**Correct Option: C. It is doubled** ### Detailed Explanation: 1. **Understanding Simple Harmonic Motion (S.H.M.):** - In S.H.M., the motion of an object is characterized by a restoring force that acts to bring the object back to its equilibrium position. This restoring force is always directed towards the equilibrium position and is proportional to the displacement from that position. - Mathematically, this relationship can be expressed using Hooke's Law, which states: \[ F = -kx \] where: - \( F \) is the restoring force, - \( k \) is the spring constant (a measure of stiffness), - \( x \) is the displacement from the equilibrium position. 2. **Analyzing the Relationship:** - According to Hooke's Law, if the displacement \( x \) is doubled (i.e., \( x \) becomes \( 2x \)), we can substitute this into the equation: \[ F' = -k(2x) = -2kx \] - This shows that the new restoring force \( F' \) is twice the original restoring force \( F \): \[ F' = 2F \] - Therefore, when the displacement is doubled, the restoring force also doubles. 3. **Why Other Options Are Incorrect:** - **Option A: It becomes zero** - This option is incorrect because the restoring force is only zero when the object is at the equilibrium position (i.e., when \( x = 0 \)). Doubling the displacement does not lead to a zero force; instead, it increases the force. - **Option B: It is halved** - This option is incorrect because halving the restoring force would imply that the displacement has decreased, which contradicts the premise of the question that states the displacement is doubled. - **Option D: It is quadrupled** - This option is incorrect because quadrupling the restoring force would imply a relationship where the force is proportional to the square of the displacement, which is not the case in S.H.M. The restoring force is directly proportional to the displacement, not to its square. ### Common Pitfalls: - **Misunderstanding Proportionality:** Students often confuse direct proportionality with other types of relationships (like quadratic). Remember that in S.H.M., the restoring force is directly proportional to the displacement. - **Forgetting the Negative Sign:** The negative sign in Hooke's Law indicates that the force acts in the opposite direction to the displacement. While this is important for understanding direction, it does not affect the magnitude of the force when considering how it changes with displacement. ### Revision Summary: - In S.H.M., the restoring force is directly proportional to the displacement from equilibrium. - Doubling the displacement results in doubling the restoring force. - The correct answer is C: It is doubled. - Remember the relationship \( F = -kx \) for understanding how force changes with displacement.