Rolling Hoop Adventure!
How much work is required to stop the hoop?
(a) 100 J
(b) 120 J
(c) 140 J
(d) 160 J
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?
Final answer: The work required to stop the hoop is 300.0 J. The hoop will travel up the incline until it reaches a height of 0.5 m before stopping and rolling back down.
Work Required to Stop the Hoop
To stop the hoop, we need to remove its kinetic energy. The kinetic energy of the hoop is given by the formula KE = 1/2 * I * ω^2, where I is the moment of inertia and ω is the angular velocity. For a hoop, the moment of inertia is I = mR^2, where m is the mass of the hoop and R is the radius.
Given that the mass of the hoop is 6.0 kg and the radius is 1.0 m, we can calculate the moment of inertia as I = 6.0 kg * (1.0 m)^2 = 6.0 kg·m^2. The hoop is rolling with a speed of 10.0 m/s, which is equivalent to an angular velocity of ω = v/R = 10.0 m/s / 1.0 m = 10.0 rad/s. Plugging these values into the kinetic energy formula, we get KE = 1/2 * (6.0 kg·m^2) * (10.0 rad/s)^2 = 300.0 J. Therefore, the work required to stop the hoop is 300.0 J.
Distance Along the Incline
When the hoop starts up the incline, it gains potential energy. As it rolls up the incline and comes to a stop, this potential energy is gradually converted back into kinetic energy. The work done against gravity to bring the hoop to a stop is equal to the change in its potential energy.
The potential energy of the hoop, when it is at the bottom of the incline, is given by the formula PE = mgh, where m is the mass of the hoop, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the incline. In this case, the height of the incline is h = R * sinθ, where R is the radius of the hoop and θ is the angle of the incline.
Given that the radius of the hoop is 1.0 m and the angle of the incline is 30°, we can calculate the height of the incline as h = (1.0 m) * sin(30°) = 0.5 m. Plugging these values into the potential energy formula, we get PE = (6.0 kg) * (9.8 m/s^2) * (0.5 m) = 29.4 J. Therefore, the hoop will travel up the incline until it reaches a height of 0.5 m before stopping and rolling back down.