The relationship between angle of incline and friction force

• bijou1
In summary, the conversation discusses a problem involving a wagon being released from rest on a 15 m long plane at an angle of 30° with friction present. The question posed is how the frictional force is affected if the angle of incline is increased. The conversation includes equations and attempted solutions, but it is ultimately determined that the approach is incorrect due to the different accelerations at different angles. The correct solution involves taking into account the moment of inertia of the wheels and determining at which angle the wheels will have greater angular acceleration.

Homework Statement

A 20 kg wagon is released from rest from the top of a 15 m long plane, which is angled at 30° with the horizontal. Assuming there is friction between the ramp and the wagon, how is this frictional force affected if the angle of the incline is increased?

Homework Equations

∑Fy = n - mgcos30° = 0
∑Fx = -ff (force of friction) + mg sin 30° = ma
up = +
down = -
→ = +
← = -
ff (force of friction) = μ (coefficient of friction) * n (normal force)

The Attempt at a Solution

I approached this problem by first finding the normal force when angle θ = 30°.
Therefore, ∑Fy = n-mgcos30° = 0
⇒n = mgcos30°
⇒n = 174 N
Then I solved ∑Fy = 0 when angle θ is increased, for example when θ = 60°
Therefore, ∑Fy = n-mgcos60° = 0
⇒n = mgcos60°
⇒n = 100 N
Then, I solved ∑Fx = mgsin30° - ff = ma when θ = 30°
⇒ff = mgsin30° - ma
⇒ff = 100 - 20a
Then, I solved ∑Fx = mgsin60° - ff = ma when θ = 60°
⇒ff = mgsin60° - ma
⇒ff = 174 - 20a
For example if I designate that "a" = 2 m/s^2, then ff = 100 - 20(2) ⇒60 N when θ = 30°
and ff = 174 - 20(2) ⇒ 134 N when θ = 60°. Therefore, when angle of incline increases, friction force increases.
However, my logic is wrong. The solution is that since ff = μ * n , and if angle of incline increases, friction force decreases, since ff = μ * 174 N when θ = 30° and ff = μ * 100 N when θ = 60°. I don't know why my approach is wrong...Any help would by greatly appreciated. Thanks.

Since it says 'wagon' I assume it is rolling. This implies static friction, not kinetic. (OK, it could be kinetic if the friction is low enough, but then why not specify a box instead of a wagon?)
bijou1 said:
ff (force of friction) = μ (coefficient of friction) * n (normal force)
That's not necessarily true for static friction. μsN is only the maximum value of the frictional force.
bijou1 said:
Then, I solved ∑Fx = mgsin30° - ff = ma when θ = 30°
⇒ff = mgsin30° - ma
⇒ff = 100 - 20a
Then, I solved ∑Fx = mgsin60° - ff = ma when θ = 60°
⇒ff = mgsin60° - ma
⇒ff = 174 - 20a
But these two accelerations will be different.
bijou1 said:
The solution is that since ff = μ * n , and if angle of incline increases, friction force decreases,
As I wrote above, that's not valid either. To get a valid answer you need to be told the moment of inertia of the wheels. If there's no moment of inertia then there's no frictional force.

bijou1 said:
I approached this problem by first finding the normal force when angle θ = 30°.
Therefore, ∑Fy = n-mgcos30° = 0
⇒n = mgcos30°
⇒n = 174 N
Then I solved ∑Fy = 0 when angle θ is increased, for example when θ = 60°
Therefore, ∑Fy = n-mgcos60° = 0
⇒n = mgcos60°
⇒n = 100 N
Right here you've solved the problem (basically). You've just shown that the normal force decreases as the angle increases. So, what must happen to the friction force as the angle increases?

bijou1 said:
I don't know why my approach is wrong...Any help would by greatly appreciated. Thanks.
Your approach is wrong because the acceleration at the different angles is not the same. You cannot just arbitrarily pick some value for "a" and assume it applies at any angle.

(Generally, you'd be asked to solve for the acceleration as a function of angle.)

As haruspex's answer illustrates, there are several hidden assumptions that you may be expected to make in solving this problem. Some of those assumptions may be physically unrealistic (or just plain wrong). For one: Is the wagon actually rolling? Or are you treating it as sliding down the ramp? (Your given solution implies that you are treating this as you would a block sliding--not rolling--down an incline, where kinetic friction is given by μ*N.)

haruspex said:
To get a valid answer you need to be told the moment of inertia of the wheels.
I withdraw that. Since you are only asked for a qualitative answer, you only need to know that the wheels have a nonzero moment of inertia, which is a reasonable assumption. So you can answer the question, but you won't get the 'book' answer. Hint: at which angle will the wheels have the greater angular acceleration?

