Solving a Pulley System: Mass M & Friction Forces

[funoras]

Homework Statement

An object of mass M=10kg is being pulled up by a system of pulleys.
a)What force do you need to exert on point A for the object to move uniformly ?
b) What force do you need to exert on point A, if the friction in each of the pullies is 0,5N and the pulleys are not massless - the mass of the 1st one is 100g, the 2nd - 200g, the 3rd - 300g, the 4th- 400g.

http://img148.imageshack.us/img148/5526/unledbde.jpg

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Homework Equations

The Attempt at a Solution

I solved the A part quite easily, got a result of F=12,5N, but I'm stuck with part B - i can't figure out the directions of the friction forces .

 

[PeterO]

Homework Statement

An object of mass M=10kg is being pulled up by a system of pulleys.
a)What force do you need to exert on point A for the object to move uniformly ?
b) What force do you need to exert on point A, if the friction in each of the pullies is 0,5N and the pulleys are not massless - the mass of the 1st one is 100g, the 2nd - 200g, the 3rd - 300g, the 4th- 400g.

http://img148.imageshack.us/img148/5526/unledbde.jpg

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Homework Equations

The Attempt at a Solution

I solved the A part quite easily, got a result of F=12,5N, but I'm stuck with part B - i can't figure out the directions of the friction forces .

ALL friction forces oppose the motion you are trying to cause - that gives you the direction.

 

[funoras]

The system is going up with some constant speed. and i drew all the friction forces in blue. Can you tell me is this corrent ?
http://img202.imageshack.us/img202/3059/unledme.jpg

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[PeterO]

The system is going up with some constant speed. and i drew all the friction forces in blue. Can you tell me is this corrent ?
http://img202.imageshack.us/img202/3059/unledme.jpg

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Looks like too many arrows to me. The friction in/at each pulley should mean that the tension on the "pulling" side of the pulley is greater than the tension on the other side.

2:30 am here - will check back tomorrow.

 

[funoras]

Yeah that makes sense . So it should be something like this. By the way,about the 4th pulley, since it is fixed (it doesn't move) , shouldn't i ignore it's mass ?
http://img204.imageshack.us/img204/3059/unledme.jpg

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[PeterO]

Yeah that makes sense . So it should be something like this. By the way,about the 4th pulley, since it is fixed (it doesn't move) , shouldn't i ignore it's mass ?
http://img204.imageshack.us/img204/3059/unledme.jpg

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Certainly you are not trying to lift #4 so it looks like it can be ignored.

Note that pulley #4 is the only one to rotate clockwise, so friction there shoul be in the opposite direction.

I would probably draw the friction at the midpoint of the contact zone - so horizontal in each case.

 

[funoras]

Okay so this is my solution
T is the tension force , and i'll assume g=10m/s^2
For the 1st pulley
$$2T_1=Mg+m_1 g+F_f$$
so $$T_1=50,75N$$
2nd pulley
$$2T_2=T_1+m_2 g +F_f$$
$$T_2=26,625N$$
3rd pulley
$$2T_3=T_2+m_3 g +F_f$$
$$T_3=15,0625N$$

and the 4th pulley it should like something like this i guess
$$F=T_3 + F_f$$
$$F=15,5625N$$

 

[PeterO]

Okay so this is my solution
T is the tension force , and i'll assume g=10m/s^2
For the 1st pulley
$$2T_1=Mg+m_1 g+F_f$$
so $$T_1=50,75N$$
2nd pulley
$$2T_2=T_1+m_2 g +F_f$$
$$T_2=26,625N$$
3rd pulley
$$2T_3=T_2+m_3 g +F_f$$
$$T_3=15,0625N$$

and the 4th pulley it should like something like this i guess
$$F=T_3 + F_f$$
$$F=15,5625N$$

The calculations look OK except that I think the tension in the RHS of each pully should be greater than T in the LHS due to friction - so rather than 50,75 for pulley 1, I think the tensions should be 50,5 and 51,0 N. And as I think about it I can't convince myself which one is the 50.5 and which is the 51.0, but I lean toward 52 on the right.

Similar arguments all the way through.

 

[ehild]

We can assume that the pulleys work in the regular way - so they rotate and the string does not slip on them: The linear velocity of the rim is the same as that of the string. That is ensured by static friction.
The other kind of friction acts in the bearings, at the axes of the pulleys. This friction causes a torque against the rotation of the pulley which is equivalent to a force of 0.5 N acting on the rim.

To get the conditions of motion with constant velocities and angular velocities, not only the forces have to be in equilibrium, but also the torques.

Considering the first pulley, the tension of the left string is T₁' and the tension of the right string is T₁. The condition for zero torque means (with R the radius of the pulley)
T₁R-T₁'R-F_fR=0, that is, T₁'=T1-0.5 N. The condition of zero net force is

T₁'+T₁-m₁g-Mg=0, subbing in T₁'=T1-0.5, 2T₁-m₁g-Mg-0.5=0.

This is the same equation Funoras used, with T1 the tension in the right-hand string.

In case of the last (fourth) pulley, T₄=F, and the equilibrium of torques require that T₄-T₃-0.5=0, so F=T₃+0.5.

Good job, Funoras!

ehild