Based on the force on the rear gear $F_t$ and the radii of the rear gear $R_g$ and rear wheel $R_w$ , how would you calculate the force the bicycle exerts on the ground when it is accelerating?
I would have thought it would be $$F_t \times (R_g/R_w)$$ but this would mean that the torque of the chain $$F_t\times R_g$$ would equal the torque of static friction $$F_t \times (R_g/R_w)\times R_w = F_t\times R_g$$That obviously cannot be the case if the wheel is angularly accelerating.
So how would you quantify the lag between the two torques?
I have a similar question about the pedal sprocket: how do you calculate the force the pedal sprocket puts on the chain based on the toque on the pedal?
Best Answer
If we take the bicycle as a whole the only force that making the bicycle accelerate is the frictional force applied on the wheels. For the moment lets forget the front wheel. Then, $$F_f = Ma$$
Since the wheel is not slipping, for the back wheel $$F_t R_g - F_fR_w = I \alpha = I\frac{a}{R_w}$$
With the use of these equations $F_f$ and $a$ can be quatified