Question

Time period of oscillation of a 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, the fractional change in time period is

• A. \( \frac{1}{8} \)
• B. \( \frac{1}{4} \)
• C. \( \frac{1}{2} \)
• D. \( \frac{1}{6} \)

Hint:

Cutting a spring reduces its length and increases stiffness, thus decreasing the time period.

Answer 1

The Correct Option is C

Step 1: Write formula of time period.
\[ T = 2\pi \sqrt{\frac{m}{k}} \]

Step 2: Understand effect of cutting spring.
When a spring is cut into \( n \) equal parts, spring constant becomes:
\[ k' = n k \]

Step 3: Apply for 4 parts.
\[ k' = 4k \]

Step 4: Write new time period.
\[ T' = 2\pi \sqrt{\frac{m}{k'}} \]
\[ T' = 2\pi \sqrt{\frac{m}{4k}} \]

Step 5: Simplify relation.
\[ T' = \frac{T}{2} \]

Step 6: Find change in time period.
\[ \Delta T = T - T' = T - \frac{T}{2} = \frac{T}{2} \]

Step 7: Fractional change.
\[ \frac{\Delta T}{T} = \frac{1}{2} \]