1. What is the gravitational portential energy, relative to the ground, of a 0.15kg baseball at the top of a 100m tall building?

A 2kg ball is attached to the bottom end of a 1m long string hanging from the ceiling of a room. The height of the room is 3m. What is the gravitational potential energy of the ball relative to

a. the ceiling

Relative to the ceiling, y = -1m

b. the floor

Relative to the floor, y=2m

c. a point at the same elevation as the ball?

Relative to the height of the ball, y=0

3. A 1000kg roller coaster is initially at the top of a rise, at point A. It then moves 135 ft, at an angle of 40deg below horizontal to a lower point, B.

a. Choose point B to be the zero level for gravitational potential energy, and find the potential energy of the roller coaster at points A and B, and the difference iin potential energy between these points.

With our choice for the zero level fro potential energy at point B, U at B equals zero. At point A, the potential energy is given by U = mgy, where y is the vertical height above zero level. with 135ft = 41.1m, this height is found as:

Therefore

b. Repeat part (a), setting the zero reference level at point A.

With our choice of the zero level at point A, we have U at a equal to zero. The potential energy at B is given by U = mgy wher y is the vertical distance of point B below A. In part (a), we found the magnitude of this distance to be 26.5m. Because this distance is now below the zero reference level, it is a negative number. So:

4. A 40-N child is in a swing that is attached to ropes 2 m long. Find the gravitational potential energy of the child relative to the child's lowest position when

a. the ropes are horizontal

We take the zero level of potential energy at the lowest point of the are. When the string is held horizontal initially, the initial position is 2m above zero level. Thus,

b. the ropes make a 30deg angle with the vertical

From the above figure, you see that at an angle of 30deg the ball is at a vertical height of (2m)(1-cos30) above the lowest point of the arc.

c. the child is at the bottom of the circular arc.

The zero level has been selected at the lowest point of the are Therefore, PE = 0 at this location.

5. A 3-kg particle moves from the origin to the position having coordinates x = 5m and y = 5m, under the influence of gravity acting in the negative y direction. Calculate the work done by gravity when the particle goes from O to C along the following paths:

a. OAC

Work along OAC = work along OA + work along AC

b. OBC

Work along OBC = work along OB + work along BC

c. OC

Your results should all be identical. Why?

Results are the same because gravitation forces are conservative.