Solving a Pulley System: Mass M & Friction Forces

• funoras
In summary, a system of pulleys is used to pull an object of mass 10kg up. The required force to move the object uniformly at point A is 12.5N. If the friction in each pulley is 0.5N and the masses of the pulleys are given, the direction of the friction force can be determined by considering that all friction forces oppose the motion. The correct direction of the friction force can be found by drawing the friction at the midpoint of the contact zone. The calculations for finding the tension force at each pulley should take into account the effects of friction, resulting in a final force of 15.5625N at the fourth pulley.
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

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 .

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funoras said:

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

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.

Last edited by a moderator:
Last edited by a moderator:
funoras said:
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

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.

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funoras said:
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

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.

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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
$$F=T_3 + F_f$$
$$F=15,5625N$$

funoras said:
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.

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-FfR=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

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1. How do you determine the tension in a pulley system?

To determine the tension in a pulley system, you must first identify all the forces acting on the system. These include the weight of the masses, the friction forces, and the tension in the ropes or cables. Then, you can use Newton's Second Law of Motion (F=ma) to calculate the tension by setting up a system of equations.

2. What factors affect the friction forces in a pulley system?

The friction forces in a pulley system are affected by the coefficient of friction between the surfaces in contact, the normal force, and any external forces acting on the system. The type of material and the smoothness of the surfaces can also impact the friction forces.

3. How does the mass of an object affect the tension in a pulley system?

The mass of an object can affect the tension in a pulley system because it contributes to the overall force acting on the system. As the mass increases, so does the force required to move it. This can impact the tension in the ropes or cables connecting the masses.

4. Can you use a pulley system to increase the mechanical advantage?

Yes, a pulley system can be used to increase the mechanical advantage. By using multiple pulleys and connecting them in the correct way, the force needed to lift a mass can be reduced. This is known as a compound pulley system and can greatly increase the efficiency of lifting heavy objects.

5. How can you minimize the friction forces in a pulley system?

To minimize the friction forces in a pulley system, you can use materials with low coefficients of friction, such as lubricants or smooth surfaces. Additionally, ensuring that the pulley system is properly aligned and maintained can also help reduce friction and increase the efficiency of the system.

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