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Coefficient of friction: smallest value to maintain mass in an equilibrium position?

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A cylinder of mass M and radius R rests on an inclined plane. It is held in place by a horizontal string that is attached to the edge of the cylinder. If the angle that the plane makes with the horizontal is theta, and the coefficient of static friction between the cylinder and the plane is u, what is the smallest value of U that will maintain this position as an equilibrium position?

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For this to be an equilibrium position, 2 things must be true:

1) Net force on the ball cylinder must be zero
2) Net torque about the cylinder must be zero.

So first, draw a careful free-body diagram of the forces on the cylinder and let Fnet = 0. To do this, I would choose as your coordinate system the directions "along the inclined plane" and "perpendicular to the inclined plane". That way you have fewer forces to resolve into components.

Here are the forces involved:
1) Gravitational force, acting downward from the centre of mass of the cylinder
2) Normal force of contact between the ramp surface and the cylinder (acting perpendicular to the ...

Solution Summary

With good explanations and calculations, the problems are solved.