# Motion of an object on the outside of a cylinder

This question considers the motion of an object of mass m sliding on the outside of a cylinder of radius R whose axis is horizontal. The motion occurs in the vertical plane, and the surface of the cylinder is rough — the coefficient of sliding friction is μ'. The diagram below shows the position of the object when it is at an angle θ from the upward vertical through the axis of the cylinder O. Model the object as a particle.

Please see the attached MS Word document for the diagram and the followup questions.

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QUESTION

This question considers the motion of an object of mass m sliding on the outside of a cylinder of radius R whose axis is horizontal. The motion occurs in the vertical plane, and the surface of the cylinder is rough — the coefficient of sliding friction is μ'. The diagram below shows the position of the object when it is at an angle θ from the upward vertical through the axis of the cylinder O. Model the object as a particle.

1. Please draw a force diagram for the forces acting on the particle, defining any forces that you introduce.

2. Why is it better to use plane polar unit vectors. Please include them on the force diagram. How do I express the acceleration of the particle in terms of these unit vectors?

3. How do I express each of the forces acting on the particle as a vector in terms of the plane polar unit vectors?

4. How would I write down a differential equation of motion for the particle, and how would I then obtain two scalar differential equations?

5. If μ < tan θ, where μ is the coefficient of static friction how would I show that the particle will slide down the outside of the cylinder?

6. Why it is not possible to use conservation of energy for this system?

7. How would I use the friction condition to show that by eliminating the unknown magnitudes of forces in the two equations ...

#### Solution Summary

The solution discusses the motion of an object on the outside of a cylinder.