 2

  • òcos x dx

0

= [sin x]p / 2

0

= sin p = 1 2

 

r2      q q

r

2   1                  æ

1 ör2

é 1     1 ù

é 1     1 ù

 

  • òk

1 2 2 dx = k q1q2 ò

2 dx = kq1q2 ç-   ÷

= –kq1q2 ê

–      ú = kq1q2 ê      –      ú

 

 

r

r1         r

p / 4

r

1

 

p / 4

è   r ør1

 

p / 4

 

 

p / 4

ë r2

r1 û

 

p

 

ë r1

r2 û

 

ò tan 2 x dx = ò(sec 2 x – 1)dx = [tan x] 0

– [x] 0

= 1 – 4

 

0                          o

General Formulae for Area and Volume.

  1. Area of square = (side)2
  2. Area of rectangle = length ´ breadth
  3. Area of triangle = 1 ´ base ´ height

2

  1. Area enclosed by a circle = p r 2 ; where r is radius
  2. Surface area of sphere = 4p r 2
  3. Surface area of cube = 6 L2 ; where L is a side of cube
  4. Surface area of cuboid = 2[L ´ b + b ´ h + h ´ L]; where L= length, b = breadth, h = height
  5. Area of curved surface of cylinder = 2p rl ; where r = radius, l = length of cylinder
  6. Volume of cube = L3
  7. Volume of cuboid = L ´ b ´ h
  8. Volume of sphere = 4 p r 3

3

  1. Volume of cylinder = p r 2l
  2. Volume of cone = 1 p r 2h

2

Introduction of Vector.

Physical quantities having magnitude, direction and obeying laws of vector algebra are called vectors.

Example : Displacement, velocity, acceleration, momentum, force, impulse, weight, thrust, torque, angular momentum, angular velocity etc.

If a physical quantity has magnitude and direction both, then it does not always imply that it is a vector. For it to be a vector the third condition of obeying laws of vector algebra has to be satisfied.

Example : The physical quantity current has both magnitude and direction but is still a scalar as it disobeys the laws of vector algebra.

 

 

Types of Vector.

 

 

  • Equal vectors : Two vectors A and B are said to be equal when they have equal magnitudes and same
  • Parallel vector : Two vectors A and B are said to be parallel when
    • Both have same
    • One vector is scalar (positive) non-zero multiple of another
  • Anti-parallel vectors : Two vectors A and B are said to be anti-parallel when
    • Both have opposite
    • One vector is scalar non-zero negative multiple of another
  • Collinear vectors : When the vectors under consideration can share the same support or have a common support then the considered vectors are
  • Zero vector (0) : A vector having zero magnitude and arbitrary direction (not known to us) is a zero
  • Unit vector : A vector divided by its magnitude is a unit vector. Unit vector for A  is  Aˆ  (read as A cap / A hat).

 

 

Since,

Aˆ =  A

A

Þ  A =  A Aˆ .

 

Thus, we can say that unit vector gives us the direction.

  • Orthogonal unit vectors : ˆi , ˆj  and kˆare called orthogonal unit vectors. These vectors must form a Right

Handed Triad (It is a coordinate system such that when we Curl the fingers of right hand from x to y then we must get the direction of z along thumb). The

 

ˆi  =

x , ˆj =

x

 

y , kˆ =  z

y            z

 

\             x  = xˆi ,

y = yˆj ,

z = zkˆ

 

  • Polar vectors : These have starting point or point of application . Example displacement and force etc.
  • Axial Vectors : These represent rotational effects and are always along the axis of rotation in accordance with right hand screw rule. Angular velocity, torque and angular momentum, etc., are example of physical quantities of this

 

 

 

 

 

 

  • Coplanar vector : Three (or more) vectors are called coplanar vector if they lie in the same plane. Two (free) vectors are always

 

 

 

Triangle Law of Vector Addition of Two Vectors.

If two non zero vectors are represented by the two sides of a triangle taken in same order then the resultant is given by the closing side of triangle in opposite order. i.e. R = A + B

Q OB = OA + AB

(1)  Magnitude of resultant vector

 

 

In D ABN cosq

= AN \ AN  = B cosq

B

 

 

sinq

BN

B

\ BN = B sinq

 

In DOBN, we have OB2 = ON 2 + BN 2

Þ R 2 = (A + B cosq )2 + (B sinq )2

Þ R 2 = A2 + B2 cos 2 q + 2AB cosq + B2 sin 2 q

Þ R 2 = A2 + B2 (cos 2 q + sin 2 q ) + 2AB cosq

Þ R 2 = A2 + B2 + 2AB cosq

Þ R =

  • Direction of resultant vectors : If q is angle between A and

| A + B|=

 

 

 

 

 

 

 

 

 

 

B, then

 

 

 

 

If R makes an angle a with A, then in DOBN, then

 

tana = BN

ON

=        BN       OA + AN

 

tana =     B sinq    

A + B cosq

Parallelogram Law of Vector Addition of Two Vectors.

If two non zero vector are represented by the two adjacent sides of a parallelogram then the resultant is given by the diagonal of the parallelogram passing through the point of intersection of the two vectors.

(1)  Magnitude

 

Since, R 2 = ON 2 + CN 2

Þ R 2 = (OA + AN)2 + CN 2

 

 

Þ R 2

= A2

  • B2
  • 2AB cosq

 

 

\  R =| R|=| A + B|=

Special cases : R = A + B when q = 0o

R = A B when q = 180o

R =                   when q = 90o

  • Direction

 

tan b = CN

ON

=     B sinq   

A + B cosq

 

Polygon Law of Vector Addition.

If a number of non zero vectors are represented by the (n – 1) sides of an n-sided polygon then the resultant is given by the closing side or the nth side of the polygon taken in opposite order. So,

 

R = A + B + C + D + E

OA + AB + BC + CD + DE = OE

 

Note : @ Resultant of two unequal vectors can not be zero.

