8. Consider a bicycle as discussed in lecture. Assume that the bike, including t

Question

8. Consider a bicycle as discussed in lecture. Assume that the bike, including the wheels and you as the rider, has mass M; the gear attached to the back wheel is massless and has radius r; and the wheels each have mass m, radius R, and moment of inertia-mR2. You can also assume that only the top length of the chain is under tension (not completely realistic, but good enough for us) Compute the following quantities for an accelerating bicycle at an instant when th the chain on the gear is F and the bike is moving at speed v a. The acceleration of the bike, and the forces due to friction between the each wheel and the road b. The power delivered to the bike in the frame of the road and in the frame of the bike. Proceed by finding the kinetic energy in the appropriate frame, taking its time derivative, and plugging in your result for acceleration from part a c. The power delivered to the bike in its own frame. Proceed by constructing the product Fj v of each force Fj and the velocity vj of the object to which it is applied, and adding them up. At the edge of the gear, for example, the relevant velocity is the velocity of the edge of the gear, not its center-of-mass velocity (which is zero in this frame), and for the friction forces you should use the speed of the edge of the wheel (which is not zero in this frame) d. The power delivered to the bicycle in the frame of the road, again from Fj v,. Now, of course, the velocities are different You will probably find that things don't add up. To track down the missing contribution, it may be helpful to pretend that this is an electric bike, where the chain is driven by a motor attached to the frame, and to assume that the motor itself is massless (so that you don't have to redo your earlier work). This is easier to deal with as a source of power than a person, who is in contact with the bike in many places and has significant mass

Explanation / Answer

8. mass of bike and rider = M

radius of gear = r

mass of each wheel = m

radius of each wheel = R

moment of inertia of each wheel, I = mR^2

Force exerted by the chain on the gear = F

bike speed = v

a. let force of friction be f

then from force balance

2f = M*a ( where a is acceleration of the bicycle)

from torque balance

F*r - f*R = I*a/R = mRa

f = (Fr/R - ma)

hence

a = 2f/M = 2(Fr/R - ma)/M

f = (Fr/R - 2m(Fr/R - ma)/M)

b. KE in the frame of road = 0.5Mv^2 + 2*0.5*I*(v/R)^2

KEr = 0.5Mv^2 + mv^2

Pr = dKEr/dt = Mvdv/dt + 2mvdv/dt = (Mv + 2mv)a = (Mv + 2mv)2(Fr/R - ma)/M

KE in bike frame = 2*0.5*I*(v/R)^2

Power = (2mv)2(Fr/R - ma)/M

c. P = F*w*r - f*w*R

now, w = v/R

hence

P = F*vr/R - (Fr/R - 2m(Fr/R - ma)/M)*v

d. Pr = 2fv = 2*(Fr/R - 2m(Fr/R - ma)/M)*v

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