1.4 Angular shm

Angular SHM involves “to and fro” angular oscillation of a body about a central position or orientation. The particle or the body undergoes small angular displacement about mean position. This results, when the body under stable equilibrium is disturbed by a small external torque. In turn, the rotating system generates a restoring torque, which tries to restore equilibrium.

Learning about angular SHM is easy as there runs a parallel set of governing equations for different physical quantities involved with the motion. Most of the time, we only need to know the equivalent terms to replace the linear counterpart in various equations. However, there are few finer differences that we need to be aware about. For example, how would be treat angular frequency “ω” and angular velocity of the oscillating body in SHM. They are different.

Restoring torque

We write restoring torque equation for angular SHM as :

$$\tau =-k\theta$$

This “k” is springiness of the restoring torque. We associate “springiness” with any force or torque which follows the linear proportionality with negative displacement. For this reason, we call it “spring constant” for all system – not limited to block-spring system. In the case of simple pendulum, we associate this “springiness” to gravity. Similarly, this property can be associated with other forces like torsion, stress, pressure and many other force systems, which operate to restore equilibrium.

Torsion constant

A common setup capable of executing angular SHM consists of a weight attached to a wire. The rigid body is suspended from one end of the wire, whereas its other end is fixed. When the rigid body is given a small angular displacement, the body oscillates about certain reference line – which corresponds to the equilibrium position.

Torsion pendulum

The body oscillates angularly. If we assume conservation of mechanical energy, then system oscillates with constant angular amplitude indefinitely. The whole system is known as torsion pendulum. In this case “k” of the torque equation is also known as “torsion constant”. Dropping negative sign,

$$k=\frac{\tau }{\theta }$$

Clearly, torsion constant measures the torque per radian of angular displacement. It depends on length, diameter and material of the wire.

Equations of angular shm

We write various equations for angular SHM without derivation – unless there is differentiating aspect involved. In general, we substitute :

1. Linear inertia “m” by angular inertia “I”
2. Force, “F” by torque “τ”
3. Linear acceleration “a” by angular acceleration, “α”
4. Linear displacement “x” by angular displacement “θ”
5. Linear amplitude “A” by angular amplitude “ ${\theta }_0$ ”
6. Linear velocity “v” by angular velocity “ $d\theta ∕dt$ ”

Importantly, symbols of angular frequency (ω), spring constant (k), phase constant (φ), time period (T) and frequency (ν) remain same in the description of angular SHM.

Angular displacement

$$\theta ={\theta }_0\sin \left(\omega t+\phi \right)$$

where “ ${\theta }_0$ ” is the amplitude, “φ” is the phase constant and “ωt + φ” is the phase. Clearly, angular displacement is periodic with respect to time as it is represented by bounded trigonometric function. The displacement “θ” varies between “ $-{\theta }_0$ ” and “ ${\theta }_0$ ”.