Help With Kinetic Energy Problem?

Question

A solid 1.0 kg ball of radius 0.50 m sits at the top of a ramp. It rolls down, ending up 10.0 m lower than where it started.

1) What is the TOTAL kinetic energy of the ball at the bottom?

2) How fast is the ball moving at the bottom of the ramp?

3) What is the ball's angular velocity and angular momentum at the bottom?

Now assume there is no friction. How would this change the above answers?

I know that for 1) you must take into account both the linear and angular KE since the ball is rolling due to friction. I just don't know how to show this and use it in order to solve for the speed of the ball. Once I have that, I know how to solve 3)

Answer 1

There is nothing in your question to say how fast the ball is moving at the bottom of the ramp. It could be stopped. That is the easiest solution.

1. zero

2. not moving

3. zero

2) with no friction, the answers are different.

1. KE = PE = 1kg*9.8*10m = 98 N

2. KE = ½mV²

  98 = ½(1)V²

solve for V

V = 14 m/s

.

Answer 2

The first 2 answers ignore the spin (rotation)of the ball.

kinetic energy = 1/2 m v^2 + 1/2(moment of inertia)*(angular velocity)^2

Answer 3

1. KE = lost gravitational potential energy = MGH

           = 1.0 x 9.81 x 10.0 = 98.1  joules

2. V^2 = u^2 + 2as

            = 0 + 2 x 9.81 x 10.0 = 196.2

    V = sqrt196.2 = 14.0 m/s

Sorry dont know the rest. These two answers are assuming theres no friction. If there was friction, I'll assume that the KE and the velocity would be lower. Coz energy will be lost as heat, and friction causes the ball to slow down, decreasing the velocity.