Question

A block is restricted to move in a smooth groove. It is connected to a spring of spring constant $k$. Neglecting the effect of gravity, find the period of small oscillation.

• A. $\frac{2\pi }{\cos \theta }\sqrt{\frac{m}{k}}$
• B. $\frac{2\pi }{\sin \theta }\sqrt{\frac{m}{k}}$
• C. $2\pi \sqrt{\frac{m{\cos }^2\theta }{k}}$
• D. $2\pi \sqrt{\frac{m}{k}}$

Answer 1

The correct option is A $\frac{2\pi }{\cos \theta }\sqrt{\frac{m}{k}}$


$F_{net}=\left(kx\cos \theta \right)\cos \theta =\left(k{\cos }^2\theta \right)x$
$\therefore a=-\left(\frac{k{\cos }^2\theta }{m}\right)x$
$\therefore T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{m}{k{\cos }^2\theta }}=\frac{2\pi }{\cos \theta }\sqrt{\frac{m}{k}}$

Why this question? Caution: When spring makes an angle$\left(\theta \right)$ with displacement, the component of displacement and component of spring force presents ${\cos }^2\theta$ in the equation.