PH8151EPUnit1

1.7. TORSION PENDULUM

The area of cross-section of the imaginary shell = circumference◊width =(2fix)◊(dx).
Therefore, force acting in the shell cross-section is

Force = (shear stress)◊(area) =

A

n

x◊

l

B

◊(2fixdx)=

2 fin◊x^2

l

dx (1.10)

The torque about the axisPQacting on the shell is

(Force)◊(Distance) =

A

2 fin◊x^2

l

dx

B

◊x=

2 fin◊x^3

l

dx (1.11)

The restoring torque acting on the suspension wireT for a given twist◊is obtained by
integrating the expression in equation (1.11)from0tor.

T=

2 fin◊

l

⁄r

0

x^3 dx=

2 fin◊

l

A

r^4

4

B

(1.12)

This can be written as

T=

A

finr^4

2 l

B

◊ (1.13)

The term in brackets is a constant called “couple per unit twist" or “ torque per unit
twist" and is denoted byc.

c=

T

◊

=

A

finr^4

2 l

B

æ T=c◊ (1.14)

Therefore, we can see that the internal restoring torque developed inside the suspension
wire is proportional to the angle of twist. Also, the sense of this torque is opposite
to that of twist because the restring torque is opposite to the moment of the external
twisting couple^4. These are the conditions for simple harmonic motion. Hence, a torsion
pendulum, when twisted and released, will execute simple harmonic oscillations where
the displacement from mean position is the angle of twist◊.

If◊is the angle of twist at timet, then the angular velocity of the oscillation isd◊/dt
and the angular acceleration isd^2 ◊/dt^2 .LetIbe the moment of inertia of the suspended
disc of torsion pendulum. Then, from Newton’s second law, the torque is also given by ,

T=I◊

d^2 ◊

dt^2

(1.15)

From equations (1.14) and (1.15),

d^2 ◊

dt^2

=

3

c

I

4

◊ æ

◊

(d^2 ◊/dt^2 )

=

I

c

(1.16)

For angular simple harmonic oscillations, the time period of one oscillation is given by

T 0 =2fi

Û

angular displacement

angular acceleration

=2fi

Û

◊

(d^2 ◊/dt^2 )

=2fi

Û

I

c

. Or,T 0 =2fi

Û

I

c

(1.17)

(^4) When the sense of the twisting angle relative to the restoring torque is also taken into account,
equation (1.14) will change toT=≠c◊. The negative sign indicates thatTand◊are opposite to each
other. But here we continue to consider only the magnitudes ofTand◊and proceed with (1.14).

PH8151 23 LICET