SHM Potential Energy and Mean Position – Rankers Physics

Equation of SHM: Practice Problem & Solution

Assertion (A): If PE of a particle executing SHM is given by \(U = x^2 - 10x + 27\), then it is executing SHM about \(x = 5\). Reason (R): At mean position, restoring force is zero.  
(1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
(2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
(3) (A) is true but (R) is false
(4) Both (A) and (R) are false

Solution Explained:

To solve this problem, we apply the core principles of Equation of SHM. Understanding the underlying formula is key to arriving at the correct answer below:

Assertion (A): Given \(U = x^2 - 10x + 27 = (x-5)^2 + 2\). For SHM, \(U = \frac{1}{2}k(x-x_0)^2 + U_0\). Comparing, mean position \(x_0 = 5\). So (A) is true. Reason (R): The restoring force \(F = -\frac{dU}{dx}\). At equilibrium (mean) position, \(F=0\). So (R) is true. (R) does not explain (A).

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Similar Practice Questions

Explore conceptually related problems to test your knowledge.

1. The velocities of a body executing SHM are v 1 and v 2 when the displacements are x1 and x2 respectively. The period is :
2. A particle is executing SHM with amplitude A and has maximum velocity V0. Its speed at displacement A/2 will be
3. Two particles execute SHM of same amplitude and frequency along the same straight line. They pass one another when going in opposite directions, each time their displacement is half of their amplitude. The phase difference between them is :
4. Assertion (A): The graph between velocity and displacement for a harmonic oscillator is a parabola. Reason (R): Velocity does change uniformly with displacement in harmonic motion.  
5. Assertion (A): Vibration of polyatomic molecules is not simple harmonic motion. Reason (R): The vibrations are superposition of SHMs of different frequency.