[Physics] Torque on Axle and Wheel

Question

I was solving a physics problem when I ran into the following conundrum.

Consider a wheel attached to an axle. The wheel has a radius of $R$, which is twice the radius of the axle, $r$. Assume both wheel and axle have the same mass. We spin the axle (through the horizontal axis) by applying a force on it. Now as a result, the axle starts moving with an angular acceleration of $\alpha$. Additionally, the wheel also starts moving with an angular acceleration of $\alpha$.

Thus, the axle has a torque equal to $I_{axle} \alpha$, while the wheel has a torque equal to $I_{wheel} \alpha$. We can simplify this as follows:

$I_{wheel} = mR^2$

$I_{axle} = mr^2$

Thus, we see that the two torques are indeed different. However, the solution that I was reading stated that the torque on a wheel would always be equal to the torque on its axle. But with the logic we just used, we prove otherwise. What have I missed?

Answer 1

Thus, we see that the two torques are indeed different. However, the solution that I was reading stated that the torque on a wheel would always be equal to the torque on its axle. But with the logic we just used, we prove otherwise. What have I missed?

It is correct the torques would be different if you were to consider the axle and the wheel as two separate objects. But they are not separate objects. They comprise one object subjected to a single torque. What is different is that the force on the wheel is less than that on the axle because torque equals the force applied at the radius, times the radius.

Thus for the wheel (outer) portion we have torque

$$T_{wheel}=F_{wheel}R$$

And for the axle (inner portion) we have torque

$$T_{axle}=F_{axle}r$$

The two torques are equal so we have

$$F_{wheel}R=F_{axle}r$$

or

$$F_{wheel}=\frac{F_{axle}r}{R}$$

So the force at the maximum radius of the wheel is less than the force at the radius of the axle to achieve the same torque. This tells us we can obtain the same torque by increasing the force tangent to the radius linearly with the radius;

We can come to the same conclusion using formulas for torque in terms of the moment of inertia times angular acceleration. First for the wheel.

$$T_{wheel}=mR^{2}α=F_{wheel}R$$

Therefore

$$F_{wheel}=mRα$$

Likewise for the axle

$$T_{axle}=mr^{2}α=F_{axle}r$$

Therefore

$$F_{axle}=mrα$$

Again since the torques are the same

$$F_{axle}r=F_{wheel}R$$

or

$$F_{wheel}=\frac{F_{axle}r}{R}$$

as before.

Hope this helps.