Simple Harmonic Motion

The displacement equation for simple harmonic motion (SHM) is: where is the amplitude, is the angular frequency, and is the phase constant. Given: From the equation, we can identify that the amplitude and the angular frequency . The velocity is given by the derivative of the displacement with respect to time: At the instant when , we can find by solving for : Using this, we calculate the velocity as: which simplifies to: Thus, the correct answer is (D).

In Simple Harmonic Motion (SHM), the velocity is related to the displacement and the amplitude by the equation: Where: - is the angular frequency, - is the displacement at any point in time, - is the amplitude of the motion. From the given data: - When , , - When , . We can use these two pieces of information to form two equations and solve for and . For and : For and : Now, solving these equations: From the first equation: Substitute into the second equation: Simplifying, we find and , which allows us to calculate the periodic time . After solving, we find that the periodic time sec.

In simple harmonic motion, the total mechanical energy is the sum of kinetic energy ( ) and potential energy ( ) at any point. The total energy in SHM is constant and is given by: where: - is the mass of the particle, - is the angular frequency, - is the amplitude. At any point, the kinetic energy is given by: where is the displacement from the mean position. The potential energy is given by:
Step 1: Equating Kinetic Energy and Potential Energy
We are given that the kinetic energy is equal to the potential energy. Therefore: Canceling common terms: Thus,
Step 2: Final Answer
The distance from the mean position at which the kinetic energy is equal to the potential energy is approximately , which corresponds to option (B).