Question

The natural frequency of a spring-mass system is 2 Hz. When an additional mass of 1 kg is added to the original mass \( m \), the natural frequency is reduced to 1 Hz. The original mass \( m \) is

• A. 1 kg
• B. \( \frac{1}{2} \) kg
• C. \( \frac{1}{3} \) kg
• D. \( \frac{1}{4} \) kg

Hint:

The natural frequency decreases with increased mass; use the ratio of frequencies to find the mass ratio in a spring-mass system.

Answer 1

The Correct Option is C

Step 1: Recall the formula for natural frequency.
The natural frequency \( f \) of a spring-mass system is given by: \[ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}, \] where \( k \) is the spring constant, and \( m \) is the mass. The frequency in Hz is related to the angular frequency \( \omega \): \[ \omega = 2\pi f = \sqrt{\frac{k}{m}}. \] Step 2: Set up equations for the two cases.
Case 1: Original mass \( m \), natural frequency \( f_1 = 2 \, \text{Hz} \): \[ \omega_1 = 2\pi f_1 = 2\pi \times 2 = 4\pi, \] \[ \omega_1 = \sqrt{\frac{k}{m}}, \] \[ 4\pi = \sqrt{\frac{k}{m}}, \] \[ (4\pi)^2 = \frac{k}{m}, \] \[ 16\pi^2 = \frac{k}{m} \quad (1). \] Case 2: Mass \( m + 1 \), natural frequency \( f_2 = 1 \, \text{Hz} \): \[ \omega_2 = 2\pi f_2 = 2\pi \times 1 = 2\pi, \] \[ \omega_2 = \sqrt{\frac{k}{m + 1}}, \] \[ 2\pi = \sqrt{\frac{k}{m + 1}}, \] \[ (2\pi)^2 = \frac{k}{m + 1}, \] \[ 4\pi^2 = \frac{k}{m + 1} \quad (2). \] Step 3: Solve for \( m \).
Divide equation (1) by equation (2): \[ \frac{16\pi^2}{4\pi^2} = \frac{\frac{k}{m}}{\frac{k}{m + 1}}, \] \[ 4 = \frac{m + 1}{m}, \] \[ 4m = m + 1, \] \[ 3m = 1, \] \[ m = \frac{1}{3} \, \text{kg}. \] Step 4: Verify the solution.
With \( m = \frac{1}{3} \, \text{kg} \):
From (1): \[ 16\pi^2 = \frac{k}{\frac{1}{3}}, \] \[ k = 16\pi^2 \times \frac{1}{3} = \frac{16\pi^2}{3}. \] From (2): \[ m + 1 = \frac{1}{3} + 1 = \frac{4}{3}, \] \[ 4\pi^2 = \frac{k}{\frac{4}{3}}, \] \[ k = 4\pi^2 \times \frac{4}{3} = \frac{16\pi^2}{3}, \] which matches. The value of \( m \) is consistent. Step 5: Select the correct answer.
The original mass \( m \) is \( \frac{1}{3} \) kg, matching option (3).