Rolling ball down a slope, how to find final velocity?

[woodie37]

Homework Statement

A ball is rolled down a rough hill with coefficient of $$\mu$$<tan$$\theta$$. The hill has a height of h. The ball has moment of inertia of I, mass of m, and radius of r, which are all constant. What is the final velocity of the ball at the bottom of the hill?

P.S. I made this question up, so I can't find an answer anywhere except from physics experts =D

Homework Equations

W = $$\Delta$$K + $$\Delta$$U where W is the work done by friction, K is the kinetic energy and U is potential energy.

The Attempt at a Solution

As the ball rolls down the hill it slips due to the fact that the frictional force is not strong enough. From an FBD, gravitational force acts on its center, and frictional force rotates the ball a little bit but the ball slips at a constant rate of slippage (idk if that's the right word).

As the ball is accelerated down wards by gravity, the frictional force decelerates the ball such that it slows down the gravitational acceleration and slightly rotates the ball...

and I have no idea how to do this...I attempted it and got an answer slightly greater than if the friction is strong enough to prevent slippage, where no energy is lost to friction.

Can someone help me solve this please? lol

 

[Andrew Mason]

Homework Statement

A ball is rolled down a rough hill with coefficient of $$\mu$$<tan$$\theta$$. The hill has a height of h. The ball has moment of inertia of I, mass of m, and radius of r, which are all constant. What is the final velocity of the ball at the bottom of the hill?

P.S. I made this question up, so I can't find an answer anywhere except from physics experts =D

Homework Equations

W = $$\Delta$$K + $$\Delta$$U where W is the work done by friction, K is the kinetic energy and U is potential energy.

The Attempt at a Solution

As the ball rolls down the hill it slips due to the fact that the frictional force is not strong enough. From an FBD, gravitational force acts on its center, and frictional force rotates the ball a little bit but the ball slips at a constant rate of slippage (idk if that's the right word).

As the ball is accelerated down wards by gravity, the frictional force decelerates the ball such that it slows down the gravitational acceleration and slightly rotates the ball...

and I have no idea how to do this...I attempted it and got an answer slightly greater than if the friction is strong enough to prevent slippage, where no energy is lost to friction.

Can someone help me solve this please? lol

What is the energy lost to friction? What is the torque on the ball? How do you relate that torque and the distance traveled to the rotational energy of the ball? How does the lost energy + rotational energy + translational energy relate to the change in gravitational potential energy?

AM

 

[woodie37]

What is the energy lost to friction? What is the torque on the ball? How do you relate that torque and the distance traveled to the rotational energy of the ball? How does the lost energy + rotational energy + translational energy relate to the change in gravitational potential energy?

AM

The energy lost due to friction is quite complicated to solve for, because as the ball slides down the hill of length h/sin$$\theta$$, it friction does NOT act on the ball for that length and instead the force of friction acts on the ball for a distance of h/sin$$\theta$$ less the distance the ball turns while rolling downhill. The torque is the product of the parrallel force and the radius of the ball less the force of friction that needs to be added to ensure the ball does not slip, which in other words, is the the product of friction and radius of the ball, r, and the torque is constant, as the force of gravity does not change, and the coefficient of friction does not either. I know how to relate the different energies by the conservation of energy, but that can only be done when the other things have been related, which are stated above, which I don't know how to do...help please!