Exciting Physics Problem: Stopping a Rolling Hollow Sphere
How much work is required to stop a hollow sphere rolling on a horizontal surface?
a) 750.5 Joules
How far along an incline will the sphere travel before stopping and rolling back down?
b) Approximately 6.63 meters
What are the differences between the results for a hollow sphere and a hoop with the same radius, mass, and initial speed?
c) The moment of inertia is less for the hollow sphere; therefore more work is required to stop it. Likewise, it rolls up the incline a shorter distance than the hoop.
Answers:
a) The work required to stop the sphere is 787.5 Joules.
b) The sphere will travel approximately 6.63 meters along the incline before stopping and rolling back down.
c) So, the moment of inertia is less for the hollow sphere; therefore more work is required to stop it. Likewise, it rolls up the incline a shorter distance than the hoop is the answer.
Let's look at the given problem step by step:
Given:- Hollow sphere mass: 7.0 kg
- Hollow sphere radius: 1.9 m
- Sphere's speed: 15.0 m/s
- Incline angle: 30°
- Initial speed up the incline: 15.0 m/s
a) Work to Stop the Sphere:
When the sphere rolls without slipping, its kinetic energy is given by the sum of translational and rotational kinetic energies:
KE = 1/2 * m * v^2 + 1/2 * I * ω^2
Substitute the given values to calculate the work required to stop the sphere which is 787.5 Joules.
b) Distance Traveled up the Incline:
Calculate the distance traveled up the incline using the conservation of mechanical energy and trigonometry.
The sphere will travel approximately 6.63 meters along the incline before stopping and rolling back down.
c) Differences from a Hoop:
The moment of inertia of a hollow sphere is greater than that of a hoop with the same mass and radius, leading to more work required to stop the hollow sphere. The hollow sphere also travels a shorter distance up the incline than a hoop with the same characteristics.