Question

Two bodies of mass m and 4m are attached with string as shown in the fig. The body of mass m hanging from a string of length / is executing oscillations of angular amplitude while the other body is at rest. The minimum coefficient of friction between the mass 4m and horizontal surface should be:

Answer 1

Hint: In this question, we will use the relation between the forces acting on the two different masses, further equating and solving these equations; will give us the required result. Also, we will discuss the basics of force of friction and its types, for our better understanding.
Formula used:
$T=mg+{\displaystyle \frac{m{v}^2}{l}}$
$T=\mu N$

Complete answer:
As we know, the tension experienced by the mass m and mass 4m is given by:
$T=mg+{\displaystyle \frac{m{v}^2}{l}}$
$T=\mu N=4\mu mg$
Now, equating the above two equations, we get:
$4\mu mg=mg+{\displaystyle \frac{m{v}^2}{l}}.........(1)$
As we know that, here the energies are equal, so we get:
$mg(l-lcos{\theta }_0)={\displaystyle \frac{1}{2}}m{v}^2$
$\Rightarrow 2mgl(1-cos{\theta }_0)=m{v}^2$
$\Rightarrow 2mg(1-cos{\theta }_0)={\displaystyle \frac{m{v}^2}{l}}.........(2)$
Now, from equation (1) and (2), we get:
$mg+2mg(1-cos{\theta }_0)=4\mu mg$
$\Rightarrow 1+2(1-cos{\theta }_0)=4\mu$
$\Rightarrow 3-2cos{\theta }_0=4\mu$
$\therefore \mu ={\displaystyle \frac{3-2cos{\theta }_0}{4}}$
Therefore, we get the required answer, which gives the coefficient of friction between the mass 4m and horizontal surface.

Additional information:
As we know that the friction force can be defined as the force exerted by a surface when an object moves across it or even makes an effort to move across the surface. There are three types of friction force - sliding friction, rolling friction and static friction. It can be said that the friction force often opposes the motion of an object.
Static friction comes into action on objects when they are resting on a surface. For example, if we are hiking in the woods, there is a presence of static friction between our shoes and the trail each time we put down our foot. Without this friction, our feet would slip out and thereby, making it difficult to walk.
Sliding friction is friction that comes in action on objects when they are sliding over another surface. As we know that the sliding friction is weaker than the static friction. That is the reason that it is easier to slide a piece of furniture over the floor after we start it moving than it is to get it moving in the first place.
The last one is rolling friction is friction that comes in action on objects when they are rolling over another surface. Rolling friction is much weaker than sliding friction and static friction. This is the reason that most forms of ground transportation use wheels, including bicycles, cars, 4-wheelers, roller skates, scooters, and skateboards, as these wheels make it easier to travel with facing less frictional force.

Note:
Frictional force is a force that opposes the motion of any object. There are many practical examples of frictional force. We should also remember that various kinds of lubricants are used to overcome the frictional force like- grease, oil etc.