Final answer:
To calculate how far a hollow sphere rolls up an incline, conservation of energy is used, equating initial kinetic energy to maximum potential energy. The height gained is found first, then the distance up the incline is calculated using the sine of the incline angle.
Step-by-step explanation:
Calculating the Distance a Rolling Hollow Sphere Climbs an Incline
To find out how far up the incline the hollow sphere rolls before reversing direction, we can use the principles of conservation of energy. When the sphere reaches the incline, its kinetic energy starts converting into gravitational potential energy. The key is that no energy is lost since the sphere rolls without slipping, meaning there is no energy lost to heat through friction.
The initial kinetic energy of the hollow sphere is the sum of translational and rotational kinetic energy:
KEinitial = (1/2)mv2 + (1/2)Iω2, where m is the mass of the sphere, v is its velocity, I is the moment of inertia of a hollow sphere (I = (2/3)mr2), and ω is the angular velocity. Since the sphere is rolling without slipping, v = rω.
At the highest point on the incline, the sphere has only potential energy: PEmax = mgh, where h is the vertical height relative to the start height.
Setting the initial kinetic energy equal to the maximum potential energy and solving for h gives us the vertical height to which the hollow sphere will climb. However, we are asked for the distance along the incline, which can be found by using the relationship between the vertical height and the incline angle: d = h / sin(θ).
The specific calculations require numerical values for the mass and radius of the sphere, which are not provided; therefore, a numerical answer can't be given.