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# Which of the following is the dimension of the coefficient of friction?$\text{A}\text{. }\left[ {{M}^{2}}{{L}^{2}}T \right]$$\text{B}\text{. }\left[ {{M}^{0}}{{L}^{0}}{{T}^{0}} \right]$$\text{C}\text{. }\left[ M{{L}^{2}}{{T}^{-2}} \right]$$\text{C}\text{. }\left[ {{M}^{2}}{{L}^{2}}{{T}^{-2}} \right]$  Verified
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Hint: Friction is force that acts between two surfaces when there is relative motion between the two surfaces. Use the formula for maximum static friction i.e. ${{f}_{\max }}=\mu N$ to find the dimensional formula of coefficient of friction.
Formula used:
${{f}_{\max }}=\mu N$

Let us first discuss about friction.

Friction is force is a force that opposes the relative motion between two surfaces that are in contact.
Suppose a body is resting on a horizontal floor. Since the body is at rest, there is no relative motion between the surfaces of the floor and the body. Hence, there is no frictional force coming into play.
Suppose now we try to bring a relative motion between the surfaces that are in contact by applying an unbalanced on the body force in horizontal direction. Since there will be a relative motion between the floor and the block, the frictional force comes into play. It will act opposite to the direction of the applied force.

Suppose we apply the force from a minimum value (i.e zero) and slowly increase it. You will observe that until some value of force (say F) the body will not move.

This is because until the force is equal to F, the frictional force acting on the body is equal to the applied force. This type of friction is called static friction. However, static friction has a limit and once the applied force is greater then the limit value of the static friction, the body will accelerate.

The maximum value of the static friction is given as ${{f}_{\max }}=\mu N$ …. (1).

Here, $\mu$ is called the coefficient of static friction and N is the normal reaction exerted by the floor on the body.
The value of $\mu$ depends on the nature of interaction between the two surfaces. To find the dimensional formula of the coefficient of friction, let us use equation (1).

From equation (1) we get,
$\mu =\dfrac{{{f}_{\max }}}{N}$ …. (2).
When observing equation (2), we get that the coefficient is a ratio of two forces, that are frictional force and the normal force.
The dimensional formula of force is $\left[ ML{{T}^{-2}} \right]$.
Therefore, the dimensional formula of $\mu$ is $\dfrac{\left[ ML{{T}^{-2}} \right]}{\left[ ML{{T}^{-2}} \right]}=\left[ {{M}^{0}}{{L}^{0}}{{T}^{0}} \right]$.
This means that the coefficient of friction is just a number and has no dimension.

So, the correct answer is “Option B”.

Note:
Note that static friction is a variable force and the value of frictional force$f=\mu N$ is the maximum value of the frictional force that acts between the two surfaces.
Also note that friction acts on the both surfaces with equal magnitudes and not only one surface. You may think that friction a person sitting in between the two surfaces. This person does not like the surfaces to move apart and hence pulls them towards him when they try to move apart.