Question
Define angular S.H.M. and obtain its differential equation.
✓
Answer
Angular S.H.M. is defined as the oscillatory motion of a body in which the torque for angular acceleration is directly proportional to the angular displacement and its direction is opposite to that of angular displacement.
i. Consider a metallic disc hanging from rigid support, when twisted, it performs an oscillatory motion for which the restoring torque acting upon it, for angular displacement $\theta$ is,
$ \tau \propto-\theta$
$\therefore \tau=-c \theta . $
ii. The constant of proportionality (c) is the restoring torque per unit angular displacement.
iii. If I is the moment of inertia of the disc, the torque acting on the disc is given by,
$\tau=\mid \alpha \text {.....(2) }$
Where $\alpha$ is the angular acceleration.
iv. From equations ( 1 ) and (2),
$ l \alpha=- c \theta$
$\therefore I \frac{ d ^2 \theta}{ dt ^2}+ c \theta=0 \ldots \ldots . .\left(\because \alpha=\frac{ d ^2 \theta}{ dt ^2}\right) $
This is the differential equation for angular S.H.M.
i. Consider a metallic disc hanging from rigid support, when twisted, it performs an oscillatory motion for which the restoring torque acting upon it, for angular displacement $\theta$ is,
$ \tau \propto-\theta$
$\therefore \tau=-c \theta . $
ii. The constant of proportionality (c) is the restoring torque per unit angular displacement.
iii. If I is the moment of inertia of the disc, the torque acting on the disc is given by,
$\tau=\mid \alpha \text {.....(2) }$
Where $\alpha$ is the angular acceleration.
iv. From equations ( 1 ) and (2),
$ l \alpha=- c \theta$
$\therefore I \frac{ d ^2 \theta}{ dt ^2}+ c \theta=0 \ldots \ldots . .\left(\because \alpha=\frac{ d ^2 \theta}{ dt ^2}\right) $
This is the differential equation for angular S.H.M.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The radius of gyration of a body about an axis at $6 \mathrm{~cm}$ from its centre of mass is $10 \mathrm{~cm}$. Find its radius of gyration about a parallel axis through its centre of mass.
→Represent graphically the displacement, velocity and acceleration against time for a particle performing linear S.H.M., when it starts from the mean position.
Write an expression for an alternating emf that varies sinusoidally with time. Show graphically variation of emf with time.
Calculate the time taken by a body performing SHM of period 2 seconds to cover half the amplitude starting from an extreme position.
If we apply a large enough electric field, we can ionize the atoms and create a condition for electric charge to flow like a conductor. The fields required for the breakdown of dielectric is called dielectric strength.
How do you calculate the shunt required to increase the range p times?
A toroidally wound coil has an inner radius of $15 \ cm$, an outer radius of $20 \ cm$ and is wound with $1500$ turns of wire. What is the magnitude of the magnetic induction at the centre of the coil when the current in the winding is $10 A$ ?
→Explain Biot Savart's Law.
Current passes through a coil shown from left to right. In which direction is th induced emf. if the current is (a) increasing with time (b) decreasing in time?
The refractive indices of glass and water w.r.t. air \(\frac{3}{2}\) and \(\frac{4}{3}\) respectively. Determine the refractive index of glass w.r.t. water.