@Resultant of three co-planar vectors may or may not be zero

@Resultant of three non co- planar vectors can not be zero.

Subtraction of Vectors.

 

 

Since,

 

A B = A + (-B)

 

and | A + B|=

 

 

Þ               | A B|=     A2 + B2 + 2AB cos (180oq )

Since, cos (180 – q ) = – cosq

Þ               | A B|=    A 2 + B 2 – 2AB cosq

 

tana1

=     B sinq    

A + B cosq

 

and

tana 2 =

B sin (180 – q )

 

A + B cos (180 – q )

 

But

sin(180 – q ) = sinq

and cos(180 – q ) = – cosq

 

Þ                 tana 2

=     B sinq    

A B cosq

 

Problem 19.   A car travels 6 km towards north at an angle of 45° to the east and then travels distance of 4 km towards north at an angle of 135° to the east. How far is the point from the starting point. What angle does the straight line joining its initial and final position makes with the east

 

(a)

50 km and tan1(5)

(b) 10 km and tan1( 5)

 

 

 

(c)

52 km and tan1(5)

(d)

52 km and tan1( 5)

 

 

 

Solution : (c)      Net movement along x-direction Sx

= (6 – 4) cos 45° ˆi

= 2 ´ 1 =

  • km

 

 

 

Net movement along y-direction Sy

= (6 + 4) sin 45° ˆj

= 10 ´ 1 = 5

2 km

 

 

 

 

Net movement from starting point | s |=                              =                                  =        km

 

Angle which makes with the east direction tanq = Y  component = 5  

X – component

 

\ q = tan1(5)

Problem 20.      There are two force vectors, one of 5 N and other of 12 N at what angle the two vectors be added to get resultant vector of 17 N, 7 N and 13 N respectively

(a) 0°, 180° and 90°              (b) 0°, 90° and 180°              (c) 0°, 90° and 90°           (d) 180°, 0° and 90°

Solution : (a)     For 17 N both the vector should be parallel i.e. angle between them should be zero.

For 7 N both the vectors should be antiparallel i.e. angle between them should be 180°

For 13 N both the vectors should be perpendicular to each other i.e. angle between them should be 90°

 

 

Problem 21.      Given that

 

A + B + C = 0 out of three vectors two are equal in magnitude and the magnitude of third vector is

 

times that of either of the two having equal magnitude. Then the angles between vectors are given by (a) 30°, 60°, 90°                             (b) 45°, 45°, 90°                   (c) 45°, 60°, 90°                (d) 90°, 135°, 135°

Solution : (d) From polygon law, three vectors having summation zero should form a closed polygon. (Triangle) since the

two vectors are having same magnitude and the third vector is  2 times that of either of two having equal magnitude. i.e. the triangle should be right angled triangle

Angle between A and B, a = 90º Angle between B and C, b = 135º Angle between A and C, g = 135º

 

 

Problem 22.      If

A = 4ˆi – 3ˆj

and B = 6ˆi  + 8ˆj

then magnitude and direction of

 

A + B will be

 

 

 

(a)

5, tan1(3 / 4)

(b)

5 5, tan1(1 / 2)

(c)

10, tan1(5)

(d)

25, tan1(3 / 4)

 

 

 

Solution : (b)

A + B = 4ˆi  – 3ˆj + 6ˆi  + 8ˆj  = 10ˆi  + 5ˆj

 

 

 

 

 

| A + B |=

 

tanq = 5 = 1

= 5

 

Þ q = tan1 æ 1 ö

 

 

10     2

ç 2 ÷

 

è ø

Problem 23.      A truck travelling due north at 20 m/s turns west and travels at the same speed. The change in its velocity be

 

 

 

 

(a) 40 m/s NW                     (b)

 

Solution : (d)     From fig.

20      m/s NW             (c) 40 m/s SW                  (d)

20       m/s SW

 

 

v1 = 20ˆj  and v 2  = -20ˆi

 

Dv = v 2v1 = -20(ˆi  + ˆj)

 

 

 

|Dv |= 20

and direction q = tan1(1) = 45° i.e. SW

 

 

Problem 24.      If the sum of two unit vectors is a unit vector, then magnitude of difference is                  [CPMT 1995; CBSE PMT 1989]

 

 

  • (b)

(c) 1 /

(d)

 

Solution : (b)       Let nˆ1  and nˆ2  are the two unit vectors, then the sum is

 

 

s
1
2

ns   nˆ1  + nˆ2  or n2 = n2 + n2 + 2n1n2 cosq

= 1 + 1 + 2 cosq

 

Since it is given that ns is also a unit vector, therefore 1 = 1 + 1 + 2 cosq

 

 

or cosq = – 1

2

or q = 120°

 

Now the difference vector is nd = n1 – n2 or n2 = n2 + n2 – 2n1n2 cos q

= 1 + 1 – 2 cos(120°)

 

d            1        2

d

\ n2 = 2 – 2(-1 / 2) = 2 + 1 = 3           Þ nd =

Problem 25.      The sum of the magnitudes of two forces acting at point is 18 and the magnitude of their resultant is 12. If the resultant is at 90° with the force of smaller magnitude, what are the, magnitudes of forces

(a) 12, 5                          (b) 14, 4                          (c) 5, 13                       (d) 10, 8

Solution : (c)      Let P be the smaller force and Q be the greater force then according to problem –

P + Q = 18………………………………………………………………………………………….. (i)

 

R =                                             = 12……………………………………………………………….. (ii)

 

tan f =  Q sinq      = tan 90 = ¥

P + Q cosq

By solving (i), (ii) and (iii) we will get

\ P + Q cosq = 0

 

P = 5, and Q = 13

…….(iii)

 

Problem 26.      Two forces

F1 = 1 N and

F2 = 2 N act along the lines x = 0 and y = 0 respectively. Then the resultant of

 

forces would be

 

(a)

