The correct option for the question regarding which quantity is directly proportional to the displacement from the equilibrium position in simple harmonic motion (S.H.M.) is: **B. Acceleration** ### Detailed Explanation 1. **Understanding Simple Harmonic Motion (S.H.M.)**: - S.H.M. is a type of periodic motion where an object moves back and forth around an equilibrium position. The motion is characterized by a restoring force that is proportional to the displacement from the equilibrium position and acts in the opposite direction. - The general equation for the displacement \( x \) in S.H.M. can be expressed as: \[ x(t) = A \cos(\omega t + \phi) \] where: - \( A \) is the amplitude (maximum displacement), - \( \omega \) is the angular frequency, - \( t \) is time, - \( \phi \) is the phase constant. 2. **Acceleration in S.H.M.**: - The acceleration \( a \) of an object in S.H.M. is given by the formula: \[ a = -\omega^2 x \] - This equation shows that acceleration is directly proportional to the displacement \( x \) from the equilibrium position. The negative sign indicates that the acceleration is directed towards the equilibrium position, which is characteristic of restoring forces in S.H.M. 3. **Why Other Options Are Incorrect**: - **A. Velocity**: - The velocity \( v \) in S.H.M. is given by: \[ v = \frac{dx}{dt} = -A\omega \sin(\omega t + \phi) \] - Velocity is not directly proportional to displacement; instead, it is related to the sine of the displacement. At maximum displacement, the velocity is zero, and at the equilibrium position, the velocity is maximum. - **C. Kinetic Energy**: - The kinetic energy \( KE \) in S.H.M. is given by: \[ KE = \frac{1}{2}mv^2 \] - Since velocity is not directly proportional to displacement, kinetic energy is also not directly proportional to displacement. It varies with the square of the velocity, which depends on the sine function of the displacement. - **D. Potential Energy**: - The potential energy \( PE \) in S.H.M. is given by: \[ PE = \frac{1}{2}kx^2 \] - While potential energy does depend on displacement, it is proportional to the square of the displacement, not directly proportional. This means that as displacement increases, potential energy increases with the square of that displacement. ### Summary of Key Points - In S.H.M., acceleration is directly proportional to displacement from the equilibrium position. - The relationship is given by \( a = -\omega^2 x \). - Velocity, kinetic energy, and potential energy do not have a direct proportionality to displacement. - Understanding the relationships between these quantities is crucial for solving problems related to S.H.M. ### Revision Summary - **Correct Answer**: B. Acceleration is directly proportional to displacement in S.H.M. - **Acceleration Formula**: \( a = -\omega^2 x \). - **Velocity**: Not directly proportional; depends on sine of displacement. - **Kinetic and Potential Energy**: Not directly proportional; kinetic energy depends on the square of velocity, and potential energy depends on the square of displacement.