Question

A particle executes simple harmonic motion under the restoring force provided by a spring. The time period is T. If the spring is divided in two equal parts and one part is used to continue the simple harmonic motion, the time period will
(a) remain T
(b) become 2T
(c) become T/2
(d) become $T/\sqrt{2}$

Answer 1

(d) become T/$\sqrt{2}$T/2√

Time period $\left(T\right)$ is given by,
$T=2\mathrm{\pi }\sqrt{\frac{m}{k}}$
where m is the mass, and
k is spring constant.

When the spring is divided into two parts, the new spring constant k₁ is given as,
k₁ = $2k$

New time period T₁:
T₁ = $2\mathrm{\pi }\sqrt{\frac{m}{2k}}=\frac{1}{\sqrt{2}}2\mathrm{\pi }\sqrt{\frac{m}{k}}=\frac{1}{\sqrt{2}}\mathrm{T}$