Question

The mass attached to a vibrating string is increased four times. What is the affect on the time period and frequency of oscillation of the mass-spring system?

Answer 1

1. The mass attached to a vibrating string is increased four times. What is the affect on the time period and frequency of oscillation of the mass-spring system?

Solution

Angular frequency of oscillations of a mass-spring system is given by:

$$\omega = \sqrt {\frac {k}{M}}$$

If the mass, attached to a vibrating, string is increased four times:

$$\omega^ {*} = \sqrt {\frac {k}{4 \cdot M}} = \frac {1}{2} \cdot \sqrt {\frac {k}{M}} = \frac {\omega}{2}$$

Then since $\omega = 2\pi f$ and since $T = 1 / f$ where $T$ is the time period,

$$T = 2 \cdot \pi \cdot \sqrt {\frac {m}{k}}$$

If the mass, attached to a vibrating, string is increased four times:

$$T ^ {*} = 2 \cdot \pi \cdot \sqrt {\frac {4 \cdot m}{k}} = 2 \cdot \left(2 \cdot \pi \sqrt {\frac {4 \cdot m}{k}}\right) = 2 \cdot T$$

Answer

Time period doubles.

Frequency of oscillation is halved.