# Pedaling a Bicycle

#### The bicycle

When you are on a bicycle and you pedal, the chain transfers the rotation to the rear sprocket and from it to the rear wheel. This rear wheel typically has a mass of 1.8 kg and a radius of 35 cm and the distance sprocket-pedals is about 12 cm with a diameter of the pedals sprocket of 10 cm and of the rear sprocket of 4 cm. The person pedaling has 70 kg and accelerates toward a speed of 11 m/s.

a) Model the bicycle and find the inertia moment of the wheel.

b) Please find the change in momentum $L$ given the bycicle acceleration.

c) What torque is produced?

d) Please find time until the speed is $v=11 m/s$ and the acceleration. Explain.

#### The wheel inertia moment

a) The wheel of bicycle can be modeled as a circular ring (hoop) of mass M and radius R

$I = MR^2 =1.8*0.35^2 =0.2205 (kg*m^2)$

b)

the linear speed is bicycle is the same as the linear speed of a point at the exterior of one wheel.

$V =\omega*R$

The change in angular momentum for one wheel (from zero speed to V) is

$\Delta(L) =I*\omega =I*V/R$

For the entire bicycle (having 2 wheels) the change in angular momentum is

$Delta(L_{tot}) =2L =2I*V/R = 2*0.2205*11/0.35 =13.86 (kg*m^2/s)$

c)

The person is exerting toque on the main sprocket. the maximum torque exerted is (D is the distance to pedal from sprocket center)

$T =F*D =G*D =m*g*D =70*9.81*0.12 =82.4 (N*m)$

Assuming there are no losses in transmission of torque the minim time to accelerate until $V=11 m/s$ is

$T =\Delta(L)/time$

time $t = \Delta(L)/T = 13.86/82.4 =0.168 sec$

#### The speed and acceleration

d)

linear acceleration is $a =V/ t= 11/0.168 =65.4 m/s^2$

Explanation: This value is significantly larger than the real acceleration of a bicycle. This happens because when accelerating, the person needs not only to change the total angular momentum of the wheels, but also to change its own linear momentum (in other words the person needs to overcome its own inertia and self accelerate its own mass). The above model with just 2 weels that need to be accelerated from rest dramatically fails.