Question

For a particle executing simple harmonic motion, the restoring force is proportional to:

• A. Velocity of the particle
• B. Square of displacement
• C. Displacement from mean position
• D. Acceleration of the particle

Hint:

Always remember the "linear" nature of SHM.
Restoring Force \(\propto -x\) and Acceleration \(\propto -x\).
If the force were proportional to \( x^2 \), the motion would be periodic but not "Simple Harmonic".

Answer 1

The Correct Option is C

Step 1: Recall what simple harmonic motion means. 
Simple Harmonic Motion (SHM) refers to a type of oscillatory motion in which a particle repeatedly moves back and forth about a fixed equilibrium position.
The defining feature of SHM is the nature of the force that acts on the particle.

Step 2: Identify the nature of the restoring force.
In SHM, the force acting on the particle always tries to pull it back toward the equilibrium position.
This force depends on how far the particle is displaced from the mean position.

Step 3: Use the mathematical relation.
The restoring force in SHM is given by:

\[ F = -k x \]

Here:

• $x$ is the displacement from the equilibrium position,
• $k$ is a constant characteristic of the system,
• The negative sign shows that the force acts opposite to the displacement.

Step 4: Explain the physical meaning.
As the particle moves farther away from the mean position, the restoring force increases in direct proportion to the displacement.
If the displacement becomes twice as large, the restoring force also becomes twice as large.
This linear relationship is what distinguishes SHM from other types of motion.

Forces that depend on velocity or vary nonlinearly with displacement do not satisfy the conditions for simple harmonic motion.

Step 5: Final conclusion.
In simple harmonic motion, the restoring force is directly proportional to the displacement from the mean position.