How Does Rolling Affect Kinetic Energy Distribution?

[FatElie]

Homework Statement

An 8 cm, 400 g sphere is released from rest at the top of a 2.1 m long, 25 degrees incline. It rolls, without slipping, to the bottom.
a. What is the sphere's angular velocity at the bottom of the incline?
b. What fraction of its kinetic energy is rotational?

Homework Equations

K rolling = (1/2) (I cm) (w^2) + (1/2) (M) (v cm^2)

The Attempt at a Solution

I know that the I cm for a sphere is 2/5 M R^2. I need to solve this equation for angular velocity but I can't seem to get the velocity of the centre of mass. I tried finding the potential energy and saying it all got converted to kinetic to find it but itgave me the wrong answer.

I found a, I just needed to substitute V cm = Rw to get the equation with only one unknown.
I'm still not sure about b though
Scratch that, got it. Thanks for your help

 

[anathema]

!

I would like to clarify some things about your solution. First, it is important to note that the equation K rolling = (1/2) (I cm) (w^2) + (1/2) (M) (v cm^2) is only applicable when the object is rolling without slipping. If there is slipping, then the equation would be different and it would be more complicated to solve.

To find the velocity of the center of mass, you can use the equation v cm = rw, where r is the radius of the sphere and w is the angular velocity. This equation is derived from the definition of angular velocity, w = v/r, where v is the linear velocity and r is the radius.

For part b, to find the fraction of kinetic energy that is rotational, you can use the equation for kinetic energy, K = (1/2)mv^2, and substitute in the values for the linear velocity and the angular velocity. Then, you can compare the rotational kinetic energy, (1/2)Icmw^2, to the total kinetic energy and find the fraction.

Lastly, it is always good practice to include units in your calculations and final answer, as it helps to ensure that your solution is correct.