ˆi + 2ˆj

(b)

ˆi + ˆj

(c)

i + 2ˆj

(d)

i + ˆj

 

 

Solution : (d)     x = 0 means y-axis Þ

F 1 = ˆj

 

 

 

y = 0 means x-axis

Þ  F 2  = 2ˆi

so resultant

F  F 1 + F 2  = 2ˆi + ˆj

 

 

 

Problem 27.      Let

A = 2ˆi  + ˆj, B = 3ˆjkˆ  and C  = 6ˆi  – 2kˆ value of

 

A – 2B + 3C would be

 

 

 

(a)

20ˆi  + 5ˆj + 4kˆ

(b)

20ˆi  – 5ˆj – 4kˆ

(c)

i + 5ˆj + 20kˆ

(d)

i + 4ˆj + 10kˆ

 

 

 

Solution : (b)

A – 2B + 3C  = (2ˆi  + ˆj) – 2(3ˆjkˆ) + 3(6ˆi  – 2kˆ)

= 2ˆi  + ˆj – 6ˆj + 2kˆ + 18ˆi  – 6kˆ

= 20ˆi – 5ˆj – 4kˆ

 

 

 

Problem 28.      A vector a

respectively

is turned without a change in its length through a small angle dq . The value of |Da | and Da

are

 

(a)

0, a dq

(b)

a dq , 0

(c) 0, 0                         (d) None of these

 

 

 

 

 

Solution : (b)     From the figure |OA|= a and |OB|= a

Also from triangle rule OB OA = AB = Da

Þ |Da |= AB

 

 

Using angle =

arc radius

 

Þ AB = a . dq

 

 

 

So |Da |= a dq

 

 

 

Da means change in magnitude of vector i.e. | OB |- | OA | Þ

a a = 0

So Da = 0

 

 

Problem 29.      An object of m kg with speed of v m/s strikes a wall at an angle q and rebounds at the same speed and same angle. The magnitude of the change in momentum of the object will be

 

(a)

2mv cosq

(b)

2mv sinq

  • 0 (d)

2 mv

 

 

 

Solution : (a)

P1  = mv sinq ˆi – mv cosq ˆj

and P 2  = mv sinq ˆi  mv cosq ˆj

 

 

So change in momentum DP P 2 – P1 = 2 mv cosq ˆj

 

|DP |= 2mv cosq

 

 

Resolution of Vector Into Components.

 

 

Consider a vector R in xy plane as shown in fig. If we draw orthogonal vectors R x

and

Ry  along x and

 

y axes respectively, by law of vector addition,

 

R RxR y

 

 

 

Now as for any vector

A =  A nˆ

so, Rx

= ˆi Rx

and Ry

= ˆjRy

 

so          R = ˆi Rx   + ˆjRy

…..(i)

 

 

 

But from fig

Rx    = R cosq

…..(ii) and

Ry   = R sinq

…..(iii)

 

 

Since R and q are usually known, Equation (ii) and (iii) give the magnitude of the components of R along x and y-axes respectively.

Here it is worthy to note once a vector is resolved into its components, the components themselves can be used to specify the vector as –

  • The magnitude of the vector R is obtained by squaring and adding equation (ii) and (iii), e.

R =

  • The direction of the vector R is obtained by dividing equation (iii) by (ii), e.

 

tanq

= (Ry / Rx )

or      q = tan -1(Ry / Rx )

 

Rectangular Components of 3-D Vector.

 

 

Þ cos g

= Rz   =            Rz                         = n R

 

 

 

where l, m, n are called Direction Cosines of the vector R

R 2 + R 2 + R 2

l 2 + m2 + n2 = cos 2 a + cos 2 b + cos 2 g =           x         y          z = 1

R 2 + R 2 + R 2

x               y               z

Note : @When a point P have coordinate (x, y, z) then its position vector OP  = xˆi  + yˆj + zkˆ

@ When a particle moves from point (x1, y1, z1) to (x2, y2, z2) then its displacement vector

®                  ˆ             ˆ             ˆ

r = (x 2 – x1 )i + (y2 – y1 ) j + (z 2 – z1 )k

Problem 30.      If a particle moves 5 m in +x– direction. The displacement of the particle will be

(a) 5 j                                        (b) 5 i                                       (c) – 5 j                                (d) 5 k Solution : (b)     Magnitude of vector = 5

Unit vector in +x direction is ˆi So displacement = 5 ˆi

 

 

Problem 31.      Position of a particle in a rectangular-co-ordinate system is (3, 2, 5). Then its position vector will be

 

(a)

i + 5ˆj + 2kˆ

(b)

i + 2ˆj + 5kˆ

(c)

i + 3ˆj + 2kˆ

  • None of these

 

Solution : (b)       If a point have coordinate (x, y, z) then its position vector OP = xˆi + yˆj + zkˆ.

Problem 32.      If a particle moves from point P (2,3,5) to point Q (3,4,5). Its displacement vector be

 

(a)

ˆi + ˆj + 10kˆ

(b)

ˆi + ˆj + 5kˆ

(c)

ˆi  + ˆj

(d)

i + 4ˆj + 6kˆ

 

Solution : (c)       Displacement vector r  = Dxˆi + Dyˆj + Dzkˆ = (3 – 2)ˆi  + (4 – 3)ˆj + (5 – 5)kˆ = ˆi  + ˆj

Problem 33.      A force of 5 N acts on a particle along a direction making an angle of 60° with vertical. Its vertical component be (a) 10 N                                                    (b) 3 N                                      (c) 4 N                                 (d) 5.2 N

Solution : (d)     The component of force in vertical direction will be F cosq = F cos 60°

= 5 ´ 1 = 2.5 N

2

 

 

 

 

 

Problem 34.      If

A = 3ˆi  + 4ˆj

and  B = 7ˆi  + 24ˆj, the vector having the same magnitude as B and parallel to A is

 

 

(a)

i  + 20ˆj

(b)

