1.2 Shm equation

Force is the cause of simple harmonic motion. However, it plays a very different role than that in translational or rotational motion. First thing that we should be aware that oscillating systems like block-spring arrangement or a pendulum are examples of systems in stable equilibrium. When we apply external force, it tends to destabilize or disturb the state of equilibrium by imparting acceleration to the object in accordance with Newton’s second law.

Secondly, role of destabilizing external force is one time act. The oscillation does not require destabilizing external force subsequently. It, however, does not mean that SHM is a non-accelerated motion. As a matter of fact, oscillating system generates a restoring mechanism or restoring force that takes over the role of external force once disturbed. The name “restoring” signifies that force on the oscillating object attempts to restore equilibrium (central position in the figure) – undoubtedly without success in SHM.

SHM is an accelerated motion in which object keeps changing its velocity all the time. The analysis of SHM involves consideration of “restoring force” – not the external force that initially starts the motion. Further, we need to understand that initial external force and hence restoring force are relatively small than the force required to cause translation or rotation. For example, if we displace pendulum bob by a large angle and release the same for oscillation, then the force on the system may not fulfill the requirement of SHM and as such the resulting motion may not be a SHM.

We conclude the discussion by enumerating requirements of SHM as :

• The object is in stable equilibrium before start of the motion.
• External destabilizing force is applied only once.
• The object accelerates and executes SHM under the action of restoring force.
• The magnitude of restoring force or displacement is relatively small.
• There is no dissipation of energy during motion (ideal reference assumption).

Force equation

Here, we set out to figure out nature of restoring force that maintains SHM. For understanding purpose we consider the block-spring system and analyze “to and fro” motion of the block. Let the origin of reference coincides with the position of the block for the neutral length of spring. The block is moved right by a small displacement “x” and released to oscillate about neutral position or center of oscillation. The restoring spring force is given by (“k” is spring constant) :

$$F=-kx$$

In the case of pendulum, we describe motion in terms of torque as it involves angular motion. Here, torque is (we shall study this relation later):

$$\tau =-mgl\theta$$

In either case, we see that “cause” (whether force or torque) is proportional to negative of displacement – linear or angular as the case be. Alternatively, we can also state the nature of restoring force in terms of acceleration,

Linear acceleration, $a=-\frac{k}{m}x$ ------ for linear SHM of block-spring system