Rolling a cylinder down a slope.

In summary, in order for the solid cylinder to roll down the slope without slipping, the minimum coefficient of friction (k) required is (1/2)tan(theta). This is found by setting the torque equation equal to the force equation and solving for k. The tangential acceleration at the edge must be equal to the acceleration of the center of mass, and the instantaneous speed of the point of contact with the surface must be zero.
  • #1
EthanB
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Homework Statement


A solid cylinder of radius R sits at the top of a slope of angle theta. When it rolls down, what is the minimum coefficient of friction (k) required to make the cylinder roll without slipping?

Homework Equations


Fx: mgsin(theta) - kmgcos(theta) = ma
Torque: kmgRcos(theta) = mR^2 * Alpha

The Attempt at a Solution



In order for the cylinder to roll without slipping, the tangential acceleration at the edge must be twice the acceleration of the center of mass (right?). Therefore, we solve for where Alpha = 2a/R.

From Torque:
kmgRcos(theta) = mR^2 * 2a/R
kgcos(theta) = 2a

Substituting for a from Fx:
kgcos(theta) = 2gsin(theta) - 2kgcos(theta)
k = (2/3)tan(theta)

The book's answer is (1/2)tan(theta).
 
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  • #2
Oops... I was using the wrong moment of inertia. It won't let me delete the post. Thanks anyway.
 
  • #3
Actually, I was using the right moment of inertia. It's a hollow cylinder. Ok, so can anyone help me out?
 
  • #4
EthanB said:
In order for the cylinder to roll without slipping, the tangential acceleration at the edge must be twice the acceleration of the center of mass (right?).
That is incorrect. For rolling without slipping the instantaneous speed of the point of contact of the cylinder with the surface of the incline must be zero (with respect to the surface). Translate that into a mathematical statement so you can apply it.
 

1. How does the mass of the cylinder affect its rolling down a slope?

The mass of the cylinder affects its rolling down a slope because it determines the amount of force needed to overcome inertia and start rolling. A heavier cylinder will require more force to start rolling compared to a lighter one.

2. What is the relationship between the angle of the slope and the speed of the rolling cylinder?

The angle of the slope affects the speed of the rolling cylinder as it determines the acceleration of the cylinder. A steeper slope will result in a greater acceleration and therefore a faster speed for the rolling cylinder.

3. How does the surface of the slope affect the rolling of the cylinder?

The surface of the slope can affect the rolling of the cylinder in several ways. A rough surface will cause more friction, making it harder for the cylinder to roll smoothly. A smooth surface will result in less friction and allow the cylinder to roll more easily. Additionally, a slope with bumps or grooves may cause the cylinder to deviate from its intended path.

4. Can the shape of the cylinder affect its rolling down a slope?

Yes, the shape of the cylinder can affect its rolling down a slope. A cylinder with a larger diameter will have a greater moment of inertia, making it more difficult to start rolling. A cylinder with a smaller diameter will have a smaller moment of inertia and will start rolling more easily.

5. How does the height of the slope affect the potential and kinetic energy of the rolling cylinder?

The height of the slope affects the potential and kinetic energy of the rolling cylinder. As the cylinder rolls down the slope, it will gain kinetic energy and lose potential energy. The higher the slope, the greater the potential energy at the top and the more kinetic energy the cylinder will have at the bottom.

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