15ˆi  + 10ˆj

(c)

20ˆi  + 15ˆj

(d)

15ˆi  + 20ˆj

 

 

 

Solution : (d)

| B|=

=               = 25

 

 

 

 

Unit vector in the direction of A will be

Aˆ = 3ˆi  + 4ˆj

5

 

æ 3ˆi  + 4ˆj ö             ˆ       ˆ

So required vector = 25ç                ÷ = 15i + 20 j

5

ç               ÷

è               ø

 

Problem 35.      Vector A makes equal angles with x, y and z axis. Value of its components (in terms of magnitude of A ) will be

 

 

(a)

  A                                       (b) A                                       (c)        A

  •   3

A

 

 

 

Solution : (a)     Let the components of A makes angles a, b and g with x, y and z axis respectively then a = b = g

 

 

cos 2 a + cos 2 b + cos 2 g = 1

Þ 3 cos 2 a = 1 Þ cosa = 1 

 

 

\ Ax = Ay = Az   = A cosa =   A  

 

Problem 36.      If  A = 2iˆ+ 4ˆj – 5kˆ the direction of cosines of the vector  A  are

 

– 5 (b)   1 , 2 and   3 (c)   4 , 0 and   4 (d)   3 , 2 and   5
45     45     45     45     45     45     45     45     45

 

  •  2 , 4 and

 

 

 

 

Solution : (a)

| A |=

\ cosa =    2 ,

=

 

cos b =  4 ,

cos g = 5 

 

 

Problem 37.      The vector that must be added to the vector ˆi – 3ˆj + 2kˆ and  3ˆi + 6ˆj – 7kˆ  so that the resultant vector is a unit vector along the y-axis is

 

(a)

i  + 2ˆj + 5kˆ

– 4ˆi – 2ˆj + 5kˆ

i + 4ˆj + 5kˆ

  • Null vector

 

 

 

Solution : (b)     Unit vector along y axis = ˆj

Scalar Product of Two Vectors.

so the required vector = ˆj – [(ˆi  – 3ˆj + 2kˆ) + (3ˆi  + 6ˆj – 7kˆ)] = – 4ˆi  – 2ˆj + 5kˆ

 

 

  • Definition : The scalar product (or dot product) of two vectors is defined as the product of the magnitude of two vectors with cosine of angle between

Thus if there are two vectors A and B having angle q between them, then their scalar product written as

 

 

 

  1. A. B is defined as

 

  1. A. B

= AB cosq

 

  • Properties : (i) It is always a scalar which is positive if angle between the vectors is acute (e., < 90°) and negative if angle between them is obtuse (i.e. 90°<q < 180°).
  • It is commutative, e. A. B = B. A

 

 

  • It is distributive, e. A.(B + C) = A. B + A. C

 

  • As by definition A. B = AB cosq

The angle between the vectors q = cos-1 é A. B ù

ê        ú

AB

ëê      úû

  • Scalar product of two vectors will be maximum when cosq = max = 1, e. q = 0o , i.e., vectors are parallel

 

 

 

 

(A. B)max

= AB

 

  • Scalar product of two vectors will be minimum when |cosq |= min = 0, e. q = 90o

(A . B)min = 0

i.e., if the scalar product of two nonzero vectors vanishes the vectors are orthogonal.

  • The scalar product of a vector by itself is termed as self dot product and is given by

(A)2 = A. A = AA cosq = A2

 

i.e., A =

 

  • In case of unit vector nˆ

nˆ.nˆ = 1 ´ 1 ´ cos 0 = 1

so nˆ.nˆ= ˆii = ˆj . ˆj = kˆ.kˆ= 1

 

 

 

  • In case of orthogonal unit vectors ˆi, ˆj

and kˆ,

ˆi . ˆj = ˆj .kˆ= kˆ.ˆi = 1 ´ 1cos 90 = 0

 

 

 

  • In terms of components

 

  1. B =(iAx + jAy  + kAz ).(iBx   + jBy   + kBz ) = [Ax Bx   + Ay By   + AZ Bz ]

 

  • Example : (i) Work W : In physics for constant force work is defined as, W = Fs cosq………………….. (i)

 

 

But by definition of scalar product of two vectors,

 

  1. F. s = Fs cosq

…….(ii)

 

 

 

 

So from eqn (i) and (ii) W = F.s

  • Power P :

i.e. work is the scalar product of force with displacement.

 

 

 

 

 

As         W = F . s            or

dW F . ds

 

[As F is constant]

 

dt              dt

 

 

or P = F .v

i.e., power is the scalar product of force with velocity.

éAs dW = P and ds = vù

 

 

 

  • Magnetic Flux f :

Magnetic flux through an area is given by df = Bds cosq

ê

êë     dt

 

……(i)

ú

dt         úû

 

 

 

But by definition of scalar product So from eqn (i) and (ii) we have

 

B . d s = Bds cos q

……(ii)

 

 

 

df = B. d s

or f = ò

  1. ds

 

 

  • Potential energy of a dipole U : If an electric dipole of moment p is situated in an electric field E or a

magnetic dipole of moment M in a field of induction B, the potential energy of the dipole is given by :

UE = – p . E and UB  = –M . B

 

Problem 38.

A = 2ˆi + 4ˆj + 4kˆ and B = 4ˆi  + 2ˆj – 4kˆ are two vectors. The angle between them will be (a) 0°    (b) 45° (c) 60°                                  (d) 90°

 

Solution : (d)

cosq =

 

  1. A. B

|A|.|B|

a1b1 + a2b2 + a3b3

|A|.| B|

= 2 ´ 4 + 4 ´ 2 – 4 ´ 4 = 0

| A|.| B|

 

\ q = cos 1(0°) Þ q = 90°

Problem 39.      If two vectors  2ˆi + 3ˆjkˆ and – 4ˆi  – 6ˆjlkˆ are parallel to each other then value of l be (a) 0                                       (b) 2                               (c) 3                            (d) 4

 

Solution : (c)      Let

A = 2ˆi  + 3ˆjkˆ and B = -4ˆi  – 6ˆj + lkˆ

 

 

 

A and B are parallel to each other

a1 = a2 = a3

i.e.    2 =   3 = 1 Þ l = 2 .

