Question

Time period of oscillation of mass \(m\) suspended from a spring is \(T\). What is the time period when the spring is cut in half and the same mass is suspended from one of the halves?

• A. \( \frac{T}{2} \)
• B. \( \frac{T}{\sqrt{2}} \)
• C. \( \sqrt{2}T \)
• D. \( 2T \)

Hint:

Spring cut in half \(\Rightarrow\) stiffness doubles.

Answer 1

The Correct Option is B

To determine the new time period of oscillation when a spring is cut in half, we need to first understand how the time period \(T\) relates to the properties of the spring and the mass.

1. The time period of oscillation of a mass \(m\) suspended from a spring is given by the formula: \(T = 2\pi \sqrt{\frac{m}{k}}\) where \(k\) is the spring constant.
2. When a spring is cut into half, the spring constant of each half becomes twice the original spring constant. This is because the spring constant \(k\) is inversely proportional to the length of the spring. Therefore, for a spring of length \(L/2\), the new spring constant \(k'\) becomes: \(k' = 2k\)
3. The new time period \(T'\) with mass \(m\) and the new spring constant \(k'\) can be calculated as: \(T' = 2\pi \sqrt{\frac{m}{k'}}\)
4. Substitute the value of \(k'\): \(T' = 2\pi \sqrt{\frac{m}{2k}}\)
5. This simplifies to: \(T' = \frac{2\pi}{\sqrt{2}} \sqrt{\frac{m}{k}} = \frac{T}{\sqrt{2}}\)

Hence, when the spring is cut in half and the same mass is suspended from one of the halves, the new time period of oscillation is \(\frac{T}{\sqrt{2}}\).

Thus, the correct option is \( \frac{T}{\sqrt{2}} \).