Doc Al said:
As haruspex's answer illustrates, there are several hidden assumptions that you may be expected to make in solving this problem. Some of those assumptions may be physically unrealistic (or just plain wrong). For one: Is the wagon actually rolling? Or are you treating it as sliding down the ramp? (Your given solution implies that you are treating this as you would a block sliding--not rolling--down an incline, where kinetic friction is given by μ*N.)
Hi, for this particular problem, the assumption is that the wagon is treated as a block "sliding" down the ramp; therefore, like you have mentioned fk (kinetic friction) = μ * n can be used. I thought I can pick a random value for acceleration and plug it into the equation to see the relationship mathematically; however, as you have mentioned that is wrong. Since I already found that n = 174 N when θ = 30°, and n = 100 N when θ = 60° (I picked a random angle value greater than 30°) and since kinetic friction and normal force are related through the equation: fk = μ * n, I can see that when θ is decreasing, normal force increases, and so frictional force increases.

haruspex said:
I withdraw that. Since you are only asked for a qualitative answer, you only need to know that the wheels have a nonzero moment of inertia, which is a reasonable assumption. So you can answer the question, but you won't get the 'book' answer. Hint: at which angle will the wheels have the greater angular acceleration?
Hi, I'm not sure but I'm guessing since angular acceleration is Δω/Δt and angular speed is Δθ/Δt, if θ is increasing, angular speed is increasing; therefore, angular acceleration must also be increasing. So, I'm guessing at θ = 60°, the wheels will have the greater angular acceleration. Sorry, I am not familiar with angular acceleration yet, so please correct me. I would like to learn...Thanks.

bijou1 said:
Hi, I'm not sure but I'm guessing since angular acceleration is Δω/Δt and angular speed is Δθ/Δt, if θ is increasing, angular speed is increasing; therefore, angular acceleration must also be increasing.
Careful! There are two different angles here. One is the angle that the wheels will describe as they rotate; the other is the angle of the incline. Don't mix them up!

bijou1 said:
So, I'm guessing at θ = 60°, the wheels will have the greater angular acceleration. Sorry, I am not familiar with angular acceleration yet, so please correct me.
Realize that the translational acceleration of the wagon and the rotational acceleration of its wheels are related. (Assuming rolling without slipping.) Also realize that this is a bit more subtle of a problem than that of a block sliding down an incline. (Since it's a different problem, you can expect a different answer--as haruspex stated.)

bijou1 said:
I can see that when θ is decreasing, normal force increases, and so frictional force increases.
Yes, but I'd like to emphasise that the correct deduction is that the maximum frictional force increases as theta decreases. This will only equal the actual frictional force as long as the block slides (and you are told it does so for theta at these two angles). If theta is reduced below 30 degrees you will come to a point where the block no longer slides. From there on the actual frictional force will decrease again.

haruspex said:
Yes, but I'd like to emphasise that the correct deduction is that the maximum frictional force increases as theta decreases. This will only equal the actual frictional force as long as the block slides (and you are told it does so for theta at these two angles). If theta is reduced below 30 degrees you will come to a point where the block no longer slides. From there on the actual frictional force will decrease again.
Since the wheels are sliding, it is assumed to have kinetic friction and so I thought it would be any constant value, not a maximum. I thought maximum frictional force referred to static friction, and since the wheels are sliding for this problem, I thought only kinetic friction mattered.

bijou1 said:
Since the wheels are sliding, it is assumed to have kinetic friction and so I thought it would be any constant value, not a maximum. I thought maximum frictional force referred to static friction, and since the wheels are sliding for this problem, I thought only kinetic friction mattered.
Yes, I agreed with all that. I was just pointing out that although the first part of your statement
when θ is decreasing, normal force increases, and so frictional force increases.
is always valid, the second part ceases to apply below the critical angle because it doesn't slide. Sorry if that was already obvious to you.

1. What is the relationship between angle of incline and friction force?

The relationship between angle of incline and friction force can be described by the law of friction, which states that the friction force is directly proportional to the normal force and the coefficient of friction, and is independent of the surface area. As the angle of incline increases, the normal force decreases, resulting in a decrease in the friction force.

2. How does the coefficient of friction affect the relationship between angle of incline and friction force?

The coefficient of friction is a measure of how much resistance there is between two surfaces in contact. As the angle of incline increases, the normal force decreases, which means that the coefficient of friction has a greater impact on the friction force. A higher coefficient of friction will result in a larger friction force, while a lower coefficient of friction will result in a smaller friction force.

3. Is there a maximum angle of incline at which friction force becomes zero?

Yes, there is a maximum angle of incline at which friction force becomes zero. This angle is known as the angle of repose and is dependent on the coefficient of friction and the surface roughness. Once the angle of incline exceeds the angle of repose, the object will begin to slide down the surface due to the lack of friction force.

4. How does the surface roughness affect the relationship between angle of incline and friction force?

The surface roughness can affect the coefficient of friction, which in turn affects the relationship between angle of incline and friction force. A smoother surface will have a lower coefficient of friction, resulting in a smaller friction force, while a rougher surface will have a higher coefficient of friction and a larger friction force.

5. Can the relationship between angle of incline and friction force be used to predict the motion of an object?

Yes, the relationship between angle of incline and friction force can be used to predict the motion of an object. By knowing the angle of incline, the coefficient of friction, and the mass of the object, the friction force can be calculated, which can then be used to determine the acceleration of the object and its motion.