# How To Find Coefficient Of Kinetic Friction (Updated 2023)

A dimensionless physical quantity the specifies the interaction of two object is called coefficients.

The value coefficient of kinetic friction is changes depending on the nature of the material used. Generally, the coefficients give the ratio of two quantities involved in the action. In this post, let us discuss how to find coefficient kinetic friction and its consequences.

## How to find coefficient of kinetic friction

Let us consider two surfaces, such that one surface is moving in contact with another one. The friction always resists the movement and finally stops the motion of the surface in the opposite direction of the motion.

A general formula to find the coefficient friction is given by the ratio of friction force and the normal reaction acting on the surfaces in a perpendicular direction.

On rearranging the above expression, we can find out the kinetic friction as well.

## How to calculate uncertainty of coefficient of kinetic friction

The uncertainty occurs due to the misalignment of the coordinate axes along the direction of the motion. Along with the normal force, the tangential force is acting on the system. This tangential force gives an account for the occurrence of uncertainty of the coefficient of the kinetic friction.

The value of the coefficient is not measured directly through the experiment. It is determined by calculating all the forces acting on the system and the angle of inclination of the object with the surface.

The general expression for the coefficient of kinetic friction is given by

Let us consider the sliding of an object in a plane. The sliding of the object is taken for the various angle of the object along the plane for different instances. Then calculate the coefficient of kinetic friction for all the angles.

The above statement tells that the value of coefficient of kinetic friction changes with change in the angle. This deviation is due to the uncertainty of kinetic friction coefficient. Let us study how to find the coefficient of kinetic friction with uncertainty.

Along with the normal force FN, the tangential force also contributes to the evolution of friction force. This leads to an error in calculating the coefficient of kinetic friction. The uncertainty measurement compensates for the error that occurred during the calculation.

The normal force is acting along the Y-axis, and the angle of misalignment be β. And the tangential force is acting along the X-axis with the misaligned angle of α. These normal and tangential forces are in contact, and the resultant force along the X and Y axes are given by

FX = FF cosα + FN sinα

FX = µK FN cosα + FN sinα

FX = FNK cosα + sinα)

Similarly for Y axis

FY = FN cosβ – FF sin β

FY = FN (cosβ – µK sinβ) By solving the resultant forces, the uncertainty in the friction is given as

In order to calculate the combined standard uncertainty measurement, the standard uncertainty function must be a standard value of input values and the partial derivatives of the frictional coefficient. The law of “propagation of uncertainty” helps us to give a standard value for the uncertainty in the friction. It is given by the equation.

Where, u is the uncertainty of the given system.

On differentiating the individual variables, we get the standard value of uncertainty in the coefficient of kinetic friction.

This gives the standard uncertainty value for the input forces acting on the system. By substituting these values in the partial derivative equation, we get the uncertainty value.

## How to calculate coefficient of kinetic friction without mass

To calculate the coefficient of kinetic friction without mass, let us consider a block moving on a flat surface. The block of mass “m” moving with acceleration “a” in the direction of the applied force. The normal force acting between the block and the surface be FN which is perpendicular to the motion of the block. We know that the friction force acting between the block and surface to retard the motion is given by the equation,

FK = µK FN

According to Newton’s second law of motion, the force acting on the moving body equals mass times the acceleration.

F = m* a

The normal force is influenced by the force of gravity given as

FN = m*g

Substituting in the equation of frictional force, we get

FK = µK m*g

Since the body is moving and the force acting on the block is kinetic friction force, Newton’s law can be modified as

FK = m* a

On equating the above two equations we get,

µK m*g = m* a

µK g = a Rearranging the equation we get,

This gives the value of the coefficient of kinetic friction.

## Determining the Coefficient of Kinetic Friction on an Inclined Plane

Kinetic Friction on an Inclined Plane

Forces Acting on the Object:

• Gravitational force:
• Normal force:
• Frictional force:

Decomposing the Gravitational Force: The gravitational force can be split into two components:

• Parallel to the incline:
• Perpendicular to the incline:

Frictional Force: When an object moves at a constant velocity on the incline:

Since , and , we get:

Example:

Suppose you have a block on a 30° incline, and you notice it starts to slide at a constant velocity without any external push. Determine the coefficient of kinetic friction.

Given:

To find:

Using the formula:

Plugging in the given value:

Thus, the coefficient of kinetic friction between the block and the incline is approximately 0.577.

