# Rotational Inertia Physics C

• Benton
In summary, the conversation discusses a solid sphere rolling without slipping on a rough surface, while a ramp rests on a smooth surface and is free to slide. It asks for the speed of the ball/ramp at the highest point, the distance the ball rolls up the incline, and the speeds of the ball and ramp after the ball rolls back down. The solution involves using conservation of momentum and energy, and the final answer for (b) is L=(8v2/15g) cscΘ. For (c), you will need to consider energy and momentum in the frame where the ramp and ball start at rest, and there may be losses due to internal friction.

#### Benton

1. Homework Statement

A solid sphere (mass of m, radius of r and I=2/5 mr2) is rolling without slipping on
a rough surface with a speed of v. A ramp (mass of 2m and angle of θ) rests on a
smooth surface and is free to slide on the surface. As the ball rolls up the ramp, the
ramp begins to move. Provide all answers in terms of the given variables and any
fundamental constants.

A. What will be the speed of the ball/ramp when the ball reaches its highest point on
the ramp?

B. What distance L will the ball roll up the incline?

C. What will be the speeds of the ball and the ramp after the ball rolls back down off
of the ramp?

## Homework Equations

Conservation of Momentum, Conservation of Energy

## The Attempt at a Solution

A. mv=(m+2m)v
mv=3mv
vf=1/3v

B.
we now have vf which is v/3 we can solve

ΣEi=1/2 Iω2+1/2mv2=

1/2(2/5mr2)(v/r)2+1/2mv2=

1/5mr2+1/2mv2=7/10mv2

ΣEf=mgh+1/2*3mvf2=

mgh+1/2*3m(v/3)2=

mgh+1/6mv2=7/10 mv2

h=8v2/15g

L=(8v2/15g) cscΘ

c. I would like guidance on how to solve c.
I know it uses energy, but I'm stuck on this one

Last edited:
Benton said:
c. I would like guidance on how to solve c.
I know it uses energy, but I'm stuck on this one
Once again, momentum and energy are conserved.

You'll need both energy and momentum for (c). It is probably easier to start in the frame where (ramp+ball) start at rest.

Technical detail for (b): It is beyond the scope of this problem, but in general you will get losses due to internal friction when a ball hits an incline like that.

## 1. What is rotational inertia in physics?

Rotational inertia, also known as moment of inertia, is a physical property of an object that describes its resistance to changes in rotational motion. It is similar to the concept of inertia in linear motion, and is dependent on the object's mass and distribution of mass around its axis of rotation.

## 2. How is rotational inertia different from linear inertia?

In linear motion, inertia is the tendency of an object to resist changes in its velocity. In rotational motion, inertia is the tendency of an object to resist changes in its rotational speed and direction. While linear inertia is dependent on an object's mass, rotational inertia is also dependent on its mass distribution.

## 3. What factors affect rotational inertia?

The factors that affect rotational inertia include the mass of the object, the distribution of mass around its axis of rotation, and the distance of the mass from the axis of rotation. Objects with a larger mass and a larger distance from the axis of rotation will have a higher rotational inertia.

## 4. How is rotational inertia calculated?

Rotational inertia is calculated using the formula I = mr², where I is the moment of inertia, m is the mass of the object, and r is the distance from the axis of rotation to the mass. This formula can be used for simple objects with a single axis of rotation, but more complex objects may require integration to calculate their moment of inertia.

## 5. How does rotational inertia affect the motion of objects?

Rotational inertia affects the motion of objects by determining how easily they can be rotated or moved in a circular motion. Objects with a higher rotational inertia will require more force to change their angular velocity, while objects with a lower rotational inertia will be easier to rotate. This property is important in understanding the behavior of objects such as spinning tops, bicycles, and planets in orbit.