**Example:** A steelball with mass in = 0.44 kg and radius r = 1.0 cm moves along a rollercoaster. Seven photogates are placed along the track at different heights hi from the base of the track; we measure the time intervals Δt_{i} for the steel ball to pass through each photogate,

Calculate the gravitational potential energy, the kinetic energy, and the total mechanical energy.

**Solution:** We have the gravitational potential energy PE_{i} = mgh_{i}. The (average) speed of the ball is **v _{i} = 2r/Δt_{i}**, as it passes through each photogate. The kinetic energy then follows

**KE = mv**. Note that the steel ball moves along the track (translational motion), but also rotates [rotational motion]. As a result, the kinetic energy of the ball has a contribution from rotation. In this situation, the contribution of rotational kinetic energy can be described by an “effective” mass m*, greater than the mass in. If the track has a width w = 0.95 cm, we find,

_{i}^{2}/2Discussion: The mechanical energy of the system (slightly) decreases due to friction.