Work and Energy Conservation: Rolling Hoop Problem
How much work is required to stop the hoop when it rolls without slipping? If the hoop rolls up an incline at 20° with an initial speed of 12.2 m/s, how far along the incline will it travel before stopping and rolling back down?
a) Work Required to Stop the Hoop
Work = Initial kinetic energy = (1/2) * Mass * Velocity^2
Calculating the work required:
Work = (1/2) * 6.7 kg * (12.2 m/s)^2
Work = 491.238 J
b) Distance Traveled on the Incline
When the hoop rolls up the incline, its gravitational potential energy increases while its translational and rotational kinetic energy decrease until it stops and rolls back down. To find how far the hoop travels, you need to calculate the height at which its kinetic energy is reduced to zero:Energy conservation equation: (1/2) * Mass * Velocity^2 + Mass * g * h = 0
Where g is the acceleration due to gravity and h is the height of the incline.