If we take a sufficiently large axle for a wheel then is it possible that the top of the wheel would have thrice the velocity of its axle?

Question

I recently learnt that the top of a wheel has velocity twice that of its axle. In all such cases of why this is so the axle is always considered to be very small in comparison to the wheel. Thus, I thought of a case where the axle has a radius $r$ and the wheel has a radius $R$ = $3r$.

Now, while rolling both of the axle and wheel would have the same angular velocity, $w$, as they would trace the same amount of angles. Let the axle and the wheel have tangential velocities, $v$ and $V$ respectively, then we can say,

$$ w = w$$ $$ \frac{v}{r} = \frac{V}{R}$$ $$ \frac{v}{r} = \frac{V}{3r}$$ $$ V = 3v $$

Now, I'm not sure if this is true so please help me understand if this true or what lapse is there in my understanding. Also, if it is true then please enlighten me if such a wheel would have any physical significance.

Answer 1

The tangential velocity at the rim of the wheel is twice the velocity of the axle with respect to the ground. This velocity does not depend on the diameter of the axle. The tangential velocity of the axle is irrelevant for the effect, but as your calculation shows, any ratio of tangential velocities can be achieved by setting the radii to have the same ratio.

Answer 2

Look at this from the frame of reference that moves with the center of the axle. In this frame, the center of the axle is stationary, the top of the axle move forward at $v$, and the top of the wheel moves forward at $V = 3v$. The ground moves backward at $V$.

Switch to a frame of reference that moves with the ground. In this frame, you add $V$ to all velocities from the first frame. The ground moves at $0$, the center of the axle moves at $V$, and the top of the wheel moves at $2V$.

So both statements are right, in the right frame of reference.

Answer 3

Oops, you seem to have misunderstood the concept. I assume you're talking about a wheel that is in a condition of pure rolling (i.e., it is not sliding at its point of contact with ground).

Now, let us take a looks at the dynamics of a rolling wheel. When a wheel is undergoing pure rolling motion, there are two velocities it has. A linear velocity, the equal velocity which every point on the wheel has and the velocity due to rotation. This is a tangential velocity ${\omega*r}$ where $r$ is the distance of the point from its center of rotation, which is the axle. So, the axle does not have any tangential velocity, it only has a linear velocity.

The point of contact with the ground has zero velocity (condition to not slide). So, the tangential velocity and linear velocity and tangential velocity of rim cancel out. Let us take $v$ as the linear velocity.

Then ${\omega*R=v}$ (where $R$ is the radius of wheel)

Now, the entire rim of the wheel will have the same tangential velocity. So, at the top, the linear velocity adds up with the rim velocity. So, ${v+\omega*R}$ which is ${2*v}$. So, it doesn't matter what the radius of the wheel is, the top of the wheel will always have twice the velocity of the axle.