The relationship between angle of incline and friction force

[bijou1]

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.

 

[haruspex]

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?)

ff (force of friction) = μ (coefficient of friction) * n (normal force)

That's not necessarily true for static friction. μ_sN is only the maximum value of the frictional force.

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.

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.

 

[Doc Al]

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?

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.)

 

[Doc Al]

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]

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?

 

[bijou1]

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.

 

[bijou1]

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.

 

[Doc Al]

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!

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.)

 

[haruspex]

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.

 

[bijou1]

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.

 

[haruspex]

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.