Question

The displacement of a particle of mass \(2g\) executing simple harmonic motion is \[ x = 8 \cos \left( 50t + \frac{\pi}{12} \right) \text{ m}, \] where \(t\) is time in seconds. The maximum kinetic energy of the particle is:

• A. \(160 \text{ J} \)
• B. \(80 \text{ J} \)
• C. \(40 \text{ J} \)
• D. \(20 \text{ J} \)

Hint:

The maximum kinetic energy in SHM is given by: \[ KE_{\max} = \frac{1}{2} m \omega^2 A^2 \] where \(A\) is the amplitude and \(\omega\) is the angular frequency.

Answer 1

The Correct Option is A

Step 1: Understanding the given equation
The displacement equation for simple harmonic motion is given as: \[ x = A \cos (\omega t + \phi) \] Comparing with the given equation, \[ A = 8 \text{ m}, \quad \omega = 50 \text{ rad/s} \] Step 2: Formula for Maximum Kinetic Energy
The maximum kinetic energy in simple harmonic motion is given by: \[ KE_{\max} = \frac{1}{2} m \omega^2 A^2 \] Here, the given mass is \(2g = 2 \times 10^{-3} \text{ kg}\). Step 3: Substituting values \[ KE_{\max} = \frac{1}{2} \times (2 \times 10^{-3}) \times (50)^2 \times (8)^2 \] \[ = \frac{1}{2} \times 2 \times 10^{-3} \times 2500 \times 64 \] \[ = 160 \text{ J} \] Thus, the correct answer is option (A) 160 J.