Question

One end of a long metallic wire of length \( L \) tied to the ceiling. The other end is tied with a massless spring of spring constant \( K \). A mass hangs freely from the free end of the spring. The area of cross section and the Young’s modulus of the wire are \( A \) and \( Y \) respectively. If the mass slightly pulled down and released, it will oscillate with a time period \( T \) equal to:

• A. \( 2\pi \sqrt{\frac{m}{K}} \)
• B. \( 2\pi \sqrt{\frac{m(YA + KL)}{YAK}} \)
• C. \( 2\pi \sqrt{\frac{mYA}{KL}} \)
• D. \( 2\pi \sqrt{\frac{m}{L}} \)

Hint:

The time period of a system involving a spring and a wire depends on the spring constant, the mass, and the Young's modulus.

Answer 1

The Correct Option is B

Step 1: Time period formula.
The time period of oscillation for a mass-spring system is given by: \[ T = 2\pi \sqrt{\frac{m}{K}} \] However, the Young’s modulus also affects the system, as it relates to the elasticity of the wire. Thus, combining the effects of the spring and the wire, we get the time period formula as shown in option (2).
Thus, the correct answer is (2).