 

b1      b2      b3           – 4     – 6      l

Problem 40.      In above example if vectors are perpendicular to each other then value of l be

(a) 25                              (b) 26                              (c) – 26                        (d) – 25

 

 

 

 

 

Solution : (c)      If A and B are perpendicular to each other then

 

  1. A. B = 0

Þ a1b1 + a2b2 + a3b3 = 0

 

So, 2(-4) + 3(-6) + (-1)(l) = 0 Þ l = -26

Problem 41.      If A = 2ˆi + 3ˆjkˆ and  B = -ˆi + 3ˆj + 4kˆ  then projection of  A  on  B  will be

 

 

(a)

  3                             (b)

   3                            (c)

(d)

 

 

 

Solution : (b)

|A|=                                    =                       =

| B |=                                      =                         =

 

 

 

A . B = 2(-1) + 3 ´ 3 + (-1)(4) = 3

 

 

 

The projection of

 

A on B

A . B   =

| B |

 

Problem 42.      A body, acted upon by a force of 50 N is displaced through a distance 10 meter in a direction making an angle of 60° with the force. The work done by the force be

(a) 200 J                                  (b) 100 J                                   (c) 300                         (d) 250 J

 

 

Solution : (d)

 

W = F . S = FS cosq

= 50 ´ 10 ´ cos 60° = 50 ´ 10 ´ 1 = 250 J.

2

 

Problem 43.      A particle moves from position

i + 2ˆj – 6kˆ  to 14ˆi + 13ˆj + 9kˆ  due to a uniform force of

i + ˆj + 3kˆ N.

If the

 

 

 

 

Solution : (a)

displacement in meters then work done will be

(a) 100 J                                 (b) 200 J                                   (c) 300 J                               (d) 250 J

S = r2 – r1

 

 

 

W = F . S

= (4ˆi  + ˆj + 3kˆ).(11ˆi  + 11ˆj + 15kˆ) = (4 ´ 11 + 1 ´ 11 + 3 ´ 15) = 100 J.

 

Problem 44.      If for two vector A and B , sum (A + B) is perpendicular to the difference (AB). The ratio of their magnitude is

  • 1 (b) 2                               (c) 3                            (d) None of these

Solution : (a)      (A + B) is perpendicular to (A B). Thus

 

(A + B) . (A B) = 0 or A 2 + B . A A . B B 2 = 0

Because of commutative property of dot product A.B = B.A

\ A2B2 = 0 or A = B

Thus the ratio of magnitudes A/B = 1

Problem 45.      A force  F  = –K(yˆi  + xˆj)  (where K  is a positive constant) acts on a particle moving in the xy  plane. Starting

from the origin, the particle is taken along the positive x– axis to the point (a, 0) and then parallel to the y-axis to the point (a, a). The total work done by the forces F on the particle is                                                            [IIT-JEE   1998]

 

(a)

  • 2 Ka 2

2 Ka 2

  • Ka 2

Ka 2

 

Solution : (c)      For motion of the particle form (0, 0) to (a, 0)

F  = –K(0ˆi + a ˆj) Þ    F  = –Kaˆj

 

 

 

Displacement

r  = (aˆi + 0 ˆj) – (0ˆi + 0 ˆj) = aˆi

 

 

 

 

So work done from (0, 0) to (a, 0) is given by W = F . r

For motion (a, 0) to (a, a)

= –Kaˆj .aˆi  = 0

 

F  = –K(aˆi + aˆj) and displacement r  = (aˆi + aˆj) – (aˆi + 0ˆj) = aˆj

 

 

So work done from (a, 0) to (a, a) W = F .r

So total work done = –Ka 2

Vector Product of Two Vector.

= –K(aˆi + aˆj).aˆj  = –Ka 2

 

 

  • Definition : The vector product or cross product of two vectors is defined as a vector having a magnitude equal to the product of the magnitudes of two vectors with the sine of angle between them, and direction perpendicular to the plane containing the two vectors in accordance with right hand screw

C = A ´ B

 

 

 

Thus, if A and B are two vectors, then their vector product written as

C = A ´ B = AB sinq nˆ

 

A ´ B is a vector C  defined by

 

 

 

 

The direction of A ´ B, i.e. C is perpendicular to the plane containing

vectors A and B and in the sense of advance of a right handed screw

rotated from A (first vector) to B (second vector) through the smaller angle between them. Thus, if a right handed screw whose axis is perpendicular to

the plane framed by A and B is rotated from A to B through the smaller angle between them, then the direction

 

of advancement of the screw gives the direction of

(2)  Properties :

 

A ´ B i.e. C

 

  • Vector product of any two vectors is always a vector perpendicular to the plane containing these two

vectors, i.e., orthogonal to both the vectors A and B, though the vectors A and B may or may not be orthogonal.

 

  • Vector product of two vectors is not commutative, e., Here it is worthy to note that

 

 

| A ´ B|=| B ´ A|= AB sinq

 

A ´ B ¹ B ´ A [but

 

= –B ´ A]

 

i.e., in case of vector

 

A ´ B and

 

B ´ A magnitudes are equal but directions are opposite.

 

  • The vector product is distributive when the order of the vectors is strictly maintained, e.

A ´ (B + C) = A ´ B + A ´ C

  • As by definition of vector product of two vectors  A ´ B = AB sinq nˆ

 

 

So        | A ´ B|= AB sinq

i.e.,

q = sin-1 é| A ´ B|ù

 

ê             ú

êë| A|| B|úû

 

 

 

  • The vector product of two vectors will be maximum when

[A ´ B]max  = AB nˆ

i.e., vector product is maximum if the vectors are orthogonal.

sinq = max = 1, i.e., q = 90o

 

  • The vector product of two non- zero vectors will be minimum when |sinq |= minimum = 0, e., q = 0o or 180o

[A ´ B]min = 0

i.e. if the vector product of two non-zero vectors vanishes, the vectors are collinear.

