(Mechanics) The Bicycle

 A student rides a bicycle on a slope of inclination θ. Due to air drag, he found that the bicycle can barely move down the slope without his pedaling. He would like to estimate the power he needs to drive the bicycle up the same slope at a uniform velocity. To achieve this, he measured that during the up-slope drive, one of his feet pedaled N cycles in a time interval T (assuming that the pedaling is continuous and at a uniform rate). He also obtained the following data: the total mass of the bicycle and the rider m, length of pedal crank L, radius of gear 1 R1 , radius of gear 2 R2 , radius of rear wheel R3 , as shown in the figure.
It is given that the air drags during the up-slope and down-slope drives have the same magnitude, and there are no slippings between the wheels and the slope during both the up-slope and down-slope drives. The energy loss due to the relative motion of the bicycle components is negligible.
(a) Derive an expression for the force needed to drive the bicycle up-slope at uniform velocity.
(b) Derive an expression for the power needed to drive the bicycle up-slope at uniform velocity.