Kinetic friction is the friction evolved between the moving solid surfaces in contact.

**How to calculate kinetic friction? One of the major questions arises when we hear the term kinetic friction. To answer this question, we are going to study various aspects of kinetic friction through this post. **

**How to calculate kinetic friction**

Let us consider an object supposed to move on a flat surface. This movement of the object is retarded by force, and the surface deforms. The deformed surface exerts an influence on the object in the perpendicular direction. This force is called normal force F_{N}. Since the object has a certain weight of mg, then the normal force is given by

F_{N}= mg

Where; m is the mass of the object.

g is the acceleration due to gravity.

The object is pushed by a force F. This applied force is restricted in the opposite direction; this force is called frictional force FK. The frictional and the applied forces are equal. That is, when the applied force increases, the frictional force also increases.

F = F_{K}

Since there is frictional force acting on the moving object, there will be a coefficient of kinetic friction, the ratio of friction, and the normal reaction. It is given by the formula;

F_{K} = µ_{K}mg

F_{K} = µ_{K} F_{N}_{}

The above equation is used to calculate the kinetic friction between the two solid surfaces.

**How to calculate kinetic friction on an incline**

When the object is moving in an inclined plane, gravity acting on the object is straight down. To calculate the kinetic friction, we need to find the slope and the angle of inclination.

Suppose the object moves in a plane and is lifted, making an angle ‘θ,’ then we can find the kinetic friction by solving all the forces acting on the object.

The forces acting on the object along the Y-axis is given by

∑F_{y} = F_{N }– mgcosθ = 0

In this case the gravitational force is opposite to the normal reaction.

F_{N }= mg cosθ

All the forces acting along the X-axis is given by

∑F_{x} = mg sinθ – F_{K }= 0

Here there is no other component of forces acting along the X direction, and the gravitational force is parallel to the applied force and the frictional force.

F_{K} = mg sinθ

The coefficient of kinetic friction is given by

Substituting the value of the coefficient of friction and the normal force, we get the equation as,

F_{K} = mg tanθ cosθ

The angle tanθ gives the slope of the inclined plane.

**How to calculate kinetic friction force without coefficient**

Is it possible to calculate the kinetic friction without knowing its coefficient?

This is a generally asked question. Yes, we can calculate the kinetic friction acting on the body without the Coefficient of friction.

To calculate the kinetic friction, let us consider a block is pushing over a surface. It is moving with the acceleration ‘a’ and the normal force acting between the surface and the body is F_{N}.

Since we need to calculate the friction without considering its Coefficient, we need to solve all the forces acting between the body and the surface.

The normal force is given by F_{N}= mg

The body is pushed in the forward direction by exerting some applied force on the body F.

Since the body has a certain mass ‘m’ and is accelerating in the forward direction with ‘a’, the net force acting on the body is given by considering Newton’s second law of motion.

F_{net} = ma

The frictional force is always acting opposite to the applied force so that the equation for the kinetic friction is given by taking the difference between the applied force and the net force.

F_{K} = F – F_{net}

**Kinetic friction formula with an angle**

When a block is sliding on the surface by making an angle ‘θ’ in a horizontal direction with mass ‘m’ is given by the formula.

F_{K} = µ_{K} mg cosθ

Where µK is the kinetic friction coefficient.

g is the acceleration due to gravity.

**How to calculate work done by kinetic friction**?

Consider a block is sliding over a surface with acceleration ‘a’ and its displacement is ‘x’. The kinetic frictional force acting between the block and the surface is given by

F_{K} = µ_{K} F_{N}_{}

The net work done on any object is given by

W = F. ds

Where F is the force and ds is the displacement.

The above equation is in the dot product; it can be resolved as

W = F_{K} x cosθ

**Case (i): The friction is evolved between the block and the surface in the opposite direction of the displacement. The angle between the friction and the displacement is ‘θ.’**

Since the angle θ = 180°, then the work done by the kinetic friction is

W = F_{K} x coss(180°)

W = -F_{K} x ( because cos(180°) = -1)

**Case (ii): If we place another block on the first block, we apply some to make the blocks move. The relative motion of the blocks is opposite to the displacement so that the friction is in the direction of the displacement. Hence the work done by the kinetic friction is given by**

W = F_{K} x cosθ

Since both displacement and friction are in the same direction, the angle between them is 0

W = F_{K} x cos0 [but, cos (0°) = +1]

W = F_{K} x

**Case (iii): Suppose the second block, which is placed on the top of the first block, is stationary, then the displacement of the second block is 0, and the relative motion is opposite to the displacement. Then the work done by the kinetic friction is given as**

W = F_{K} x cosθ

W = F_{K} (0) cosθ

W = 0

By analyzing the above three cases, we can say that the work done by the kinetic friction either be negative, positive, or zero.

**How to calculate the magnitude of the kinetic frictional force**

The kinetic friction is given by

**F _{K} = µ_{K} F_{N}**

The frictional force always balances the gravitational pull. The value we get from the above equation gives the maximum value. Above that value the friction cannot be increased. If the object exerts more friction above the maximum value, then the object has to move in the upward direction, which is contradicts to the laws of motion.

