KJ , the negative sign, as we

know indicating that the acceleration is directed oppositely to displacement x. Putting c

m

= w

2_{, where w is the angular velocity of the particle, the equation takes the form}

d x dt

2

2 = –w2x = –mx ...(1)

where m is a constant, equal to w2. Or, since d x

dt

2

2 = –m if x = 1, we may define m as the

acceleration per unit displacement of the particle.

Multiplying both sides of the equation by 2dx

dt, we have 2 2 2 dx dt d x dt = –w 2 _{◊ 2x} dx dt,

integrating which with respect to t, we have

dx dt

F

HG

IKJ

2 = –w2x2 + A ...(2)

where A is a constant of integration.

Since at the maximum displacement (or amplitude) a of the oscillator (or the oscillation), the velocity dx

dt = 0, we have

0 = –w2a2 + A, hence, A = w2a2

Substituting this value of A in relation (2), therefore, we have

dx dt

F

HG

IKJ

2 = –w2x2 + w2a2 = w2 (a2 – x2),

Hence, the velocity of the particle at any instant t, is given by

dx dt =

w

a x 2

_-

2 _...(3) Putting equation (3) as dx a x dt

2

_-

2

= w

and integrating again with respect to t, we have

sin-1x

a = wt + f

or x = a sin (wt + f) ...(4)

This gives the displacement of the particle at an instant t in terms of its amplitude ‘a’ and its total phase (wt + f), made up of the phase angle wt f is called the initial phase, phase constant or the epoch of the particle, usually denoted by the letter e. This initial phase or epoch arises because of our starting to count time, not from the instant that the particle is in some standard position, like its mean position or one of its extreme positions, but from the instant when it is anywhere else in between.

Thus, if we start counting time when the particle is in its mean position, i.e., when x = 0 at

t = 0, we have f = 0 and, therefore, x = a sin wt

And, if we start counting time when the particle is in one of its extreme positions, i.e., when

x = a at t = 0, we have a = a sin(0 + f) = a sin f, i.e., sin f = 1 or f =

p

2. So that, x = asin

w p

t

+

F

HG

2

IKJ

= a cos wt. Thus, a simple harmonic motion may be expressed either in terms of a sine or a cosine

function. The time-period of the particle,

T = 2

p

displacement

acceleration

Since T is quite independent of both a and f, it is clear that the oscillations of the particle are isochronous, i.e., take the same time irrespective of the values of a and f.

The number of oscillations (or vibrations) made by the particle per second is called its frequency of oscillation or, simply, its frequency, usually denoted by the letter n. Thus, fre- quency is the reciprocal of the time-period, i.e.,

n = 1 2 1 2 T c m

=

w

=

p

p

Since w is the angle described by the particle per second, it is also referred to as the angular frequency of the particle.

4.4 ENERGY OF A HARMONIC OSCILLATOR

P.E. at displacement x is given by

7

= 1 2 1 2 1 2 2 2 2 m x m c m x cx

w =

F_HG

I_KJ

=

The maximum value of the potential energy is thus at x = a is

7

= 1 2

2

ca

K.E. of the particle at displacement x = 1 2

1 2

2 2 2 2 2

m

w

(a

-

x )

=

c a(

-

x )

The maximum value of K.E. is at x = 0 and is also equal to = 1

2

1 2

2 2 2

m

w

a

=

ca

Total energy of the particle at displacement x i.e., E = K.E. + P.E.

= 1 2 1 2 1 2 1 2 2 2 2 2 2 2 2 2 m

w

(a

-

x )

+

m

w

x

=

m

w

a

=

ca

Maximum value of K.E. = maximum value of P.E. = total energy E = 1

2

1 2

2 2 2

m

w

a

=

ca .