## How to Find the Coefficient of Kinetic Friction with Acceleration

Kinetic Friction with Acceleration

When an object slides over a surface, it experiences a resistive force due to the surface. This resistive force is called kinetic friction. The magnitude of the kinetic frictional force () is given by:

Where:

• is the coefficient of kinetic friction.
• is the normal force (or the force acting perpendicular to the surface). In many cases, this is equal to the weight of the object if the surface is horizontal.

If an object is moving on a horizontal surface and no other horizontal forces are acting on it, then the net force () acting on the object due to friction is:

Using Newton’s second law (), where is the mass of the object and is its acceleration, we can equate the above equations to find:

To solve for , you can rearrange this equation:

Example:

Suppose we have a block of mass 10 kg sliding on a horizontal surface. The block has an acceleration of 2 m/s² in the direction of motion. Given that the gravitational acceleration () is approximately 9.81 m/s², we want to find .

First, calculate the normal force ():

Then, use the formula for :

So, the coefficient of kinetic friction between the block and the surface is approximately 0.204.

## How to Find the Coefficient of Kinetic Friction Without Friction Force

kinetic friction without friction force

In real-world scenarios, you may not always have a direct measure of the frictional force between two surfaces, but there might still be a need to determine the coefficient of kinetic friction (). One way to derive is by analyzing the motion of an object on an incline.

When an object is sliding down an incline without acceleration (i.e., at constant velocity), the net force acting on it is zero. This means that the component of gravity pulling it down the incline is balanced by the frictional force resisting its motion.

Let’s dive into the mathematics of this:

1. Gravitational Force Parallel to the Incline

The component of gravitational force acting parallel to the incline can be found using:

Where:

• is the mass of the object.
• is the acceleration due to gravity (approximately near the Earth’s surface).
• is the angle of inclination.
1. Frictional Force

The frictional force acting on the object can be represented as:

Where is the normal (perpendicular) force. For an incline, the normal force is given by:

Thus, the frictional force is:

1. Balancing the Forces

At constant velocity:

Substituting in our expressions:

From this, we can solve for :

### Worked-out Example

Let’s say an object is observed to slide down an incline at a constant velocity, and the angle of the incline, , is measured to be 30°.

Using the derived formula:

(rounded to three decimal places)

Thus, the coefficient of kinetic friction, , between the object and the incline is approximately 0.577.

NOTE: this method assumes no other forces (like air resistance) are acting on the object, and that the object moves at a constant velocity down the incline.

## How to Find the Coefficient of Kinetic Friction Using Velocity and Distance

In many experimental or real-world scenarios, you might have information about the initial velocity of an object and the distance it traveled before coming to a stop due to friction. This data can be invaluable in determining the coefficient of kinetic friction () between the object and the surface it’s sliding upon.

kinetic friction with velocity and distance

Let’s understand the principles behind this:

1. Work Done by Frictional Force

The work done by the frictional force over the distance () is equal to the change in kinetic energy of the object.

Where:

• is the normal (perpendicular) force. On a horizontal surface, , where is the mass of the object and is the acceleration due to gravity (approximately ).
1. Change in Kinetic Energy

The object’s initial kinetic energy (when it has velocity ) is:

Since the object comes to a stop, its final kinetic energy is zero. Thus, the change in kinetic energy is:

1. Equating Work and Change in Kinetic Energy

For the object to come to a stop:

Substituting in our expressions:

From this equation, we can solve for :

### Worked-out Example

Imagine an object sliding on a horizontal surface. It has an initial velocity of and comes to a stop after traveling . Let’s determine the coefficient of kinetic friction, , between the object and the surface.

Using the derived formula:

(rounded to three decimal places)

Thus, the coefficient of kinetic friction, , between the object and the surface is approximately 0.127.

NOTE: This method is based on the principle of conservation of energy. It assumes that the only force doing work on the object (leading to a change in its kinetic energy) is the frictional force, with no other forces (like air resistance) at play.

## How to Find the Coefficient of Kinetic Friction Using Mass and Force

When an object is in motion on a horizontal surface and you know the force being applied to it and its mass, you can determine the coefficient of kinetic friction () between the object and the surface. Let’s delve into the process step by step.

