Question

System is released after slightly stretching it. Find angular frequency of its oscillations:

• A. \(5\)
• B. \(10\sqrt{5}\)
• C. \(2\sqrt{5}\)
• D. \(5\sqrt{5}\)

Hint:

For two masses connected by a spring on a smooth surface, always use {reduced mass} to find the angular frequency.

Answer 1

The Correct Option is D

Concept:
Two blocks connected by a spring on a smooth horizontal surface execute simple harmonic motion
when slightly displaced.

The system oscillates about its centre of mass.
The effective mass of the system is the reduced mass
.
Angular frequency is given by: \[ \omega = \sqrt{\frac{k}{\mu}} \] where \(\mu\) is the reduced mass.

Step 1: Calculate Reduced Mass
Masses: \[ m_1 = 2\,\text{kg}, \quad m_2 = 3\,\text{kg} \] Reduced mass: \[ \mu = \frac{m_1 m_2}{m_1 + m_2} = \frac{2 \times 3}{2 + 3} = \frac{6}{5}\,\text{kg} \]
Step 2: Substitute in Angular Frequency Formula
Spring constant: \[ k = 150\,\text{N/m} \] \[ \omega = \sqrt{\frac{k}{\mu}} = \sqrt{\frac{150}{6/5}} = \sqrt{125} \] \[ \omega = 5\sqrt{5}\,\text{rad/s} \] \[ \boxed{\omega = 5\sqrt{5}\,\text{rad/s}} \]