  • The self cross product, e., product of a vector by itself vanishes, i.e., is null vector

A ´ A = AA sin 0o nˆ = 0

  • In case of unit vector nˆ´ nˆ= 0 so that ˆi ´ ˆi  = ˆj ´ ˆj  = kˆ´ kˆ = 0
  • In case of orthogonal unit vectors, ˆi, ˆj,kˆin accordance with right hand screw rule :

 

 

ˆi ´ ˆj  = kˆ,

ˆj ´ kˆ = ˆi

and

kˆ´ ˆi  = ˆj

 

And as cross product is not commutative,

 

ˆj ´ ˆi  = –kˆ

kˆ´ ˆj  = -ˆi

and

ˆi         ˆj

ˆi ´ kˆ = -ˆj

kˆ

 

  • In terms of components

 

A ´ B =  Ax

Bx

Ay          Az

By         Bz

= ˆi(Ay Bz   Az By ) + ˆj(Az Bx   Ax Bz ) + kˆ(Ax By   Ay Bx )

 

 

  • Example : Since vector product of two vectors is a vector, vector physical quantities (particularly representing rotational effects) like torque, angular momentum, velocity and force on a moving charge in a magnetic field and can be expressed as the vector product of two It is well – established in physics that :
  • Torque t = r ´ F

 

  • Angular momentum
  • Velocity v = w ´ r

 

L = r ´ p

 

 

 

  • Force on a charged particle q moving with velocity v in a magnetic field B is given by F = q(v ´ B)

 

  • Torque on a dipole in a field t E = p ´ E and t B = M ´ B

Problem 46.      If  A = 3iˆ + ˆj + 2kˆ and  B = 2iˆ – 2ˆj + 4kˆ  then value of | A ´ B |  will be

 

 

 

 

(a)

8                               (b)

8                               (c)

8                            (d) 5

 

 

 

 

Solution : (b)

ˆi          ˆj

 

 

A ´ B =   3        1

2      – 2

kˆ

2  = (1 ´ 4 – 2 ´ -2)ˆi + (2 ´ 2 – 4 ´ 3)ˆj + (3 ´ -2 – 1´ 2)kˆ

4

= 8ˆi – 8ˆj – 8kˆ

 

 

 

 

\ Magnitude of A ´ B =| A ´ B|=                                                 = 8

 

 

Problem 47.      In above example a unit vector perpendicular to both

 

A and B will be

 

 

(a)

+   1  (ˆi – ˆjkˆ)

(b)

  1  (ˆi – ˆjkˆ)

(c) Both (a) and (b)            (d) None of these

 

 

 

 

Solution : (c)

 

nˆ =   A´ B   =

| A´ B|

=   1  (ˆi – ˆjkˆ)

 

 

There are two unit vectors perpendicular to both  A  and  B  they are nˆ = ±   1  (ˆi – ˆjkˆ)

 

 

Problem 48.      The vectors from origin to the points A and B are of the triangle OAB be

A = 3ˆi – 6ˆj + 2kˆ and

B = 2ˆi + ˆj – 2kˆ respectively. The area

 

 

(a)

5 17

2

sq.unit                (b)

2 17

5

sq.unit                (c)

3 17

5

sq.unit            (d)

5 17

3

sq.unit

 

Solution : (a)       Given OA = a  = 3ˆi  – 6ˆj + 2kˆ and OB = b = 2ˆi  + ˆj – 2kˆ

 

 

ˆi              ˆj

 

 

\ (a ´ b) =       3      – 6

2        1

kˆ

2     = (12 – 2)ˆi  + (4 + 6)ˆj + (3 + 12)kˆ

– 2

 

 

 

= 10ˆi  + 10ˆj + 15kˆ Þ | a ´ b |=

=               = 5

 

 

 

Area of DOAB = 1 | a ´ b |= 5 17

sq.unit.

 

2

Problem 49.      The angle between the vectors

2

A and B is q . The value of the triple product

 

A.(B ´ A ) is

 

(a)

Solution : (b)     Let

A2 B

A.(B´ A) = A.C

(b) Zero                           (c)

A2 B sinq

(d)

A2 B cosq

 

 

 

Here C = B ´ A Which is perpendicular to both vector A and B         \ A . C = 0

 

Problem 50.      The torque of the force  F  = (2ˆi  – 3ˆj + 4kˆ)N

acting at the point r

= (3ˆi + 2ˆj + 3kˆ) m about the origin be

[CBSE PMT 1995]

 

(a)

i  – 6ˆj + 12kˆ

(b)

17ˆi – 6ˆj – 13kˆ

(c)

– 6ˆi + 6ˆj – 12kˆ

(d)

– 17ˆi + 6ˆj + 13kˆ

 

 

 

 

Solution : (b)

 

 

t = r ´ F

ˆi        ˆj

= 3     2

2 – 3

kˆ

3    = [(2 ´ 4) – (3 ´ -3)] ˆi  + [(2 ´ 3) – (3 ´ 4)]ˆj + [(3 ´ -3) – (2 ´ 2)]kˆ = 17ˆi  – 6 ˆj – 13 kˆ

4

 

 

 

Problem 51.      If

 

A ´ B = C, then which of the following statements is wrong

 

 

 

(a)

 

C ^ A

(b)

 

C ^ B

(c)

 

C ^(A + B)

(d)

 

C ^(A ´ B)

 

 

Solution : (d)     From the property of vector product, we notice that C must be perpendicular to the plane formed by vector

A and B . Thus C is perpendicular to both A and B and (A + B) vector also must lie in the plane formed

 

 

 

by vector A and B . Thus C must be perpendicular to (A + B)

also but the cross product

 

(A ´ B) gives a

 

 

vector C which can not be perpendicular to itself. Thus the last statement is wrong.

