Final answer:
The hollow sphere's increased moment of inertia causes it to travel a greater distance and take more time to come to rest on an inclined plane as compared to a solid sphere or marble.
Step-by-step explanation:
When the preceding problem's marble is replaced with a hollow sphere, the new result shows an increase in both the distance traveled up the plane and the time taken before coming to rest. The hollow sphere has greater moment of inertia when compared to a solid sphere or marble, which means that it resists changes to its rotational state more significantly. This resistance to rotational acceleration results in a longer travel duration and greater distance before stopping when ascending an incline.
To calculate the distance (x) and the time (t), we use the equations derived from kinematics and rotational dynamics. For a hollow sphere, we have:
v² = v₀² − 2acmx ⇒ x = (v₀²) / (2g sin θ)
t = (v − v₀) / acm ⇒ t = v₀ / (g sin θ)
In these equations, v is the final velocity (zero at rest), v₀ is the initial velocity, g is the acceleration due to gravity, θ is the incline angle, and acm is the component of gravitational acceleration along the incline. For a hollow sphere rolling without slipping, the acceleration down an incline is g/(1+k²/r²) sin θ, with k being the radius of gyration and r the sphere's radius; k²/r² for a hollow sphere is 2/3, altering the acceleration experienced compared to a solid sphere for which k²/r² is 2/5.