Coefficient of Friction

The coefficient of friction, µ, is a measure of the amount of friction existing between two surfaces. A low value of the coefficient of friction indicates that the force required for sliding to occur is less than the force required when the coefficient of friction is high. The value of the coefficient of friction is given by

[Tex]μ = \frac{F}{N} [/Tex]

Where,

F is the frictional force and,

N is the normal force.

Derivation for coefficient of friction

Perpendicular force reduced

When an object is placed on an incline, the force perpendicular between the surfaces is reduced, according to the angle of the incline. The force required to overcome friction (F_r) equals the coefficient of friction (μ) times the cosine of the incline angle (cos θ) times the weight of the object (W). There are mathematical tables that give the values of cosines for various angles.

F_r = μ W Cosθ 

Gravity contributes to sliding

Note that when an object is on an incline, the force of gravity contributes to causing the object to slide down the ramp or incline. Let’s call that force (F_g), and it is equal to the weight of the object (W) times the sine of the angle (sin θ)

F_g = W sin θ

Tangent of angle determines coefficient

If you put the ramp at a steep enough angle, F_g will become greater than F_r and the object will slide down the incline. The angle at which it just starts to slide is determined from the equation:

μ W cos θ = W sin θ

Dividing both sides of the equation by W and cos θ, we get the equation for the static coefficient of friction  

μ = tan θ

where tan θ is the tangent of angle θ and equals [Tex]\frac{Sin\theta}{Cos\theta}  [/Tex].

For example: If you put a book on a ramp and changed the angle of the ramp until the book started to slide and then measured the angle of the ramp, you could determine the coefficient of friction between the book and the ramp. If the angle was 30 degrees, then the tangent of 30 degrees is about 0.58. That would be the static coefficient of friction in this case. Even if you increased the weight on the book, it would still slide at 30 degrees.

Sample Problems on Coefficient of Friction

Question 1: An object having a mass of 10 kg is placed on a smooth surface. Static friction between these two surfaces is given as the 15 N. Find the coefficient of static friction?

Solution:

Given

m = 10 kg

F = 30 N

We know that,

Normal force, N = mg

So, N = 10× 9.81 = 98.1 N

The formula for coefficient of static friction is,

μ = F/N

μ = 30/98.1

μ = 0.305

Question 2: The normal force and the frictional force of an object are 50 N and 80 N respectively. Find the coefficient of friction?

Solution:

Given

N = 50 N

F = 80 N

The formula for coefficient of friction is

μ = F/N

μ = 80/50

μ = 1.6 N

Question 3: A particle of mass 5 kg is at limiting equilibrium on a rough plane which is inclined at an angle of 30 degrees to the horizontal. Find the coefficient of friction between the particle and the plane.

Solution:

Resolving up the plane:

[Tex]F – 5g sin30^{\circ}=0[/Tex]

Resolving perpendicular to the plane:

[Tex]R = 5g cos30^{\circ}[/Tex]

In limiting equilibrium, so  [Tex]F =\mu R[/Tex]

[Tex]5g \times Sin30^{\circ} = \mu \times 5g \times cos30^{\circ}[/Tex]

[Tex]\mu = \frac{sin30^{\circ}}{cos30^{\circ}}=tan30^{\circ}=\frac{1}{\sqrt{3}}= 0.577[/Tex]

Question 4: A 5 kg box on a horizontal table is pushed by a horizontal force of 15 N as shown on the right. If the coefficient of friction is 0.4, will the box move?

Solution: 

The sketch shows the forces acting on the box. Note that the weight of a box of mass 5 kg is 5g where  

Since the vertical forces are in equilibrium,  

Hence, the maximum possible friction is  

The pushing force, 15 N, is less than this and so cannot overcome the friction.

The box will not move.

Friction is a force between two surfaces that are sliding, or trying to slide, across each other. For example, when you try to push a book along the floor, friction makes this difficult. Friction always acts in the opposite direction to the movement of the object i.e. it opposes the motion of an object. Friction always slows a moving object down.

The amount of friction depends on the materials from which the two surfaces are made. The rougher the surface, the more friction is produced. Friction also produces heat. If you rub your hands together quickly, you will feel them get warmer.

Advantages of Friction

• Friction is responsible for many types of motion
• It helps us walk on the ground
• Brakes in a car make use of friction to stop the car
• Asteroids are burnt in the atmosphere before reaching Earth due to friction.
• It helps in the generation of heat when we rub our hands.

Disadvantages of Friction

• Friction produces unnecessary heat leading to the wastage of energy.
• The force of friction acts in the opposite direction of motion, so friction slows down the motion of moving objects.
• Forest fires are caused due to the friction between tree branches.
• A lot of money goes into preventing friction and the usual wear and tear caused by it by using techniques like greasing and oiling.