Frictional Force Equation Doesn't Make Sense

In summary: The normal force is always perpendicular to the surface, and the frictional force is always parallel to the surface. In summary, the normal force and frictional force are related through the coefficient of friction and the angle of the inclined plane, but their directions are perpendicular to each other.
  • #1
FredericChopin
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Frictional Force is mathematically defined as:

Ff = μ*m*g*cos(θ)

, where μ is the coefficient of friction, m is the mass of the object, g is the acceleration due to gravity and θ is the angle of the inclined plane.

But in terms of direction, this makes no sense!

Suppose there is an object on an inclined plane. It is going to travel down the plane because its Weight Force is pulling it towards the Earth. At the same time, a Frictional Force will act in the opposite direction to the motion of the object.

The Frictional Force, however, is calculated using the object's Normal Force times cos(θ) (times μ), which acts perpendicular to the surface of the plane. But this means that the Normal Force is not acting in the same axis as the Frictional Force, so how can Normal Force times cos(θ) (times μ) be used to calculate Frictional Force?

Let me elaborate: In order to calculate Normal Force in the same axis as Frictional Force, you should use μ*m*g*sin(θ), not μ*m*g*cos(θ). Would it not make more sense this way?

I'm either very frustrated or very confused. Maybe a diagram with your answer can help.

Thank you.
 
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  • #2
But it makes sense if you think that you're breaking up the gravitational force into two components one parallel to the motion and one perpendicular to the motion. Its the perpendicular component magnitude that dictates the friction not its direction.

It makes sense if you consider motion on a horizontal plane vs motion on an incline of say 45 degree vs motion sliding down a vertical plane. In the first case its the full weight of the object(cos(0)*weight vs cos(45)*weight vs cos(90)* weight (or zero frictional force).
 
  • #3
The magnitude of the frictional force can be related to the magnitude of the normal force by ##f\leq \mu N##. When slipping occurs, we have ##f= \mu N##. If a block is sliding down an inclined plane of angle ##\theta## (i.e. slipping relative to the surface of the inclined plane), Newton's 2nd law gives for the motion along the plane and perpendicular to the plane, respectively, ##mg\sin\theta - f = ma## and ##mg\cos\theta - N =0 ## (I have set up my coordinates so that the x-axis is along the incline and the y-axis is perpendicular to the incline). We can then easily find an expression for the magnitude of friction ##f## and we find that ##f = \mu mg\cos\theta##. In summary, we are relating the magnitudes of the normal force to the frictional force, not their directions! The direction of ##f## is of course along the incline.
 
  • #4
A New Question

Thank you jedishrfu and WannabeNewton; that cleared a lot of things up. But this leaves a different question.

Why is Normal Force proportional to Frictional Force? In other words:

"...we find that f=μmgcosθ...".

, but why?

I think this would require more of an explanation than an equation.
 
  • #5
I believe you are asking for a justification of why ##f\leq \mu N## holds true for sliding friction. Keep in mind that it is an approximate model of sliding friction and is known as Coulomb friction; it is justified via experiment. http://en.wikipedia.org/wiki/Friction#Laws_of_dry_friction
 
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  • #6
WannabeNewton said:
I believe you are asking for a justification of why ##f\leq \mu N## holds true for sliding friction. Keep in mind that it is an approximate model of sliding friction and is known as Coulomb friction; it is justified via experiment. http://en.wikipedia.org/wiki/Friction#Laws_of_dry_friction

Good point, there is also an initial frictional force that must be overcome to start something sliding that is higher than the sliding frictional force.
 
  • #7
Brilliant. Thank you guys. :smile:
 
  • #8
FredericChopin said:
Thank you jedishrfu and WannabeNewton; that cleared a lot of things up. But this leaves a different question.

Why is Normal Force proportional to Frictional Force? In other words:

"...we find that f=μmgcosθ...".

, but why?

I think this would require more of an explanation than an equation.

Considering "Cause and Effect", I would express that as the magnitude of the Frictional Force is directly proportional to the magnitude of the Normal Force. The resulting equation is the same, but in my view it's the amount of the Normal Force which produces the force of friction.
 
  • #9
FredericChopin said:
Frictional Force is mathematically defined as:

Ff = μ*m*g*cos(θ)

, where μ is the coefficient of friction, m is the mass of the object, g is the acceleration due to gravity and θ is the angle of the inclined plane.

But in terms of direction, this makes no sense!

Suppose there is an object on an inclined plane. It is going to travel down the plane because its Weight Force is pulling it towards the Earth. At the same time, a Frictional Force will act in the opposite direction to the motion of the object.

The Frictional Force, however, is calculated using the object's Normal Force times cos(θ) (times μ), which acts perpendicular to the surface of the plane. But this means that the Normal Force is not acting in the same axis as the Frictional Force, so how can Normal Force times cos(θ) (times μ) be used to calculate Frictional Force?

Let me elaborate: In order to calculate Normal Force in the same axis as Frictional Force, you should use μ*m*g*sin(θ), not μ*m*g*cos(θ). Would it not make more sense this way?

I'm either very frustrated or very confused. Maybe a diagram with your answer can help.

Thank you.

You are confused because μ*m*g*cos(θ) is not the frictional force. It represents the maximum frictional force that can be sustained without the block beginning to slide down the plane. The block will begin to slide when the angle is just large enough for mg sin(θ) to equal μ*m*g*cos(θ). Otherwise, the frictional force will be less than μ*m*g*cos(θ), and the block will not slide.
 

What is the frictional force equation?

The frictional force equation is a mathematical formula that represents the force of friction between two surfaces in contact. It is typically written as Ff = μN, where Ff is the frictional force, μ is the coefficient of friction, and N is the normal force.

Why doesn't the frictional force equation make sense?

Some people may find the frictional force equation confusing because it seems to contradict our everyday experience. For example, we may expect that a heavier object would have a greater frictional force, but according to the equation, the weight of an object does not directly affect the frictional force.

What is the coefficient of friction?

The coefficient of friction is a dimensionless quantity that represents the amount of friction between two surfaces. It is a property of the materials in contact and depends on factors such as surface roughness and the presence of lubricants. A higher coefficient of friction means there is more resistance to motion between the surfaces.

How is the normal force related to the frictional force?

The normal force is the force that two surfaces exert on each other when they are in contact. In the frictional force equation, the normal force is multiplied by the coefficient of friction to determine the frictional force. This means that a higher normal force will result in a higher frictional force.

Are there any limitations to the frictional force equation?

Yes, the frictional force equation is a simplified model that does not take into account all the factors that can affect friction. In real-world situations, factors such as temperature, surface roughness, and the motion of the objects can also impact the frictional force. Additionally, the equation is only applicable for objects moving at a constant speed, not for objects that are accelerating or decelerating.

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