Question

A gramophone record is revolving with angular velocity . A coin is placed at a distance r from the centre of the record. The static coefficient of friction is $\mu$. The coin will revolve with the record if
A. $r = \mu g{\omega ^2}$
B. $r < \dfrac{{{\omega ^2}}}{{\mu g}}$ .
C. $r \leqslant \dfrac{{\mu g}}{{{\omega ^2}}}$
D. $r \geqslant \dfrac{{\mu g}}{{{\omega ^2}}}$.

Answer 1

In this question we will use the concept of static friction. While solving this problem, we have to remember the concept of static friction. The force of friction which comes into play between two bodies before one body actually starts moving over the other is called static friction $({f_{\text{S}}})$, ${f_S}$ is defined as, ${f_{\text{S}}}{\text{(static friction)}} \leqslant {\mu _S}mg$.

Formula used: F = $mr{\omega ^2}$, Force of friction = $\mu mg$, ${f_{\text{S}}}{\text{(static friction)}} \leqslant {\mu _S}mg$.

Complete step-by-step answer:
Given that, the gramophone record is revolving with angular velocity = $\omega$ and the static coefficient of friction =$\mu$.
We know that a force required to make a body move along a circular path with uniform speed is called centripetal force.
Centripetal force = mass $\times$ centripetal acceleration.
F = $mr{\omega ^2}$ …………………………(i)
And, whenever a body moves or tends to move over the surface of another body, a force comes into play which acts parallel to the surface of contact and opposes the relative motion. This opposing force is called the force of friction.
Force of friction = $\mu mg$……….(ii)
Here, the centripetal force provided by the revolving record is the static friction.
So, the coin will revolve with the record of centripetal force $\leqslant$ force of friction.
Therefore,
$\Rightarrow$ centripetal force $\leqslant$ force of friction.
$\Rightarrow$ $mr{\omega ^2} \leqslant \mu mg$
$\Rightarrow$ $r \leqslant \dfrac{{\mu g}}{{{\omega ^2}}}.$
Hence, we can see that the coin will revolve with the record if $r \leqslant \dfrac{{\mu g}}{{{\omega ^2}}}.$
Therefore, the correct answer is option (C).

Note: Whenever we ask such types of questions, we have to remember some basic points of static friction. First we have to find out the given details in the question. Then we will use the relationship of static friction with the coefficient of static friction and normal reaction(mg), i.e. ${f_{\text{S}}} \leqslant {\mu _S}mg$. then by using this relation, we can easily find out the required condition, that has been asked in the question.