Question

A uniform metal chain is placed on a rough table such that one end of the chain hangs down over the edge of the table. When one-third of its length hangs over the edge, the chain starts sliding. Then, find the coefficient of static friction.
A. \[\dfrac{3}{4}\]
B. \[\dfrac{1}{4}\]
C. \[\dfrac{2}{3}\]
D. \[\dfrac{1}{2}\]

Answer 1

Hint:In order to proceed with this question let’s have a look at static friction. Static friction is defined as the limiting frictional force above which the object starts to slide or move on a surface and it will be equal to the weight of the hanging chain.

Formula Used:
To find the coefficient of static friction the formula is,
\[\mu = \dfrac{F}{N}\]
Where, F is force and N is normal force.

Complete step by step solution:
Consider a uniform metal chain that is placed on a rough table such that one end of the chain hangs down over the edge of the table. When one-third of its (metal chain) length hangs over the edge, the chain starts sliding. Then we need to find the coefficient of static friction.

Initially, we have the length here, which is,
Length of the chain \[ = L\]
And when it becomes one-third, then,
\[{L^1} = \dfrac{L}{3}\]
The chain starts sliding then,
\[\dfrac{L}{3} = \dfrac{{\mu L}}{{\mu + 1}}\]
Here, \[\mu \] is the coefficient of frictional force.
\[\mu + 1 = 3\mu \]
\[\Rightarrow 2\mu = 1\]
\[ \therefore \mu = \dfrac{1}{2}\]
Therefore, the value of the coefficient of static friction is \[\dfrac{1}{2}\].

Hence, Option D is the correct answer.

Note:There is an alternative method to find this solution.
The length of the chain \[ = L\]
The mass of the chain \[ = m\]
When the one-third of its length hangs over the edge then,
\[{L^1} = \dfrac{L}{3}\]
The mass distributed along the length is \[\dfrac{m}{3}\].
When the force pulls the chain downwards due to gravity then, \[F = \dfrac{{mg}}{3}\]
The length of the chain on the table is given by \[\dfrac{{2L}}{3}\].
The normal force on the chain on the table is, \[N = \dfrac{{2mg}}{3}\]
Since, \[\mu \] is the coefficient of frictional force
The frictional force is,
\[F = \mu N\]
\[\Rightarrow \mu = \dfrac{F}{N}\]
Substituting the value of F and N in the above equation, we get
\[\mu = \dfrac{{\left( {\dfrac{{mg}}{3}} \right)}}{{\left( {\dfrac{{2mg}}{3}} \right)}}\]
\[\therefore \mu = \dfrac{1}{2}\]
Therefore, the value of coefficient of static friction is \[\dfrac{1}{2}\].