Question

A sphere of radius $r$ is kept on a concave mirror of radius of curvature $R.$ The arrangement is kept on a horizontal table (the surface of concave mirror is frictionless and sliding not rolling). If the space displasded from its equilibrium position and left then it executes $SHM$. The period of oscillation will be

• A. $2\pi \sqrt{\left(\frac{(R-r)1.4}{g}\right)}$
  
• B. $2\pi \sqrt{\left(\frac{R-r}{g}\right)}$
• C. $2\pi \sqrt{\left(\frac{r^R}{a}\right)}$
• D. $2\pi \sqrt{\left(\frac{R}{gr}\right)}$

Answer 1

The correct option is B $2\pi \sqrt{\left(\frac{R-r}{g}\right)}$

After displacing the sphere, it will experience force towards the mean position which will be provided by component of gravitaional force
From the figure we can write $\theta =\frac{x}{R}$
Tangential acceleration,
$a_t=-g\sin \theta =g\theta$
$a_t=-g\frac{x}{(R-r)}$
Motion is $SHM$, with time period
$T=2\pi \sqrt{\frac{Displacement}{Acceleration}}$
$2\pi \sqrt{\frac{\frac{x}{gx}}{(R-r)}}$
$2\pi \sqrt{\frac{R-r}{g}}$
Final answer: (b)