Shown above is a very old type of bicycle. A rider (with mass m = 65 kg) sits on

Question

Shown above is a very old type of bicycle. A rider (with mass m = 65 kg) sits on the seat (which is a distance x = 0.4 m behind the handlebars) and turns the pedals (which are a distance L = 0.5 m apart). The pedals are directly attached to the axle of the front wheel (mass of the wheel is 10 kg) so that each time the pedals make one revolution the front wheel also makes one revolution. Assume both wheels roll without slipping. In this problem lets assume that the rider pushes only one pedal at a time (the force their foot puts on the other pedal is zero). It is hard to turn pedals faster than about 100 rev/min so lets assume that this is the maximum angular velocity of the pedals. We are going to calculate how long it takes the rider to attain maximum speed. a) What is the maximum speed of the bicycle? b) Now the rider is pedaling as hard as they can to get the bicycle up to its maximum speed. Take the entire bicycle and the person to be a single object and draw a free body diagram for that object. Neglect rolling friction on the rear tire and neglect air resistance. c) Use Newton?s 2nd Law to write an equation for the translational acceleration of the bicycle. Neglect the mass of the bicycle frame and the rear wheel. d) Almost all of the mass of the front wheel is at the rim (at the outside diameter) of the wheel so estimate the moment of inertial of the front wheel. e) Make a reasonable estimate for the maximum force that the rider can apply to one pedal. f) Think of the front wheel as a separate object and draw an extended free-body diagram for the front wheel when the rider is pushing as hard as they can straight down on one pedal and the bicycle is accelerating forward. g) Use the Newton?s 2nd Law for rotational motion to write an equation for the angular acceleration of the front wheel. h) Now use the two Newton?s 2nd Law equations that you have written (together with the ?roll without slipping? equation) to solve for the acceleration of the bicycle when the rider is trying as hard as possible. i) Find the time the rider takes to get to their maximum speed. j) If the rider just stops pedaling now and puts all of their weight on the seat, find the normal forces by the ground on the two wheels. (Hint, think of your object as the bicycle and draw the extended free- body diagram for the bicycle including forces by the road, by the earth, and by the rider).

Explanation / Answer

100 rev /min =10.47 rad/sec for small wheel as rolling is there v =wr =10.47*0.2 =2.09 m/s hence max speed of cycle =v =2.09 m/s

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