How much work is required to stop the hoop and how far will it travel on the incline?

a) How much work is required to stop the hoop?

b) If the hoop starts up a surface at 30° to the horizontal with a speed of 10.0 m/s, how far along the incline will it travel before stopping and rolling back down?

a) Answer:

The work required to stop the hoop is 600 joules.

b) Answer:

The distance traveled by the hoop on the inclined surface until it comes to rest is 20.38 meters.

To stop the hoop: All of its kinetic energy needs to be converted into other forms of energy, such as heat due to friction. Therefore, the work required to stop the hoop is equal to its initial kinetic energy:

Kinetic Energy (KE) = 1/2 mv^2 + 1/2(mr^2)v^2/r^2

= 1/2 (6.0 kg)(10.0 m/s)^2 + 1/2(6.0 x 1^2) x 100/1

= 600 J

Therefore, 600 J of work is required to stop the hoop.

On the inclined surface: The initial kinetic energy of the hoop is the same as in part (a), so it is still 600 J. As the hoop rolls up the incline, some of its kinetic energy will be converted into potential energy, decreasing its speed.

The work done by the force of gravity on the hoop as it rolls up the incline is equal to the change in potential energy, which is given by:

Potential Energy (PE) = mgh = 600 J

h = 600 / (6 x 9.81) meters

h = 10.19 meters

Let the hoop roll a distance of x meters on the inclined surface.

x = h/sin30

x = 10.19 x 2 = 20.38 meters

Therefore, the distance traveled on the inclined surface is 20.38 meters.