Question

A particle is executing a simple harmonic motion. Its maximum acceleration is $\alpha$ and maximum velocity is $\beta$ Then, its time period of vibration will be

• A. $ \frac{ \beta^2}{ \alpha}$
• B. $ \frac{ 2 \pi \beta}{ \alpha} $
• C. $ \frac{ \beta^2}{ \alpha^2}$
• D. $ \frac{ \alpha}{ \beta} $

Answer 1

Answer: (B)

lf A and $ \omega $ be amplitude and angular frequency of vibration, then
$ \alpha = \omega^2 A $ ...(i)
and $ \beta = \omega A $ ...(ii)
Dividing eqn. (i) by eqn. (ii), we get
$ \frac{ \alpha}{ \beta} = \frac{ \omega^2 A }{ \omega \, A} = \omega $
$\therefore $ Time period of vibration is
T = $ \frac{ 2 \pi}{ \omega} = \frac{ 2 \pi}{ ( \alpha / \beta)} = \frac{ 2 \pi \beta}{ \alpha} $