Question

The time period of mass suspended from a spring is $T$. If the spring is cut into four equal parts and the same mass is suspended from one of the parts, then the new time period will be

• A. T/4
• B. T
• C. T/2
• D. 2T

Answer 1

The Correct Option is C

To determine the new time period when the spring is cut into four equal parts, we need to understand the relationship between the spring constant and the time period of oscillation.

The time period T of a mass-spring system is given by the formula:

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

where:

• m is the mass.
• k is the spring constant.

When a spring of original length L is cut into four equal parts, the length of each part becomes \frac{L}{4}. The spring constant is inversely proportional to the length of the spring, which means for each smaller spring:

k_{\text{new}} = 4k

Now, suspending the same mass m to one of these smaller springs, the time period T_{\text{new}} can be calculated using:

T_{\text{new}} = 2\pi \sqrt{\frac{m}{k_{\text{new}}}}

Substituting k_{\text{new}} = 4k, we get:

T_{\text{new}} = 2\pi \sqrt{\frac{m}{4k}} = 2\pi \sqrt{\frac{1}{4} \cdot \frac{m}{k}} = \frac{1}{2} \cdot 2\pi \sqrt{\frac{m}{k}} = \frac{T}{2}

Thus, the new time period when the same mass is suspended from one part of the cut spring is \frac{T}{2}.

Therefore, the correct answer is T/2.