1. Frictional Force

The frictional force acting against the motion of an object on a horizontal surface can be given by:

Where:

• is the normal (perpendicular) force. On a horizontal surface, , where is the mass of the object and is the acceleration due to gravity (approximately ).
1. Net Force Acting on the Object

If a force () is being applied to the object to keep it moving at a constant velocity on the horizontal surface, the net force is zero (since there’s no acceleration). This means that the applied force is balanced by the frictional force:

1. Finding

Using the above equations, we can express in terms of :

From this equation, we can solve for :

### Worked-out Example

Let’s consider an object with a mass of being pushed on a horizontal surface. To keep the object moving at a constant velocity, a force of is applied. Determine the coefficient of kinetic friction, , between the object and the surface.

Using the derived formula:

(rounded to three decimal places)

Thus, the coefficient of kinetic friction, , between the object and the surface is approximately 0.204.

NOTE: This approach assumes that the object is moving at a constant velocity, which means there’s no acceleration and the net force acting on it is zero. This is crucial because it lets us equate the applied force with the frictional force.

## Does the calculation of kinetic friction without mass give the same value of coefficient obtained by considering the mass?

Yes, the value of the coefficient of kinetic friction with or without considering the mass is the same.

Since friction is a quantity that is independent of the absolute mass of the system, the mass does not affect the value of the friction involved in the process. Hence the coefficient of kinetic friction remains unchanged with or without considering the mass of the object.

## Does the nature of the material influence the coefficient of kinetic friction?

The coefficient of kinetic friction is a numerical value that gives the evidence for the presence of friction force between the objects.

Since friction is influenced by the nature of the material, it is so evident that its coefficient is also largely influenced by the nature of the material.

## What is necessary to find the coefficient of kinetic friction of a moving object?

Without the coefficient of kinetic friction, it is quite difficult to measure the force that makes the object to hinders its motion.

The friction is always proportional to the normal perpendicular reaction between the surfaces. This proportionality relation is specified by the dimensionless quantity called the coefficient. The coefficient of kinetic friction measures the absolute value of the friction force that stops the moving object.

## Can the value of the coefficient of kinetic friction be greater than 1?

Generally, the value of the kinetic friction coefficient ranges from 0 to 1. Sometimes it gives a value of coefficient exceeds 1.

If the influence of friction force is stronger than the perpendicular reaction between the two moving surfaces, the value of kinetic friction coefficient exhibit the value greater than 1. Maximum frictional force makes the object to restrict its motion so that automatically the coefficient of kinetic friction increases proportionally.

## Does greater coefficient of kinetic friction lead to energy dissipation?

The dissipation of energy due to friction can be described in terms of energy conservation Law.

A greater coefficient of kinetic friction means the friction force is stronger than the applied force. The challenging task is to keep the body in motion in the presence of friction. Hence it takes much force to keep the body in motion. The maximum force exerted to keep the body in motion causes the kinetic energy dissipation released in the form of heat.

## What is the coefficient of friction?

A: The coefficient of friction is a dimensionless quantity that represents the ratio of the force of friction between two objects to the force pressing them together.

## How can I calculate the coefficient of friction?

A: The coefficient of friction can be calculated by dividing the force of friction by the normal force acting on the object.

## What is the difference between kinetic and static friction?

A: Kinetic friction occurs when two objects are in relative motion, while static friction occurs when there is no relative motion between the two objects, i.e., the objects are at rest.

## What is the formula for the coefficient of kinetic friction?

A: The formula for the coefficient of kinetic friction is μk = Fk/N, where μk is the coefficient of kinetic friction, Fk is the force of kinetic friction, and N is the normal force.

## How can I find the coefficient of kinetic friction for a moving object on a flat surface?

A: To find the coefficient of kinetic friction for a moving object on a flat surface, you can use the equation μk = tan(θ), where θ is the angle between the force of kinetic friction and the force perpendicular to the surface.

## What is the equation for calculating the force of kinetic friction?

A: The equation for calculating the force of kinetic friction is Fk = μkN, where Fk is the force of kinetic friction, μk is the coefficient of kinetic friction, and N is the normal force.

## How can I find the coefficient of static friction?

A: The coefficient of static friction can be found by dividing the maximum force of static friction by the normal force.

## What is the relationship between the coefficients of static and kinetic friction?

A: The coefficient of static friction is generally greater than the coefficient of kinetic friction for a given pair of surfaces.

## How can I solve a friction problem using the coefficient of friction?

A: To solve a friction problem using the coefficient of friction, you can set up equations based on the friction equation and other relevant equations, and solve for the unknown variables using algebraic methods.