Need help for homework! Please answer these questions based on Calculus

They say once you learn to ride a bike, you never forget. In case you never learned, the basic principle is you apply force to a pedal which turns a gear that sets the bicycle in motion. (That, and try not to fall.) Three vectors govern this situation.

F: the force applied by the rider's foot to the pedal

r: the "radius" vector from the gear's center to the point on the pedal where

the force is being applied

a: a unit vector along the rotation axis of the gear

1) To represent these vectors we will need a co-ordinate system. Place the origin at the point where the pedal crank meets the gear. Let the z-axis run along the rotation axis of the gear. Let the x-axis run parallel to the ground with its positive direction agreeing with the forward motion of the bicycle. Finally the y-axis points upward toward the sky. This makes at least one of our vectors ridiculously easy.

2) The pedal crank is 9 inches long. A pedal is attached to the last two inches of the pedal crank.

The pedal is 4 inches wide. Assume the force is applied to the center of the pedal.

a) Let the pedal crank be parallel to the ground. Sketch a picture of the pedal, pedal crank, and axis of rotation on the xz-plane. Use this to help you find r when the pedal crank is parallel to the ground.

b) Assuming that the magnitude of F is fixed (by the strength of the cyclist) and the force is being applied straight down, find the torque . (For simplicity of notation use F to denote F.) Is the torque parallel to the axis of rotation of the gear?

c) Since the gear can only revolve around a, we are only interested in the effective part of the torque, which is the magnitude of the component of in the direction of a.

eff = | a|

Find the effective torque in the case of (b).

3) When riding a bicycle, the pedal crank does not stay in the same position. To express the changing nature of r, define to be the angle that the pedal crank makes with the vertical, measured clockwise.

a) Find r when the pedal crank is at an angle of from the vertical.

b) Using the same force as in (2), find eff(), the effective torque as a function of .