A bicyclist is pedaling along a road that is sloped upward with angle theta. At which of these road elevation angles must the bicycle wheels exert the greatest force back on the road to move the bicycle forward at the same constant speed?
To determine the road elevation angle at which the bicycle wheels exert the greatest force back on the road, we need to consider the forces acting on the bicycle.
When the bicycle is on an upward slope, there are three main forces acting on it: the force of gravity, the normal force, and the force of friction.
1. Force of gravity (Fg): This force pulls the bicycle and the cyclist downward. It is equal to the product of the mass of the bicycle and the acceleration due to gravity (mg).
2. Normal force (Fn): This force acts perpendicular to the surface of the road and balances the force of gravity. It is equal in magnitude but opposite in direction to the force of gravity.
3. Force of friction (Ff): This force acts parallel to the surface of the road and opposes the motion of the bicycle. It is responsible for the bicycle's forward movement.
To find the angle at which the bicycle wheels exert the greatest force back on the road, we need to analyze the components of the gravitational force and determine when the force of friction is maximum.
Let's break down the weight component of the force of gravity parallel to the road surface:
Fg_parallel = Fg * sin(theta)
Where theta is the road elevation angle.
The force of friction depends on the coefficient of friction (μ) between the tires and the road. The maximum force of friction is given by:
Ff_max = μ * Fn
Now, since Fn is equal in magnitude to Fg (balancing it), we can rewrite the equation as:
Ff_max = μ * Fg
The force of friction is also what propels the bicycle forward. So, the greatest force of friction occurs when the force of friction is at its maximum. For this to occur, the force of gravity parallel to the road (Fg_parallel) must be maximized.
To maximize Fg_parallel, sin(theta) must be maximized, which occurs when theta is equal to 90 degrees (θ = 90°). At this angle, the bicycle is completely vertical, meaning it is facing straight uphill.
Therefore, the road elevation angle at which the bicycle wheels exert the greatest force back on the road to move the bicycle forward at the same constant speed is 90 degrees (θ = 90°).