How Much Can Block A Weigh Without Block C Slipping Off?

[ashimashi]

Problem:
Block A is suspended from a vertical string. The string passes over a pulley and is attached to block B. The mass of block B is 1.8 kg. The mass of block C is 0.75 kg. If the coefficient of kinetic friction is 0.100 between block B and the ramp and the coefficient of static friction between block B and C is 0.900, determine the maximum mass of block A, such that block C does not slip off.

Homework Equations

f = ma

The Attempt at a Solution

I drew a FBD for block C and was able to calculate its force of friction which is: 5.61 N (μmgcosθ)

I also found out that the force of gravity for Block B is (M)(G)(COSθ). But I don't know what the force of Block C on B would be. If I could find that, I will be able to find the Force of Normal for Block B and solve for the force of friction for block B as well.

After that, I am not too sure what I must do next. I am thinking of solving for acceleration but not too sure if that will help.

Any help or any push in the right direction would be very helpful.

Thanks

 

[BvU]

Hello Ashi, and welcome to PF.
Your $\mu\, mg\,\cos\theta$ is the maximum force block B can exercise on block C without block C slipping, right ?

Can you calculate the friction force block B exercises on block C in the case that block A is so light that block C and B do not move ?

 

[ashimashi]

Thank you BvU!

Okay I think I got it but I am not too sure:

I made a mistake on the Force of friction between block B and C. It should have been (mgμcosθ)/m where m is the mass of block c.

So force of friction must be greater than acceleration for block c to slip off.

Therefore: a = fnet/total mass = force of friction of block c

force of friction of block c = 7.48 N

total mass = M + 2.55

fnet = Mg - (2.55)(g)(0.1)(cos32) - (2.55)(g)(sin32)

acceleration = [Mg - (2.55)(g)(0.1)(cos32) - (2.55)(g)(sin32)] / [M + 255][Mg - (2.55)(g)(0.1)(cos32) - (2.55)(g)(sin32)] / [M + 255] = 7.48

M = 14.0KG

I believe that is the right answer but I am not 100% sure. Can someone please check my work and let me know if they believe my answer is correct.

 

[BvU]

I made a mistake on the Force of friction between block B and C. It should have been (mgμcosθ)/m where m is the mass of block c.

No: $\mu\, g\,\cos\theta$ has the dimension m/s^2, an acceleration. So it cannot be a force.

So force of friction must be greater than acceleration for block c to slip off.

You mean smaller ?

Therefore: a = fnet/total mass = force of friction of block c

An acceleration can not be a force.

force of friction of block c = 7.48 N

Where does the 7.48 come from ? Before, you had 5.616 as the maximum force of friction, right ? (I did ask rather explicitly).
However, with $\mu\, g\,\cos\theta =$ 7.49 m/s² (not a force, but an acceleration) you do have a value for the maximum acceleration of block C (and it is NOT this 7.49 m/s²! (*)). And thereby a value for the maximum acceleration of all three of A, B and C.

An earlier small mistake I need to point out:

the force of gravity for Block B is (M)(G)(COSθ)

No: just mg.

So back to the drawing board: free body diagrams for C and for B !

(*) Can you calculate the friction force block B exercises on block C in the case that block A is so light that block C and B do not move ? (Didn' t I ask for that already?)

 

[HalfLostAndSearching]

for the help!

As a scientist, you have correctly identified the forces acting on the system and have attempted to solve for the force of friction and acceleration. To find the maximum mass of block A, you will need to use the concept of equilibrium. This means that the net force on each block must be zero in order for the system to remain in place and block C to not slip off. Using this idea, you can set up equations for the forces acting on each block and solve for the maximum mass of block A. Keep in mind that the mass of block A will affect the normal force on block B and the force of friction between blocks B and C. Additionally, you may need to use the equations for friction to account for the static and kinetic coefficients of friction. Overall, continue to use your knowledge of forces and Newton's laws to solve for the maximum mass of block A in this pulley and ramp problem.