# Rotational Kinetic Energy of sphere

• latitude
In summary, a 4 kg solid sphere rolls up an incline with an angle of 30 degrees without slipping. At the bottom of the incline, its center of mass has a translational speed of 5 m/s. The total kinetic energy of the sphere at the bottom is 70 J. It travels 1.43 m up the incline before coming to rest and starting to roll back down. The answer does not depend on the mass.
latitude

## Homework Statement

A solid sphere of mass 4 kg rolls w/o slipping UP an incline with an angle of 30 degrees. the radius of the sphere is 0.5 m and its moment of Inertia is I = 2/5(m)R^2. At the bottom of the incline the center of mass of the sphere has a translational speed of 5 m/s.
a) What is the total kinetic energy of the sphere at the bottom of the incline?
b) How far does the sphere travel up the incline before coming to rest and starting to roll back down?
c) Does the answer to b) depend on the mass?

## The Attempt at a Solution

a) K = Krotational + Ktranslational
= 1/2Iw^2 + 1/2mv^2
w = v/R = 5/0.5
= 1/2(2/5)(4)(o.5^2)(10^2) + 1/2(4)(5^2)
= 70 J

b) Kf + Uf = Ki + Ui
0? (Not sure about this, because not sure if there is still some rotational kinetic energy?) + mgh = 70 J + 0J
(5)(9.8)(h) = 70
h = 1.43 m

h = xsin30
1.43 = xsin30
x = 2.86 m

c) So this is where I screw up... because I know from my theory classes that the answer SHOULDNT depend on the mass because all spheres roll down the same regardless of mass... but mine does so I think it's wrong. Thanks everyone!

b.) When the sphere starts to roll back down, it means that it has lost all its kinetic energy to potential energy

c.) no, because when you do b, you will notice that m cancels out from the equation. Do part b and you will udnerstand.

latitude said:
Since the sphere is rolling without slipping, if it moves, it has to spin. That's why both the translational movement and rotation stops at the same time.

c) So this is where I screw up... because I know from my theory classes that the answer SHOULDNT depend on the mass because all spheres roll down the same regardless of mass... but mine does so I think it's wrong. Thanks everyone!

How do you know that your answer does depend on the mass? This is what happens if you plug in numerical values early in the problems. As mentioned by Oerg, do it taking m as the mass and see whether it cancels out.

## 1. What is rotational kinetic energy?

Rotational kinetic energy refers to the energy possessed by an object due to its rotation around an axis.

## 2. How is rotational kinetic energy calculated?

The formula for calculating rotational kinetic energy is KE = 1/2 * I * ω^2, where KE is the kinetic energy, I is the moment of inertia, and ω is the angular velocity.

## 3. What factors affect the rotational kinetic energy of a sphere?

The rotational kinetic energy of a sphere is affected by its mass, radius, and angular velocity. The moment of inertia also plays a role, which is influenced by the shape and distribution of mass within the sphere.

## 4. What are some real-life applications of rotational kinetic energy?

Rotational kinetic energy is used in various applications, such as spinning tops, gyroscopes, and flywheels in engines. It is also important in sports, such as figure skating and gymnastics, where athletes utilize rotational kinetic energy to perform spins and flips.

## 5. Can rotational kinetic energy be converted into other forms of energy?

Yes, rotational kinetic energy can be converted into other forms of energy, such as thermal, electrical, or potential energy, depending on the situation. For example, when a flywheel slows down, its rotational kinetic energy is converted into heat energy due to friction.

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