Question

A mass of 1 kg is executing SHM. Its displacement is given by \( x = 6.0 \cos(100t + \pi/4) \) cm. What is the maximum kinetic energy?

• A. 3 J
• B. 6 J
• C. 9 J
• D. 18 J

Hint:

Always ensure consistency in units. If mass is in kg and frequency in rad/s, amplitude must be in meters to obtain energy in Joules. Convert cm to m before calculation.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the maximum kinetic energy of a mass undergoing Simple Harmonic Motion (SHM), given its mass and displacement equation.
Step 2: Key Formula or Approach:
The general equation for SHM is \( x = A \cos(\omega t + \phi) \).
Comparing this with the given equation, we can identify the amplitude ($A$) and angular frequency ($\omega$).
The maximum kinetic energy ($K_{max}$) in SHM is given by: \[ K_{max} = \frac{1}{2} m A^2 \omega^2 \]
Step 3: Detailed Explanation:
Given:
Mass, \( m = 1 \text{ kg} \)
Displacement equation: \( x = 6.0 \cos(100t + \pi/4) \text{ cm} \)
From the equation:
- Amplitude, \( A = 6.0 \text{ cm} = 0.06 \text{ m} \)
- Angular frequency, \( \omega = 100 \text{ rad/s} \)
Substitute these values into the maximum kinetic energy formula:
\[ K_{max} = \frac{1}{2} \times (1 \text{ kg}) \times (0.06 \text{ m})^2 \times (100 \text{ rad/s})^2 \] \[ K_{max} = \frac{1}{2} \times 1 \times (0.0036) \times (10000) \] \[ K_{max} = \frac{1}{2} \times 36 = 18 \text{ J} \]
Step 4: Final Answer:
The maximum kinetic energy is 18 J.