Question

A particle of mass $1\text{g}$ executes an oscillatory motion on the concave surface of a spherical dish placed on a horizontal plane. If the motion of the particle begins from a point on the dish at height $1\text{cm}$ from the horizontal plane and coefficient of friction is $0.01$, find the total distance covered by the particle before it comes to rest (Assume the radius of curvature of the concave surface to be very large).

• A. $10\text{cm}$
• B. $28\text{cm}$
• C. $58\text{cm}$
• D. $100\text{cm}$

Answer 1

The correct option is D $100\text{cm}$

We know that, work done by all non-conservative forces
$W_{nc}=\Delta ME$

Here, friction is the only non-conservative force and initial mechanical energy of the particle, $E=mgh$
where $h$ is the height above horizontal plane.

The particle keeps oscillating on the concave surface till its initial potential energy $mgh$ is lost in work done against friction force $f=\mu N=\mu mg\cos \theta$.



If $d$ is the distance traversed, the work done against friction
$W_f=\mu mg\cos \theta d=\Delta ME=mgh$
Here $\theta \approx 0$ [since radius of curvature of surface is given to be very large]
$\Rightarrow \mu mgd=mgh$
$\Rightarrow d=\frac{h}{\mu }=\frac{1}{0.01}=100\text{cm}$

Hence, the correct answer is option (d).