# Wheel versus Sphere: Traveling Linearly Down a Track

Assume the following:

• A tire-like wheel and sphere are both of diameter d and mass m
• They are both made of the same material
• They are both resting at the apex of an inclined track
• An equal amount of force is applied to both objects to send it from rest to moving downhill
• The wheel and sphere will travel linearly

How do the shapes and the shapes alone of these two objects impact it's speed, acceleration, and distance traveled? Clearly the surface area of the contact patch will differ greatly and thus the tire is likely to experience more friction, but I'm curious if the shapes have an impact on their travel.

Questions to Specify the Problem

"Do they roll without slipping?" – fqq

I'd say that the answer is probably enveloped within the solution. In other words, I don't know! Does the shape make a difference?

• Do they roll without slipping?
– fqq
Jun 8, 2016 at 16:59
• @fqq Great question. I added it as part of this answer; thank you. Jun 8, 2016 at 17:01
• en.wikipedia.org/wiki/… See the cute graphic to the right.
– Gert
Jun 8, 2016 at 17:17

## 2 Answers

Assuming rolling without slipping, this can easily be solved by means of Energy Conservation.

Let's assume the incline has a height $h$. During the travel down the incline, potential energy $U$ is then converted to kinetic energy $K$:

$K=U$

$K=mgh$

The kinetic energy $K$ is a combination of translational and rotational energy:

$\frac12mv^2+\frac12 I\omega^2=mgh$

With $I$ the moment of inertia of the rolling object about its rotation axis.

Rolling without slipping means:

$$v=\omega R$$ Note however that a wheel and a sphere, even of equal $R$ and $m$, do not have the same inertial moment $I$. Let 1 be the wheel and 2 be the sphere, then:

$$mgh=\frac12 mv_1^2+\frac12 mI_1R^2v_1^2$$ And: $$mgh=\frac12 mv_2^2+\frac12 mI_2R^2v_2^2$$

Inserting the resp. values of $I_1$ and $I_2$ then allows to calculate $v_1$ and $v_2$. It turns out that $v_1<v_2$, because a wheel, all other things being equal, has a higher inertial moment than a sphere.

Wikipedia has a nice graphic illustrating this problem.

The mass of the wheel is located farther from the axis of rotation, thus the moment of inertia is greater for the wheel. So it rotates slower with the same rotational energy and therefore the ball rolls faster down the slope than the wheel.