# Rotational Motion-Up an Incline

• abspeers
In summary, three objects - a solid sphere, thin hoop, and solid disk, all with the same radius - are allowed to roll down an inclined plane without slipping and then up another inclined plane. The question is which object will travel the furthest up the second incline, neglecting frictional forces. While a previous question found that the sphere reaches the bottom first, the correct answer for the furthest distance traveled is the object with the greatest combination of kinetic and potential energy at the start of the motion. Therefore, the solution is not based on the kinetic energy at the bottom of the first incline, as initially suspected.
abspeers

## Homework Statement

A solid sphere (S), a thin hoop (H), and a solid disk (D),
all with the same radius, are allowed to roll down an
inclined plane without slipping. At the bottom, they run into another inclined plane and begin rolling upwards. Which will travel the furthest up the inclined plane, neglecting all frictional forces?

Thin Hoop KE=1/2mv^2 + 1/2(MR^2)w^2

Solid Sphere KE=1/2mv^2 + 1/2(2/5MR^2)w^2

Solid Disk KE=1/2mv^2 + 1/2(1/2MR^2)w^2

## Homework Equations

1/2mv^2 + 1/2Iw^2 = mgh

## The Attempt at a Solution

A previous question had asked which reaches the bottom first and I had found that to be the sphere, followed by the disk, and finally the thin hoop. I suspect that the sphere would travel the furthest up the incline because it has the greatest KE at the bottom of the first incline, but I am unsure if this assumption is correct.

Thank you for the help!

Welcome to PF!

Hi abspeers! Welcome to PF!
abspeers said:
… I suspect that the sphere would travel the furthest up the incline because it has the greatest KE at the bottom of the first incline, but I am unsure if this assumption is correct.

You're right, it's not correct …

never mind the KE at the bottom, its the KE and PE at the start that matters, isn't it?

Your assumption is correct. The sphere will travel the furthest up the incline due to its greater kinetic energy at the bottom of the first incline. This is because the sphere has both translational and rotational kinetic energy, while the hoop and disk only have rotational kinetic energy. Therefore, the sphere has a greater total kinetic energy and will be able to overcome the gravitational potential energy to travel further up the incline. Additionally, the moment of inertia for a solid sphere is smaller than that of a disk or hoop, meaning it requires less energy to rotate and therefore has more energy available for translational motion.

## What is rotational motion?

Rotational motion is the movement of an object around an axis or center point. It can be described in terms of its angular displacement, angular velocity, and angular acceleration.

## What is an incline?

An incline is a surface that is angled or sloped, rather than flat. In the context of rotational motion, an incline is often used to describe a ramp or surface that an object is moving up or down.

## How does rotational motion on an incline differ from flat ground?

On an incline, the force of gravity is acting in a different direction, causing the object to accelerate down the incline. This affects the object's angular velocity and angular acceleration, as well as the amount of work and energy required to move the object.

## What factors affect rotational motion up an incline?

The angle of the incline, the mass and distribution of mass of the object, and the coefficient of friction between the object and the incline's surface can all affect rotational motion up an incline.

## How is rotational motion up an incline calculated?

To calculate rotational motion up an incline, you will need to use principles of rotational kinematics and dynamics, as well as trigonometry. This involves finding the object's angular velocity and acceleration, calculating the net torque acting on the object, and using the equations of motion to determine the object's motion along the incline.

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