## What is Friction

Before we look into how to find friction force, I will first explain what a frictional force is…

**Definition:** Frictional force is a force that opposes the motion between two surfaces in contact with each other. The coefficient of static friction, μ is defined as the ratio of the limiting frictional force (F) to the normal reaction (R). The frictional force is measured in Newtons. The formula for frictional force is F=μR. Where F is the force of friction, μ is the coefficient of friction, and R is the normal reaction.

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## Formula for Frictional Force

Coefficient of static friction (μ) = Frictional force (F) / Normal reaction (R)

**We can rewrite the above formula coefficient of friction as μ = F / R**

And the formula for fractional force F=μR

The normal reaction is equal to the weight, W of the block. That is to say R = W = mg

And from the above equation, m is the mass of the object in kilogram, and g is the acceleration due to gravity. The si unit of frictional force is in Newtons. Remember that a normal reaction is equal to the weight of the object and they both have the same unit which is also in Newtons.

Additionally, the coefficient of friction has no unit.

We need to also know that for workdone by friction is W = F x h = μR x h = μRh

Where h is the height.

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## How to Find Friction Force

An object may be pulled horizontally or along an inclined plane. Depending on the situation and the question you are solving, we have different formulas to apply in order to find friction force.

For example, when an object slides down along an inclined plane, we can easily find the coefficient friction as:

From the first diagram in the video above, as the object slides vertically downward, the following conditions will arise:

- Along the x-axis which is perpendicular to the plane: cosθ = x/W, we can now make x subject of the formula to obtain: x = W x cosθ
- On the y-axis which is parallel to the plane: sinθ = y/W. We make y subject of the formula to obtain: y = W x sinθ

For us to be able to find the coefficient of friction (μ), we will need to slowly increase the angle of inclination (θ) so that the object will slide down the plane.

As the object begins to slide down the plane, the weight pulling it down (wsinθ) will be equal to the frictional force (F=μR) acting upward along the plane.

Therefore, we can now say that: F = wsinθ

But since F = μR, the above equation will now become:

μR = wsinθ

Additionally, we can see from the figure that the normal component R is equal to the perpendicular component of the weight (wcosθ). Therefore, R = wcosθ

From our formula for the force of friction which is **μ = F / R**. And by looking at F = wsinθ, and R = wcosθ. We can conclude that μ = F / R = wsinθ / wcosθ

Therefore, we can also write the formula for coefficient of friction,** μ = wsinθ / wcosθ **

Additionally, the other formula for for coefficient of friction, μ = tanθ = y/x = wsinθ / wcosθ

**We also refer to the angle of inclination (θ) as angle of friction.**

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## Solved Problems: How to Calculate Frictional Force

Here are a few problems to help you understand how to calculate frictional force:

### Problem 1

A wooden block of mass 1.6 kilograms rests on a rough horizontal surface. If the limiting frictional force between the block and the surface is 8 Newtons. Calculate the coefficient of friction (g = 10 m/s^{2}).

#### Answer

**Data:** The information from the question

Frictional force, F = 8 N

Mass of the wood, m = 1.6 kg

Acceleration due to gravity, g = 10 m/s^{2}

**Unknown:** What we need to find

Coefficient of friction, μ** **= ?

**Formula:** The equations that will help us to solve the problem

First step: Find the normal reaction by using the formula (R = W = mg)

Second step: Find the coefficient of static friction, **μ = F / R**

##### Solution

Normal reaction, R = W = mg = 1.6 x 10 = 16 N

To find the coefficient of friction, we use the formula

**μ = F / R = 8 / 16 = 0.5 **

### Problem 2

A concrete block of mass 25 kg is placed on a wooden plank inclined at an angle of 32^{0} to the horizontal. Calculate the force parallel to the inclined plane that will keep the block at rest if the coefficient of friction between the block and the plank is 0.45.

#### Answer

**Data:** The information from the question

Mass of the body, m = 25 kg

Acceleration due to gravity, g = 10 m/s^{2}

Normal reaction, R = W = mg = 25 x 10 = 250 N

Angle of inclination, θ = 32^{0}

μ = 0.45

##### Solution

To calculate the force parallel to the inclined plane, we will need to find Wsinθ and F=μR=μWcosθ. We will then subtract F=μR=μWcosθ from Wsinθ.

We can write the above formula mathematically as follows:

**F _{p} = Wsinθ – μWcosθ = (250 x sin32^{0}) – (0.45 x 250 x cos32^{0}) = 132.48 – 95.41 = 37.07 N **

### Problem 3

Calculate the magnitude of the force required to just move a 20 kg object along a horizontal surface if the coefficient of friction is 0.2.

#### Answer

**Data:** Information from the question

Coefficient of friction, μ = 0.2

Mass of the object, m = 20 kg

Acceleration due to gravity, g = 10 m/s^{2}

**Unknown:** What we need to find

Normal reaction, R = ?

Frictional force, F = ?

**Formula:**

Normal Reaction, R = mg

Friction Force, F = μR

##### Solution

Normal Reaction, R = Weight, W = mg = 20 x 10 = 200N

The friction force, F = μR = 0.2 x 200 = 40N

Therefore, the magnitude of the force of the object along the horizontal surface is 40 Newtons.

### Problem 4

If an object just begins to slide on a surface inclined at 30^{0} to the horizontal. What is the coefficient of static friction?

#### Answer

**Data:** Information from the above question

The angle of inclination, θ = 30^{0}

**Unknown:** What we need to find

Coefficient of static friction, μ = ?

**Formula: **The equation that will help us solve the problem

μ = tanθ

##### Solution

μ = tanθ = tan30^{0} = 1/(√3)

Therefore, the coefficient of static friction is 1/√3

**Note:** The coefficient of friction for an inclined plane is the tangent of the angle of the plane when an object on the plane just begins to slide

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