To find the coefficient of friction, the normal reaction and the friction involved in the necessary quantities. But how to find coefficient friction given velocity and distance?
The velocity and the distance contribute to the friction evolving between the surfaces. The impact of these two quantities can be resolved. By knowing the velocity and the distance moved by the object, the coefficient of friction can be calculated.
How to find coefficient of friction given velocity and distance moved by the object
To find the coefficient of friction given velocity and distance, let us consider an object of mass ‘m’ moving with a velocity ‘v’ at a distance ‘d’ from the initial position. The friction force Ff is retards the motion of the object in the direction opposite to the movement. The normal reaction Fn is acting perpendicular to the motion of the object. The motion of the object is influenced by the gravity ’g’, which results in a normal reaction.
We can find the coefficient friction given velocity and distance by two methods. Let us it discuss one by one.
Method 1: considering the work done by the friction
The work done by the friction on the object is given by
W = PE+ KE+ Eloss
Since the object is moving, the stored potential energy is zero, and the energy loss is negligible during the process. So the work done can be rewritten as
Where m is the mass of the object and v is the velocity at which the object is moving.
This work done is equal to the friction force times the distance, so we can write the equation as
The friction force acting on the object is given by
Ff = µFn
Where µ is the coefficient of friction and Fn is the normal reaction.
The normal reaction is equal to the net weight of the object given by Fn=mg
So that the friction is given by the equation,
Ff = µmg …..(2)
Since the above two equations of friction are the same, we can equate them
Rearranging the equation we get,
Now consider that initially the object is moving with velocity v0, with the time its velocity changes and finally it is moving with velocity vf covering the distance d, then the coefficient of friction is given by
Method 2: By using the Kinematics
The kinematic equation of motion for given velocity and distance is,
vf2 = v02 + 2ad
Where, vf2 is the final velocity of the moving object.
v02 is the initial velocity of the moving object.
a is the acceleration, and d is the distance travelled by the object.
Here we consider the object is moving with a constant velocity so that the final velocity vf2 become zero. Hence we can modify the equation as
0 = v02 + 2ad
-v02 = 2ad
From Newton’s second law of motion, the relation between the acceleration and the force is given by,
F = m*a
Substituting in the above equation, we get
The force acting on the object cannot have a negative value. We can take the magnitude of the equation as,
Since in this case, we have considered the force acting on the object is friction force so that substituting the formula of the friction, we get the equation as
Generally, we can write the equation as
On rearranging the terms, we get the coefficient of friction as,
It is clear that the coefficient of friction arrived from both methods are the same. Using the above formula, we can find the coefficient of friction given velocity and distance.
Solved Problems On Coefficient Of Friction
An object of mass of 2kg is moving with a velocity of 12ms-2, the friction force acting on the body make the object stop at a distance of 22m. Find the coefficient of friction and hence calculate the friction force. The acceleration due to gravity g given as 10ms-2.
Solution:
Given: Mass of the object m = 2kg
Velocity v = 13ms-2
Distance covered by the object d = 22m
Acceleration due to gravity g = 10ms-2
The formula to find coefficient of friction given velocity and distance is
Substituting the values of the given terms in the above equation
µ = 0.38
The formula to calculate friction is
Ff = µmg
Ff = 0.38×10×2
Ff = 7.68N.
Find the coefficient of friction given velocity and distance as 28ms-2 and 34m respectively and hence find the normal reaction and the friction force. (Given: Mass of the object is 4kg and Acceleration due to gravity is 10 ms-2).
Solution:
The velocity is 17ms-2
The distance travelled by the object is34m.
The coefficient of friction for given velocity and distance is given by the formula
Substituting the values in the expression,
µ = 0.425
The normal reaction is given by FN = m*g
FN = 4 × 10
FN = 40 N.
The friction force acting on the object is Ff = µ FN
Ff = 0.425 × 40
Ff = 17 N.
A body of mass of 12kg is moving on a rough surface. It travelled a distance of 72m, and then its motion is hindered by the friction force of 45N. Calculate the coefficient of friction and hence find the velocity at which the body is moving.
Solution:
Mass of the body m = 12kg
The distance traveled by the body d = 72m
The friction force acting on the body Ff = 45N
Acceleration due to gravity g = 9.8 ms-2.
The coefficient of friction for given friction is given by the formula
Substituting values in the above equation
µ = 0.382
To find the velocity, let us consider the equation
On rearranging the terms, we get the equation for the velocity as,
v2 = 2µgd
Substituting the values
v2 = 2× 0.382× 9.8× 72
v2 = 539.07
Taking the square root, we get
The velocity at which the body is moving is v = 23.21 ms-2.
The coefficient of friction is 0.46, and the mass of the object is 7kg. The object is moving with a constant velocity of 46 ms-2. Calculate the distance travelled by the object after friction retards the object’s motion.
Solution:
The coefficient of friction µ = 0.46
Mass of the object m = 7kg
Velocity of the object v = 16 ms-2
Acceleration due to gravity g = 9.8 ms-2
Rearranging the expression
d = 28.39m.
The object covers a distance of 28.39m before it stops its motion.
A block of mass 5kg is moving with the initial velocity of 12ms-2. After a time t, its velocity is increased by 19ms-2 and covers a distance of 33m, then its motion is stopped by the friction. Find the coefficient of friction given velocity and distance and hence find the friction required to stop the motion of the object.
Solution:
The initial velocity of the object v0 = 12 ms-2
The final velocity of the object vf = 19ms-2
The distance covered by the object d = 33m
Mass of the object m = 5kg
The coefficient of friction for given initial and final velocity is given by the expression
µ = 0.33
The friction required to stop the motion of the object is
Ff = µmg
Ff = 0.33× 5× 9.8
Ff = 16.17 N.
Also Read:
- How to find velocity at impact
- How to measure velocity in astrophysics
- How to find velocity in kinetic energy
- How to find launch velocity
- How to find distance in velocity time graph
- Initial velocity formula
- How to find velocity between two points
- How to calculate velocity in loop quantum gravity
- How to find final velocity with acceleration and distance
- How can you find acceleration from a velocity time graph
I am Keerthi K Murthy, I have completed post graduation in Physics, with the specialization in the field of solid state physics. I have always consider physics as a fundamental subject which is connected to our daily life. Being a science student I enjoy exploring new things in physics. As a writer my goal is to reach the readers with the simplified manner through my articles.