Problem 52.      If a particle of mass m is moving with constant velocity v parallel to x-axis in xy plane as shown in fig. Its angular momentum with respect to origin at any time t will be

 

(a)

mvb kˆ

(b)

  • mvb kˆ

(c)

mvbˆi

(d)

mvˆi

 

Solution : (b)     We know that, Angular momentum

ˆi           ˆj           kˆ

 

 

 

L = r ´ p

in terms of component becomes L = x

px

y           z

py            pz

 

 

 

 

As motion is in xy plane (z = 0 and Pz = 0 ), so L = k (xpy ypx )

 

Here x = vt, y = b,

px  = mv

and py   = 0

 

 

\ L

Lami’s Theorem.

= k [vt ´ 0 – b mv] = –mvb kˆ

 

 

 

 

In any

D A BC

 

with sides a, b, c

 

sina

a

= sin b

b

= sin g

c

 

i.e., for any triangle the ratio of the sine of the angle containing the side to the length of the side is a constant.

For a triangle whose three sides are in the same order we establish the Lami’s theorem in the following manner. For the triangle shown

 

 

 

a + b + c = 0

Þ            a + b = –c

[All three sides are taken in order]…………………………………. (i)

…..(ii)

 

 

Pre-multiplying both sides by a

 

 

 

a ´ (a + b) = –a ´ c         Þ

 

0 + a ´ b = –a ´ c Þ

 

a ´ b = c ´ a

…..(iii)

 

 

Pre-multiplying both sides of (ii) by b

 

 

 

 

 

b ´ (a + b) = – b ´ c

 

Þ b ´ a + b ´ b = –b ´ c Þ

 

a ´ b = –b ´ c Þ

 

a ´ b = b ´ c

…..(iv)

 

From (iii) and (iv), we get Taking magnitude, we get

 

a ´ b = b ´ c = c ´ a

 

 

|a ´ b |=|b ´ c |=|c ´ a |

 

Þ ab sin(180 – g ) = bc sin(180 – a) = ca sin(180 – b )

Þ ab sin g = bc sina = ca sin b

 

 

Dividing through out by abc, we have Þ

 

Relative Velocity.

sina

a

sin b

b

sin g

c

 

 

  • Introduction : When we consider the motion of a particle, we assume a fixed point relative to which the given particle is in motion. For example, if we say that water is flowing or wind is blowing or a person is running with a speed v, we mean that these all are relative to the earth (which we have assumed to be fixed).

Now to find the velocity of a moving object relative to another moving object, consider a particle P whose

 

®

position relative to frame S is rPS

®

®

while relative to S¢ is rPS¢ . If the position of frames S¢ relative to S at any time is

 

r S¢S

then from fig.

 

®              ®            ®

 

rPS rPS¢ + rS¢S

Differentiating this equation with respect to time

 

 

®

drPS

dt

®

drPS¢

dt

®

  • drS¢S

dt

 

 

®               ®              ®                                                                ®             ®

 

or                   vPS  vPS¢ + vS¢S

[as

v = dr /dt ]

 

 

®               ®             ®

or                   vPS¢ = vPS – vS¢S

®

  • General Formula : The relative velocity of a particle P1 moving with velocity v1 with respect to another

®                                           ®                  ®           ®

particle P2 moving with velocity v2 is given by, v r12 = v1 – v2

  • If both the particles are moving in the same direction then :
r
1        2

u    = u  u

12

  • If the two particles are moving in the opposite direction, then :
r
1         2

u     = u + u

12

  • If the two particles are moving in the mutually perpendicular directions, then:
r

u     =

12

 

®                  ®

  • If the angle between and

be q, then u

= [u 2 + u 2 – 2u u

cosq ]1 / 2 .

 

u1       u 2

r12              1         2           1   2

 

 

 

®

  • Relative velocity of satellite : If a satellite is moving in equatorial plane with velocity vs

®

and a point

 

on the surface of earth with

®

ve  relative to the centre of earth, the velocity of satellite relative to the surface of earth

®          ®

 

vse

= vs v e

 

So if the satellite moves form west to east (in the direction of rotation of earth on its axis) its velocity relative to

earth’s surface will be vse = vs ve

And if the satellite moves from east to west, i.e., opposite to the motion of earth, vse = vs – (-ve ) = vs + ve

 

 

®

  • Relative velocity of rain : If rain is falling vertically with a velocity vR

®

and an observer is moving

 

horizontally with speed  vM        the velocity of rain relative to observer

®                   ®     ®

 

will be

vRM

= vR vM

 

which by law of vector addition has magnitude

 

vRM =

 

direction q = tan-1(vM / vR ) with the vertical as shown in fig.

  • Relative velocity of swimmer : If a man can swim relative to water with velocity

 

 

®

v and water is flowing

 

®

relative to ground with velocity vR

®

velocity of man relative to ground vM

will be given by:

 

®         ®           ®                         ®               ®       ®

v = vM vR , i.e., vM = v + vR

So if the swimming is in the direction of flow of water,

vM = v + vR

And if the swimming is opposite to the flow of water,

vM = v vR

  • Crossing the river : Suppose, the river is flowing with

velocity u . A man can swim in still water with velocity u . He is

r                                                                                                                                                    m

standing on one bank of the river and wants to cross the river Two cases arise.