Average K.E. of the particle = m

w

a ca

2 2 2 4 1 4

=

Average P.E. of the particle = 1 4

2 2

m

w

a = 1 4

2

4.5 THE SIMPLE PENDULUM

The simple pendulum is a heavy point mass suspended by a weightless inextensible and flexible string fixed to a rigid support. But these conditions defines merely an ideal simple pendulum which is difficult to realize in practice. In laboratory

instead of a heavy point mass we use a heavy metallic spherical bob tied to a fine thread. The bob is taken spherical in shape be- cause the position of its centre of gravity can be precisely defined. The length (l) of the pendulum being measured from the point of suspension to the centre of mass of the bob. In Fig. 4.1, let S be the point of suspension of the pendulum and O, the mean or equilib- rium position of the bob. On taking the bob a little to one side and then gently releasing it, the pendulum starts oscillating about its mean position, as indicated by the dotted lines. At any given in- stant, let the displacement of the pendulum from its mean position

SO into the position SA be q. Then, the weight mg of the bob, acting

vertically downwards, exerts a torque or a moment –mgl sin q about the point of suspension, tending to bring it back to its mean

position, the negative sign of the torque indicating that it is oppositely directed to the displace- ment (q). If d dt 2 2

q

be the acceleration of the bob, towards O, and I, its M.I. about the point of suspen-

sion (S), the moment of the force or the torque acting on the bob is also equal to Id

dt 2 2

q

. We, therefore, have Id dt 2 2

q

_{= – mgl sinq}

Now, expanding sin q into a power series, in accordance with Maclaurin’s theorem, we have sin q =

q q-

+q

3 5

3 5

! ! .... if, therefore, q be small, i.e., if the amplitude of oscillation be small, we may neglect all other terms except the first and take sin q = q, so that

Id dt 2 2

q

_{= – mglq} d dt 2 2

q

₌

_-

mgl I

q

.

or, since M.I. of the bob (or the point mass) about the point of suspension (S) is ml2, we have

d dt 2 2

q

₌

-

mgl

_{= -}

_{= mq}

ml g l 2

q

q

– where g

l

= m

, the acceleration per unit displacement.

The acceleration of the bob is thus proportional to its angular displacement q and is directed towards its mean position O. The pendulum thus executes a simple harmonic motion and its time-period is, therefore, given by

T = 2

p

1 2

p

1 2

p

m=

g

=

l

l g

The displacement here being angular, instead of linear, it is obviously an example of an angular simple harmonic motion. It is also evident from above expression that the graph between l and T2 will be a straight line with a slope equal to 4

2

p

g .

4.6 DRAWBACKS OF A SIMPLE PENDULUM

Though simple pendulum method is the simplest and straightforward method for determina- tion of ‘g’, it suffers from several defects:

(i) The conditions defining an ideal simple pendulum are never realizable in practice. (ii) The oscillations in practice have a finite amplitude i.e., the angle of swing is not vanish-

ingly small.

(iii) The motion of the bob is not purely translational. It also possesses a rotatory motion about the point of suspension.

(iv) The suspension thread has a finite mass and hence a definite moment of inertia about point of suspension.

(v) The suspension thread is not inextensible and flexible. Hence it slackens when the limits of swing are reached. Thus effective length of the pendulum does not remain constant during the swing.

(vi) Finite size of the bob, yielding of the support and the damping due to air drag also need proper corrections.

(vii) The bob also has a relative motion with respect to the string at the extremities of its amplitude on either side.

Most of the defects are either absent or much smaller in the case of a rigid or compound pendulum.

4.7 THE COMPOUND PENDULUM

Also called a physical pendulum or a rigid pendulum, a compound pendulum is just a rigid body, of whatever shape, capable of oscillating about a horizontal axis passing through it.

The point in which the vertical plane passing through the c.g. of the pendulum meets the axis of rotation is called its point or centre of suspension and the distance between the point of suspension and the c.g. of the pendulum measures the length of the pendulum.

Thus figure shows a vertical section of a rigid body or a compound pendulum, free to rotate about a horizontal axis passing through the point or centre of suspension S. In its normal position of rest, its c.g., G, naturally lies vertically below S, the distance S and G giving the length l of the pendulum.

Let the pendulum be given a small angular displacement q into the dotted position shown, so that its c.g. takes up the new position G¢ where

SG¢ = l. The weight of the pendulum, mg, acting vertically downwards at G¢ and its reaction at the point of suspension S constitute a couple (or a

torque), tending to bring the pendulum back into its original position. Moment of this restoring couple = – mgl sin q, the negative sign indi- cating that the couple is oppositely directed to the displacement q. If I be the moment of inertia of the pendulum about the axis of suspension

(through S) and d

dt

2 2

In document Practical Physics (Page 134-138)