Chapter.
01 (Harmonic Oscillations) BS & ADS (Physics)
Torsional Oscillator
A mass or disk suspended from a fixed support by a thin torsion wire when twisted
about its axis in a horizontal plane is called torsional oscillator or torsional pendulum.
Torsional Oscillator
Consider a torsion wire whose one end is attached with a
centre of disc and other end is fixed to a solid support as shown in
figure. The rest position of disc is called mean position. Now take a
point P on the rim of disc and draw a radial line OP. This radial line is
called reference line. Rotate the disc in horizontal plane through an
angle θm such that reference line OP reaches at line OQ. The wire gets
twisted and torque is produced. The line OQ on disc moves toward
reference line OP when wire is allowed to untwist. The twisting of the
wire creates a restoring torque. The line OP oscillates between OQ
and OR under this restoring torque.
The magnitude of applied torque is directly proportional to rotation angle θ .
τ ∝θ
τ =kθ
Where k is constant of proportionality called torsional constant. The twisted wire will result
in the restoring torque on the disc which is equal in magnitude but opposite in direction to the
applied torque.
τ =−kθ (1)
The negative sign means direction of torque is opposite to the direction of angular
displacement
According to Newton’s second law of angular motion
τ =I α
2
d θ
τ =I 2 (2)
dt
Comparing Eq. (1) and (2)
Zohaib Akram Khan Lecturer (Physics)
MPhil (Physics) G.C Khanewal
Chapter. 01 (Harmonic Oscillations) BS & ADS (Physics)
2
d θ
−kθ=I 2
dt
2
d θ k
2 +
θ=0
dt I
This is called equation of motion of torsional oscillator.
Angular Velocity
The equation of motion of torsional oscillator is
2
d θ k
2 +
θ=0
dt I
dt dt( )
d dθ −k
=
I
θ
dω −k
= θ
dt I
dω dθ −k
= θ
dt dθ I
−k
ω dω= θdθ
I
Integrating on both sides
−k
∫ ω dω= I
∫θ d θ
()
2 2
ω k θ
=− +C
2 I 2
ω 2=− ( kI )θ + 2C
2
(3)
Where, 2C is integration constant. Its value can be determined by applying boundary
conditions. ω=0 when θ = θ m
(0)2=− ( kI ) θ +2 C
2
2 C=− ( kI )θ 2
m
Zohaib Akram Khan Lecturer (Physics)
MPhil (Physics) G.C Khanewal
Chapter. 01 (Harmonic Oscillations) BS & ADS (Physics)
Put this value in Eq. (3)
ω=
2
( kI )(θ −θ )
2
m
2
ω=± (√ kI ) √(θ −θ )
2
m
2
Maximum velocity
The angular velocity of torsional oscillator is maximum at mean position (θ = 0)
ω max=± (√ kI ) √(θ −0) 2
m
ω max=±θ m
(√ kI )
Minimum velocity
The angular velocity of torsional oscillator is minimum at extreme position (x = θm )
ω min =± (√ kI ) √(θ −θ )
2
m
2
m
ω min =0
Displacement Harmonic Oscillator
The angular velocity of torsional oscillator is
ω=± (√ kI ) √(θ −θ )
2
m
2
dθ
dt
=± (√ kI )√(θ −θ )2
m
2
−dθ
√(θ 2
m −θ )2
= (√ kI ) dt
Here considers only negative part because motion is started from extreme position.
Integrating on both sides
Zohaib Akram Khan Lecturer (Physics)
MPhil (Physics) G.C Khanewal
Chapter. 01 (Harmonic Oscillations) BS & ADS (Physics)
cos
−1 θ
θm
= (√ kI ) t+ φ
Where φ is integration constant. Put ω= (√ kI ) called angular frequency. The term ω here is
not angular velocity.
θ
=¿cos (ωt +φ ¿
θm
θ=θmcos (ωt +φ ¿
This equation is called angular displacement of torsional oscillator. Here θm is maximum
angular displacement called amplitude of angular oscillation.
Time Period
The time taken to complete one oscillation is called time period of torsional oscillator
2π
T=
ω
T =2 π (√ kI )
Frequency
Number of oscillations completed by torsional oscillator in one second is called its frequency.
It is reciprocal of time period.
Frequency is given by
1 1
f= =
(√ kI )
t
2π
f=
1
2π √ k
I
Application
The rigidity modulus of a material can be found by taking the material in the form of
a wire and setting up a torsional pendulum using any regular body of suitable mass.
Zohaib Akram Khan Lecturer (Physics)
MPhil (Physics) G.C Khanewal