  • To cross the river over shortest distance : That is to cross the river straight, the man should swim making angle q with the upstream as

®              ®     ®              ®                                                                                                  ®            ®

 

Here OAB is the triangle of vectors, in which

OA = vm , AB = u r . Their resultant is given by OB = u . The

 

direction of swimming makes angle q with upstream. From the triangle OBA, we find,

 

cos q = ur

um

Also

sina = ur

u m

 

where a is the angle made by the direction of swimming with the shortest distance (OB) across the river. Time taken to cross the river : If w be the width of the river, then time taken to cross the river will be given by

 

 

t = w =        w

1     u

 

  • To cross the river in shortest possible time : The man should swim perpendicular to the bank. The time taken to cross the river will be:

t    = w

2      u

m

 

In this case, the man will touch the opposite bank at a distance AB

down stream. This distance will be given by:

 

AB = ur t 2

=u   w

u
r

m

or      AB = ur   w

um

 

Problem 53.      Two trains along the same straight rails moving with constant speed 60 km/hr respectively toward each other.

If at time t = 0 , the distance between them is 90 km, the time when they collide is

(a) 1 hr                                     (b) 2 hr                                     (c)  3 hr                                 (d) 4 hr Solution : (a)     The relative velocity vrel. = 60 – (-30) = 90 km / hr.

 

Distance between the train s

rel.

= 90 km, \ Time when they collide = srel.

vrel.

= 90 = 1 hr.

90

 

Problem 54.   Two cars are moving in the same direction with the same speed 30 km/hr. They are separated by a distance of 5 km, the speed of a car moving in the opposite direction if it meets these two cars at an interval of 4 minutes, will be

(a) 40 km/hr                              (b) 45 km/hr                              (c) 30 km/hr                         (d) 15 km/hr

Solution : (b)     The two car (say A and B) are moving with same velocity, the relative velocity of one (say B) with respect to

 

the other

A, vBA

 

= v B  vA

= v v = 0

 

So the relative separation between them (= 5 km) always remains the same.

Now if the velocity of car (say C) moving in opposite direction to A and B, is v C

 

 

velocity of car C relative to A and B will be vrel. = vC v

 

relative to ground then the

 

 

 

But as v is opposite to vC so

vrel

= vc – (-30) = (vC + 30)km / hr.

 

 

So, the time taken by it to cross the cars A and B

t   d  

Þ   4 =       5    

Þ vC = 45 km / hr.

 

vrel

60     vC  + 30

 

Problem 55. A steam boat goes across a lake and comes back (a) On a quite day when the water is still and (b) On a rough day when there is uniform current so as to help the journey onward and to impede the journey back. If the speed of the launch on both days was same, in which case it will complete the journey in lesser time

(a) Case (a)                                                                       (b) Case (b)

(c) Same in both                                                               (d) Nothing can be predicted

Solution : (b)  If the breadth of the lake is l and velocity of boat is vb. Time in going and coming back on a quite day

 

tQ    + l   = 2l

…..(i)

 

vb          vb           vb

 

 

Now if va is the velocity of air- current then time taken in going across the lake,

 

t1 =        l      

vb + va

[as current helps the motion]

 

and time taken in coming back t 2 =               l     

vb va

So tR = t1 + t2 =            2l             

vb [1 – (va / vb )2 ]

 

[as current opposes the motion]

 

…..(ii)

 

 

From equation (i) and (ii)

t R  =           1         > 1

tQ          [1 – (va / vb )2 ]

[as 1

2

v
v
<
2

   a          1]

b

i.e.

tR > tQ

 

i.e. time taken to complete the journey on quite day is lesser than that on rough day.

Problem 56.      A man standing on a road hold his umbrella at 30° with the vertical to keep the rain away. He throws the umbrella and starts running at 10 km/hr. He finds that raindrops are hitting his head vertically, the speed of raindrops with respect to the road will be

  • 10 km/hr (b) 20 km/hr                                          (c) 30 km/hr                     (d) 40 km/hr

Solution : (b)     When the man is at rest w.r.t. the ground, the rain comes to him at an angle 30° with the vertical. This is the direction of the velocity of raindrops with respect to the ground.

Here v rg = velocity of rain with respect to the ground

 

 

vmg =

velocity of the man with respect to the ground.

 

 

 

and

vrm = velocity of the rain with respect to the man,

 

We have

vr g

 

= vr m + vmg

……(i)

 

Taking horizontal components equation (i) gives vr g sin 30° = vm g

= 10 km / hr

 

 

or    vr g

=  10  = 20 km / hr

sin 30°

 

Problem 57.      In the above problem, the speed of raindrops w.r.t. the moving man, will be

 

(a)

10 /

2 km / h

  • 5 km/h (c)

10 3 km / h

(d) 5 /

3 km / h

 

 

 

Solution : (c)     Taking vertical components equation (i) gives vrg cos 30° = vrm = 20 3

2

= 10

3 km / hr

 

Problem 58. Two cars are moving in the same direction with a speed of 30 km/h. They are separated from each other by 5 km. Third car moving in the opposite direction meets the two cars after an interval of 4 minutes. What is the speed of the third car

(a) 30 km/h                             (b) 35 km/h                             (c) 40 km/h                         (d) 45 km/h Solution : (d) Let v be the velocity of third car, then relative velocity of third car w.r.t. the either car is

v – (– 30) = (v+30) km/h.

Now (u + 30) ´ (4/60) = 5 Þ v = 45 km/h

 

 

Problem 59.      To a person, going eastward in a car with a velocity of 25 km/hr, a train appears to move towards north with a

velocity of 25      km/hr. The actual velocity of the train will be

 

  • 25 km/hr (b) 50 km/hr                           (c) 5 km/hr                         (d) 5      km/hr

 

 

Solution : (a)

vT   =                       =

=                     =             = 25 km/hr

 

 

Problem 60.      A boat is moving with a velocity 3i + 4j with respect to ground. The water in the river is moving with a velocity – 3i – 4j with respect to ground. The relative velocity of the boat with respect to water is                 [CPMT 1998]

(a) 8j                                         (b) – 6i – 8j                             (c) 6i +8j                            (d) 5

Solution : (c)      Relative velocity = (3i + 4j) – (– 3i – 4j) = 6i + 8j

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