Does Ramp Steepness Affect Sphere's Descent Time?

[Zmuffinz]

Homework Statement

A Block slides down a frictionless ramp and a sphere rolls without sliding down a ramp of the same angle. the block and sphere have the same mass, start at rest from point A and descend through point B

a) Consider the sphere rolling down the incline without sliding. If the roll is now repeated with an incline that is less steep but of the same height as the first incline, is the sphere's time to reach the botton greater than, less than, or the same as perviously? Justify your answer

Homework Equations

The Attempt at a Solution

I think the sphere would take more time in rolling down the ramp than the first time but then if the height is the same the gravitational pull doesn't change so it would be the same? I don't know slightly confused

 

[queenofbabes]

Consider the component of the gravitational pull acting along the slope. It is this component that is responsible for the sphere rolling down. How is this component related to the steepness of the slope?

 

[tomcue]

Your reasoning is correct. The sphere's time to reach the bottom of the incline will be the same regardless of the incline's angle, as long as the height remains the same. This is because the sphere's acceleration down the incline is determined by the force of gravity, which is constant and does not depend on the incline's angle. Therefore, the sphere's time to reach the bottom will also be constant. However, it is important to note that this assumes a perfectly round and frictionless sphere. In real-world situations, there may be slight variations in time due to factors such as rolling resistance and imperfections in the sphere's shape.