Kevin S. Huang

2014: Problem 8

Topic: Oscillatory Motion
Concepts: Effective spring constant, Mass-spring system

Solution:

From the period formula for a mass-spring system, we have

T_0=2\pi\sqrt{\frac{M}{k}}

By cutting the spring in half, the spring constant gets doubled since adding two half-springs in parallel should result in the original spring,

\frac{1}{k_{\text{half}}}+\frac{1}{k_{\text{half}}}=\frac{2}{k_{\text{half}}}=\frac{1}{k}

k_{\text{half}}=2k

Oscillating on an inclined plane doesn’t affect the period since it is independent of $g$ . Thus, the new period is

T’=2\pi\sqrt{\frac{M}{k_{\text{half}}}}=2\pi\sqrt{\frac{M}{2k}}=\frac{T_0}{\sqrt{2}}

so the answer is D.