Angular Kinetics

Question

A mountain biker tries to start pedalling in the mud. The total mass (80 kg, bike + biker) is distributed equally between the front and rear wheels. There is a loss of 10% of moment of force between pedal and the rear wheel. The diameter of wheels is 70 cm, and the static friction coefficient between tyre and the mud is 0.3. The pedal angle is 25° from the vertical and has a length of 20 cm. The force exerted by the cyclist is vertical. What is the maximal force that the biker can produce without skidding?

Answer 1

Let

m = 80 kg be the weight of the biker plus bike,

g = 9.81 m/s² be gravitational acceleration,

f = 0.3 be coefficient of friction,

R = 70 cm = 0.7 m be the diameter of the wheels,

r = 20 cm = 0.2 m be the length of the pedal,

α = 25° be the angle of the pedal from the vertical,

F (unknown) be the force the biker exerts,

k = 10% be the loss of the moment of force between the pedal and rear wheel.

The portion of the total weight pushing down on the real wheel is

mg/2

That causes force of friction

mgf/2

Skidding would occur is the force of friction were exceeded by the the force

exerted on the wheel by the biker.

Let's calculate that.

The moment of force exerted by the biker on the pedal is

Frsin(α)

That transfers a moment of force on the rear wheel of magnitude

Frsin(α)(1-k)

That exerts force at the edge of the wheel of magnitude

Frsin(α)(1-k)/R

As explained above, that force can be at most as big as the force of friction

Frsin(α)(1-k)/R = mgf/2

From that we get the maximum force on the pedal

F = mgfR/2rsin(α)(1-k)

Substituting actual numbers

F = 80×9.81×0.3×0.7/(2×0.2×sin(25°)×0.9) = 1083 N

The maximal force, 1083 N, which exceeds his weight, so he need not be careful.