How does a mass-spring system exhibit SHM?

A mass-spring system exhibits Simple Harmonic Motion (SHM) due to the restoring force proportional to the displacement.

In a mass-spring system, when the mass is displaced from its equilibrium position, it experiences a restoring force. This force tries to bring the mass back to its original position. The nature of this force is such that it is directly proportional to the displacement but acts in the opposite direction. This relationship is described by Hooke's Law, which states that the force (F) exerted by a spring is equal to the negative of the spring constant (k) times the displacement (x) from the equilibrium position, or F = -kx.

The negative sign indicates that the force is a restoring force, i.e., it acts in the opposite direction to the displacement. This is a key characteristic of Simple Harmonic Motion (SHM). In SHM, the restoring force on an object is directly proportional to its displacement from its equilibrium position but in the opposite direction.

When the mass is displaced and let go, it starts oscillating about the equilibrium position. The motion is periodic, meaning it repeats after regular intervals of time. The mass moves back and forth due to the restoring force provided by the spring. The magnitude of the velocity of the mass is maximum at the equilibrium position and it is minimum (zero) at the extreme positions. The acceleration of the mass is zero at the equilibrium position and maximum at the extreme positions.

The energy in a mass-spring system oscillating in SHM is conserved, meaning it constantly switches between potential energy stored in the spring when it is compressed or extended, and kinetic energy of the moving mass. The total energy remains constant if there is no energy loss due to friction or other non-conservative forces.

In conclusion, a mass-spring system exhibits Simple Harmonic Motion because the restoring force provided by the spring, which is proportional to the displacement, causes the mass to oscillate about an equilibrium position in a periodic manner.