Analyzing acceleration of block on a ramp connected to a pulley

[I_Try_Math]

Homework Statement

A block of mass 3 kg slides down an inclined plane at an angle of with a massless tether attached to a pulley with mass 1 kg and radius 0.5 m at the top of the incline (see the following figure). The pulley can be approximated as a disk. The coefficient of kinetic friction on the plane is 0.4. What is the acceleration of the block?

Relevant Equations

$\tau = I\alpha$

Initially I thought a good strategy for solving the problem would be to find the torque on the pulley to get alpha (angular acceleration) and then use alpha to find the tangential acceleration of the pulley which is equal to the block's acceleration. I'm not sure if this is correct.

Let $F_{r}$ be the force down the ramp due to gravity.
Let $F_{f}$ br the force of friction opposing motion down the ramp.

Then the force on the pulley is $F_{r} - F_{f}$.

$F_{r} - F_{f} = 25\tan45 - \mu_km_{block}g\sin45$

$F_{r} - F_{f} = 25\tan45 - 0.4(3)(9.8)\sin45$

Is my equation for the force on the pulley incorrect? Or maybe my mistake is something later on in my work?

 

[kuruman]

What about the tension in the rope acting on the mass? It's the same tension that generates the torque on the pulley.

 

[I_Try_Math]

What about the tension in the rope acting on the mass? It's the same tension that generates the torque on the pulley.

I'm not exactly sure where I'm going wrong but I'm getting very confused. The way I'm visualizing the forces, if $F_{pulley}=T$ then $T=F_{r}-F_{f}$ which would give a net force down the ramp of zero going against Newton's laws.

 

[haruspex]

if $F_{pulley}=T$ then $T=F_{r}-F_{f}$

Why? Can’t the block accelerate?

 

[kuruman]

I'm not exactly sure where I'm going wrong but I'm getting very confused. The way I'm visualizing the forces, if $F_{pulley}=T$ then $T=F_{r}-F_{f}$ which would give a net force down the ramp of zero going against Newton's laws.

If you actually mean the net force on the pulley, then that force is zero because the pulley is and reamains at rest. That's because the axle, which is firmly attached to the incline which is firmly attached to the Earth, keeps the center of the pulley from accelerating. For that reason the linear acceleration of the pulley is of no interest to us. Of interest is the pulley's angular acceleration for which you have written no equation so far.

 

[I_Try_Math]

If you actually mean the net force on the pulley, then that force is zero because the pulley is and reamains at rest. That's because the axle, which is firmly attached to the incline which is firmly attached to the Earth, keeps the center of the pulley from accelerating. For that reason the linear acceleration of the pulley is of no interest to us. Of interest is the pulley's angular acceleration for which you have written no equation so far.

The tension can't be equal to the force on the pulley else you would have acceleration without a force, no? Can't figure out where I'm going wrong.

 

[haruspex]

View attachment 348602
The tension can't be equal to the force on the pulley else you would have acceleration without a force, no? Can't figure out where I'm going wrong.

What matters for acceleration of the block is the forces on the block:, $T, F_r, F_f$.
What matters for acceleration of the pulley is the torques on the pulley.
You are not explaining why you think $F_p=F_r-F_f$.

 

[Orodruin]

View attachment 348602
The tension can't be equal to the force on the pulley else you would have acceleration without a force, no? Can't figure out where I'm going wrong.

You are assuming that the forces add up to zero when you assume that $T = F_r - F_f$ and therefore implicitly assuming that the acceleration is zero. There is no a priori reason for $T = F_r - F_f$ outside of equilibrium.

Have you solved any problems involving an Atwood machine? Those are basic for building the understanding of problems like this one.

 

[I_Try_Math]

What matters for acceleration of the block is the forces on the block:, $T, F_r, F_f$.
What matters for acceleration of the pulley is the torques on the pulley.
You are not explaining why you think $F_p=F_r-F_f$.

What about$F_p=F_r-F_f - T$? If that's correct I'm still not sure how to solve for the tension. Edit: I believe I might see a way to solve it actually. I'll give it a try.
Edit 2: Nope that didn't work.

 

[I_Try_Math]

You are assuming that the forces add up to zero when you assume that $T = F_r - F_f$ and therefore implicitly assuming that the acceleration is zero. There is no a priori reason for $T = F_r - F_f$ outside of equilibrium.

Have you solved any problems involving an Atwood machine? Those are basic for building the understanding of problems like this one.

I believe I've spent time on some Atwood machine problems in the past but it's been a little while. So I'm pretty rusty apparently. And frankly I'm not sure why I assumed the forces added up to zero but the tension is what's bringing me the most confusion. Although I think I may now have an idea on how to solve it.

 

[Orodruin]

What about$F_p=F_r-F_f - T$? If that's correct I'm still not sure how to solve for the tension.

It is not correct. The only force acting on the pulley is the tension T (apart from forces at the support as already mentioned above).

Part of the issue here is your “free body diagram”, which is not really a free body diagram. A FBD draws a single system and all of the forces acting on that system only.

Edit: I believe I might see a way to solve it actually. I'll give it a try.

Good luck!

 

[I_Try_Math]

So if the force on the pulley is equal to the tension in the string then I suppose it all comes down to figuring out how to calculate the tension. It must be less than $F_{r}-F_{f}$ I guess but other than that i'm not sure what else I can definitively say about it.

 

[Orodruin]

Make FBDs for the block and for the pulley. What is the acceleration of the block? What is the angular acceleration of the pulley? Can they be related?

 

[I_Try_Math]

So I think I finally figured out how to relate the tension to the angular acceleration but I'm getting a different answer than the book. Did I calculate $F_{r} - F_{f}$ incorrectly or something?

 

[I_Try_Math]

So I think I finally figured out how to relate the tension to the angular acceleration but I'm getting a different answer than the book. Did I calculate $F_{r} - F_{f}$ incorrectly or something?
View attachment 348634

Yeah looks like I made a basic trig mistake as far as I can tell? Not very encouraging. Anyway here's a video that helped me solve it in case it might help someone in the future.

 

[Lnewqban]

Initially I thought a good strategy for solving the problem would be to find the torque on the pulley to get alpha (angular acceleration) and then use alpha to find the tangential acceleration of the pulley which is equal to the block's acceleration. I'm not sure if this is correct.

I would suggest a better strategy; yours seems to be limited.
Free body diagrams are your friends.

Naturally, the block, if free of any restriction, would free fall (vertically) at acceleration g.
Its restrictions preventing it from doing so are:

1) Geometrically, it is forced to slide down an inclined plane at an angle from the natural vertical trajectory.

2) The friction force acting between the surfaces in contact of the block and of the inclined plane, which acts in a direction opposed to the relative movement.

3) The rotational inertia of the pulley (simplified as a massive disk) to accelerate its rotation (induced by the falling block).

Note that the massless tether offers no restriction to the slide of the block; it simply connects it with the rotating pulley.