Very difficult mechanics problem with friction

I am unable to guess your notation.My notation is bad, I'm sorry about that. ##v_{a,dx}## is the x component of the velocity of the right point of the cylinder in contact with the ground. ##v_{a,sx}## is the x component of the velocity of the left point of the cylinder that is in contact with the ground. ##v_{CdM}## is the velocity of the centre of the cylinder, so it's the same as ##v##, in the picture is it the point C. ##a## is the distance between the centre of the cylinder and the line of the contact between the cylinder and the ground. R is the radius of the cylinder. I hope it's
  • #1
Bestfrog

Homework Statement


A hollow cylinder with mass m and radius R stands on a horizontal surface with its smooth flat end in contact the surface everywhere. A thread has been wound around it and its free end is pulled with velocity v in parallel to the thread. Find the speed of the cylinder. Consider two cases: (a) the coefficient of friction between the surface and the cylinder is zero everywhere except for a thin straight band (much thinner than the radius of the cylinder) with a coefficient of friction of μ, the band is parallel to the thread and its distance to the thread a<2R (the figure shows a top-down view); (b) the coefficient of friction is μ everywhere. Hint: any planar motion of a rigid body can be viewed as rotation around an instant centre of rotation, i.e. the velocity vector of any point of the body is the same as if the instant centre were the real axis of rotation.

The figure: http://i64.tinypic.com/2ijrv53.jpg

Homework Equations



The Attempt at a Solution


I have no idea were to start! I know that it is against the forum rules.. but can someone give me only a hint to set up the solution?
 
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  • #2
The figure
 

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  • #3
Bestfrog said:
give me only a hint to set up the solution?
I take it we are to assume steady state, i.e. the cylinder moves to the right at constant speed.
For a), how many forces are there? What can you say about their magnitudes and directions? What equations can you write relating them?
 
  • #4
haruspex said:
I take it we are to assume steady state, i.e. the cylinder moves to the right at constant speed.
For a), how many forces are there? What can you say about their magnitudes and directions? What equations can you write relating them?

There are three forces. One that pull the thread (its magnitude is ##F_c = m \frac{v^2}{R}##). The other 2 are the frictions forces acting on the other direction (in the left), ##f= dm.g. \mu## with ##dm## the infinitesimal mass of the cylinder in contact with the friction line. I think that the forces make a torque (is it possible?)
 
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  • #5
Bestfrog said:
its magnitude is ##F_c = m \frac{v^2}{R})##
I have no idea how you arrive at that. It is not a centripetal force. The speed is constant, so it can be any value for the same force. Anyway, its magnitude does not matter.
Bestfrog said:
The other 2 are the frictions forces acting on the other direction
Not quite.
Kinetic friction acts to oppose the relative motion of the surfaces in contact. As the cylinder moves to the right and rotates, what is the direction of the relative motion between the cylinder and the table at those two points?
 
  • #6
haruspex said:
I have no idea how you arrive at that. It is not a centripetal force. The speed is constant, so it can be any value for the same force. Anyway, its magnitude does not matter.
Ok I'm totally wrong :D

Not quite.
Kinetic friction acts to oppose the relative motion of the surfaces in contact. As the cylinder moves to the right and rotates, what is the direction of the relative motion between the cylinder and the table at those two points?
So the situation is this
 

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  • #7
Bestfrog said:
So the situation is this
Yes. Can you find the actual components of the relative velocities at the frictional points in terms of v, R, a and ω?
 
  • #8
haruspex said:
Yes. Can you find the actual components of the relative velocities at the frictional points in terms of v, R, a and ω?
I'm struggling but I can't find any idea. For relative velocities you mean with respect to the ground?
For the velocity (not angular) I'm thinking that all the velocity vectors at a certain distance from the centre are equal to the vector of ##v_{CdM}## (i.e. At a distance ##R-a## the sum of the two vectors of the two points of the cylinder in contact with the ground is ##v_{CdM}##). So ##v_{a,dx} + v_{a,sx} = v_{CdM}##
 
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  • #9
Bestfrog said:
For relative velocities you mean with respect to the ground?
Yes.
Bestfrog said:
For the velocity (not angular) I'm thinking that all the velocity vectors at a certain distance from the centre are equal to the vector of ##v_{CdM}## (i.e. At a distance ##R-a## the sum of the two vectors of the two points of the cylinder in contact with the ground is ##v_{CdM}##). So ##v_{a,dx} + v_{a,sx} = v_{CdM}##
I am unable to guess your notation.
Consider the motion of the cylinder as the sum of a linear motion, speed u say, and a rotation at rate ω. What equation relates some or all of u, ω, R, a and v?
Consider, e.g., the right-hand frictional point of the cylinder. That has x and y components of motion, x being parallel to the thread.
What contributions to its x and y velocities come from u (easy)?
Now ignore u. What contributions to its x and y velocities come from ω?
 

1. What is the definition of "friction" in mechanics?

Friction is the force that opposes the relative motion of two surfaces in contact with each other. It is caused by microscopic irregularities on the surfaces and can be affected by factors such as surface roughness, weight, and the material of the surfaces.

2. Why are mechanics problems involving friction considered difficult?

Mechanics problems involving friction are considered difficult because friction is a complex force that can be difficult to quantify and predict. It is also affected by various factors that can make it challenging to accurately calculate the motion of objects in a system.

3. How do you approach solving a mechanics problem with friction?

The first step in solving a mechanics problem with friction is to identify all the forces acting on the objects in the system, including friction. Then, use Newton's laws of motion and other relevant equations to set up and solve the equations of motion. It is important to carefully consider the direction and magnitude of the friction force in the calculations.

4. Can friction be completely eliminated in a mechanics problem?

No, it is not possible to completely eliminate friction in a mechanics problem. Friction is a natural and unavoidable force that exists between all surfaces in contact. However, it is possible to reduce its effect by using smoother surfaces or lubricants.

5. Are there any real-life applications where understanding friction in mechanics is important?

Yes, understanding friction in mechanics is essential in many real-life applications. For example, engineers need to consider friction when designing machines or vehicles to ensure they function properly and efficiently. Friction is also important in sports, as athletes need to understand and use friction to their advantage in activities such as running, swimming, and rock climbing.

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