# How to calculate force acting on axle from hub motor with known torque?

In some electric bikes, a motor is mounted to the hub of the front wheel. When turned on, this hub motor applies a torque, turning the wheel, reacting against the ground, with a resultant force acting against the inertia of bike / rider / air resistance etc.

How can I work out the force applied to the centre of the axle, given a motor acting with specific torque t and wheel or radius r, assuming the tyre does not overcome the friction force of the ground?

Lets say we fix the wheel, axle, and motor to a fork, that is anchored (so nothing can move). Lets assume the fork cannot flex.

Forgive my crude drawing, but this might look something like this: The Force of the motor acts on the rim, then the reaction force from the road acts on the fork (I think). The net foce is to the right hand of the page (where the motor is at the centre of the wheel).

Assuming the motor has a Torque of 40Nm, what is the force on the fork centred at the axle?

(I'm trying to determine if my fork can support a front motor, so the next stage will be to look at whether this fork can support the force at that distance)

Since the system is fixed in place, there is no acceleration or rotation. Therefore any torque developed by the motor is completely countered by friction with the ground, and that force is transmitted to the fork. So this is just $$\frac{40\rm{Nm}} {311\rm{mm}} = 129\rm{N}$$. If the wheel could rotate and the frame could accelerate, that force would be reduced.
• I gave your answer a plus, but you need to correct it by changing the denominator to 340mm+311mm=651mm, not 311mm. $\tau=F*r$, $F=\frac{\tau}{r}$ May 19 at 22:16