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# Bicycle: Five equations relating to horizontal force, horizontal acceleration, net torque, angular acceleration

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A bicycle contains a big chain ring with 52 teeth and a radius rbc = 16 cm, with a pedal arm length rp = 19 cm. Attached to the rear wheel is a cog having a choice of gears ranging from 12 teeth to 21 teeth, and a corresponding cog radius rc which is proportional to the number of teeth (determine this value using the dimensions of the big chain ring). The rear wheel has a radius rw = 35 cm, a mass mw = 1.5 kg, and a moment of inertia Iw = mwrw2. The total weight of the bike is mb = 12 kg. A cyclist with a mass Mc = 70 kg is riding the bike. The combined moment of inertia of the big chain ring, pedal arm, and pedals is Ip.

When the pedal arms are horizontal to the ground the cyclist puts all of his weight on one of the pedals. The cyclist is starting from rest. Assume the wheels don't slip with respect to the ground, you will determine the acceleration of the bike when the chain is in the cog with 12 teeth.

(a) Write down five equations; one equating the net horizontal force to the horizontal acceleration, one relating the net torque on the rear wheel to it angular acceleration (?w), one relating the net torque on the chain ring/crankarm/pedal/ and its angular acceleration (?p), one relating the acceleration a and ?w, and one relating ?w and ?p.
(b) Solve the five equations symbolically and derive an expression for the acceleration in terms of rw, rp, rc, rbs, Fp, mw, Mc, mb, Iw, and Ip.
(c) Assuming Ip << Iw, simplify your previous expression, and then plug in the appropriate numbers to get a numerical value of your acceleration.
(d) Determine the acceleration numerically when the chain is in the cog with 21 teeth.
(e) If you put on a rear tire with a larger radius but the same mass, would the acceleration increase or decrease?
(f) How much of a force must be exerted on the pedal in order to get the rear wheel to peel out (slip). Assume the coefficient of static friction is ?s = .7.

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