Question

When riding a 10-speed bicycle up a hill, a cyclist shifts the chain to a larger-diameter gear attached to the back wheel. Why is this gear preferred to a smaller gear? a. The torque exerted by the chain on the gear is larger. b. The force exerted by the chain on the gear is larger. c. You pedal more frequently to travel the same distance. d. Both a and c are correct.

Answer 1

To solve this problem we will use the concepts of the moment of rotational inertia, angular acceleration and the expression of angular velocity.

The rotational inertia is expressed as follows:

$I = \sum mr^2$

Here,

m = Mass of the object

r = Distance from the rotational axis

The rotational acceleration in terms of translational acceleration is

$\alpha = \frac{a}{R}$

Here,

a = Acceleration

R = Radius of the circular path of the object

The expression for the rotational speed of the object is

$\omega = \frac{\Delta \theta}{\Delta t}$

Here,

$\Delta \theta$ is the angular displacement of the object

The explanation by which when climbing a mountain uphill is changed to a larger pinion, is because it produces a greater torque but it is necessary to make more pedaling to be able to travel the same distance. Basically every turn results in less rotations of the rear wheel. Said energy that was previously used to move the rotation of the wheel is now distributed in more turns of the pedal. Therefore option a and c are correct.

This would indicate that the correct option is D.