The magnitude of the frictional force gives the solution to overcome from the above problem, because the magnitude of frictional force and the normal force are proportional to each other.

|F_{K}| = µ_{K} |F_{N}|

Here the frictional force is always equal to the maximum value of the applied force.

F_{K} = F_{max}

**How to find Coefficient of kinetic friction using energy**

Let us consider a block is moving along a surface of another object of mass ‘m’. Initially, the block is at stationary state, when some force is applied, it begins to move. But at the stationary state, the stored potential energy is acting on the block. The overall energy acting on the block is given by using the work-energy theorem.

W + PE_{0} + KE_{0} = PE_{F} + KE_{F} + Energy loss

Initially, the block is at rest. When we apply some force, the block begins to move, and the initial kinetic energy is given by

Where m is the mass of the block and v0 is the initial velocity.

Since the body is overcome from the stationary state, the stored potential energy is zero.

When the block displaces at a certain distance, kinetic friction is evolved. Thus the final potential energy on the moving object is zero. But the final kinetic energy of the moving body is lost due to the friction. Therefore the kinetic energy is also zero. The loss in kinetic energy is released in the form of heat. It is given by

Energy loss = F_{K}.x

Where x is the distance at which the block is moving.

F_{K} is the kinetic frictional force

Thus the work-energy equation can be written as

KE_{0} = Energy loss

The kinetic friction acting between any two rough surfaces is given by

F_{K} = µ_{K} F_{N}_{}

Since the net weight acting on the body is m*g, the normal force is given by

F_{N} = m*g

F_{K} = µ_{K} m*g

Substituting the value of F_{K }in the equation in the above, we get

Rearranging the terms, we get the coefficient of kinetic frictional force as,

**Solved Problems on the Kinetic friction**

**How to calculate kinetic friction between the road and the car tire if the normal force acting on the car and the road is 34N, and the kinetic frictional Coefficient is 0.36.**

**Solution:**

Given: Kinetic frictional coefficient µ_{K} = 0.36

Normal force F_{N} = 34N

The formula to calculate the kinetic friction is

F_{F} = µ_{K} *F_{N}

Substituting the given value

F_{F} = 0.36 * 34

F_{F} = 12.24N.

**An object of mass 4kg is sliding over an inclined plane. The angle of inclination of the plane is 35°. Calculate the frictional force acting and hence find out the kinetic frictional coefficient and normal force acting on them.**

**Solution:**

Given: the angle of inclination θ = 35°

Mass of the block m = 4kg

The acceleration due to gravity g = 9.8 m/s^{2}.

The kinetic friction at an inclined plane is given by the formula

F_{K} = mg tanθ cosθ

F_{K} = 4* 9.8 * tan(35°) cos(35°)

F_{K} = 4* 9.8 (0.700) (0.819)

F_{K} = 22.473 N.

**To calculate the coefficient of kinetic friction**

We know that tanθ = µ_{K}

tan(35°) = 0.700

µ_{K} = 0.700

The normal force acting between the two surfaces is given by

F_{N} = m*g

F_{N} = 4 * 9.8

F_{N} = 39.2 N.

**How to calculate kinetic friction coefficient? If the angle between the two objects is given as 27° and hence find out the kinetic friction.**

**(Given: mass of the object is 5kg and acceleration due to gravity is 10 m/s**^{2}.)

^{2}.)

**Solution:**

The kinetic friction F_{F} = µ_{K} *F_{N}

But here, only the angle is given. So we need to calculate the coefficient of kinetic friction µ_{K}

µ_{K} = tanθ

µ_{K} = tan(27°)

µ_{K} = 0.51

The normal force acting on the objects is given by

F_{N} = m*g

F_{N} = 5 * 10 = 50N.

Substituting the value of normal force and the coefficient of friction in the kinetic friction formula,

F_{F} = 0.51 *50

F_{F} = 25.5 N.

**A wooden block is moved at a distance of 3m, and friction acting between the block and the surface is 12N. Calculate the work done by the kinetic friction force, if **

**(i) the displacement and the friction are acting opposite to each other. **

**(ii) the displacement and the friction are acting in the same direction.**

**Solution:**

(i)If the displacement and the friction are acting opposite to each other.

The work done by the kinetic friction if the friction and displacement acts in the opposite direction

W = -F_{K} x

W = – (12*3)

W = -36J

(ii) If the displacement and the friction acts in the same direction.

The work done by the kinetic friction is given by

W = F_{K} x

W = 12 * 3

W = 36J

**An object of mass 2kg has displaced a distance of 7m by applying some force. The object is moving at a velocity of 4m/s2. Then how to calculate the kinetic friction coefficient? Hence find out the kinetic friction also. (Take acceleration due to gravity g as 10 m/s2)**

** Solution:**

Given: Mass of the object m = 2kg

Displacement of the object x = 7m

Velocity of the object v = 4m/s^{2}.

The coefficient of kinetic friction is given by

µ_{K} = 0.11

The kinetic frictional force is given by

F_{F} = µ_{K}*F_{N}

The normal force acting between the object and the surface is

F_{N} = m*g

F_{N} = 2*10

F_{N} = 20N.

Substituting the value of coefficient kinetic friction and the normal force we get,

F_{F} = 0.11*20

F_{F} = 2.2N.