Question

A particle executes S.H.M. along x-axis. The force acting on it is given by (A, k>0)

• A. A cos (kx)
• B. A $ e^{ - kx}$
• C. Akx
• D. - Akx

Answer 1

The Correct Option is D

To determine the correct force expression for a particle executing Simple Harmonic Motion (S.H.M) along the x-axis, let's explore the fundamental characteristics of S.H.M.

In S.H.M., the restoring force acting on a particle is directly proportional to its displacement from the mean position and is always directed towards the mean position. This is mathematically expressed as:

F = -kx

Where:

• F is the restoring force,
• k is the force constant or spring constant (a positive value),
• x is the displacement from the equilibrium position.

Here, the negative sign indicates that the force is always in the opposite direction of the displacement, reinforcing that it is a restoring force.

Analyzing the given options, we need the choice that represents this relationship:

• A cos(kx): This expression represents a periodic function, not a linear proportionality to displacement, so it is incorrect for expressing the restoring force in S.H.M.
• A e^{-kx}: This term relates to an exponential decay and does not fit the linear relationship required for S.H.M. Thus, it is incorrect.
• Akx: This represents a linear relationship but lacks the necessary negative sign to indicate the force direction, so it is not correct.
• - Akx: This option correctly represents the restoring force in S.H.M., where the force is proportional to the displacement and directed towards the equilibrium position.

Therefore, the correct answer is: - Akx