Question

A particle is undergoing circular motion with angular frequency $\omega$ as shown

If the shadow of the particle on the wall is performing SHM such that its acceleration's maximum value is 'a', find the acceleration of the particle and the radius of the circle.

• A. $a,\frac{9}{2{\omega }^2}$
• B. $2a,\frac{a}{{\omega }^2}$
• C. $a,\frac{a}{{\omega }^2}$
• D. Data is sufficient

Answer 1

The correct option is C

$a,\frac{a}{{\omega }^2}$

The particle doing circular motion has a centripetal acceleration. The component of this acceleration parallel to the wall is the

acceleration of the shadow.

In the above diagram the acceleration of the shadow as is given by ac $cos\theta$. It will be maximum when $cos\theta$ = 1 i.e., in the extreme position. Here, $a{s}_{max}=ac$

Given in the problem is that max acceleration of the shadow is a so $a{s}_{max}=a=ac$ The acceleration of the particle doing circular motion is 'a' Also as we know, $m{a}_c=m{\omega }^{2}r$

$\Rightarrow ma=m{\omega }^{2}r\Rightarrow {\omega }^{2}r=a$

$R=\frac{a}{{\omega }^2}$