A bicyclist is coasting straight down a hill at a constant speed. The mass of the rider and bicycle is 80 kg and the hill is inclines at 15 degrees with respect to the horizontal. Air resistance opposes the motion of the cyclist. Later, the bicyclist climbs the same hill at the same constant speed (so same force of air resistance). How much force (directed parallel to the hill) must be applied by friction to the bicycle in order for the bicyclist to climb the hill?
To find the force of friction required for the bicyclist to climb the hill, we need to first understand the forces acting on the bicycle.
When the bicyclist is coasting down the hill, there are three main forces acting on the bicycle: the force of gravity, the force of air resistance, and the force of friction.
1. The force of gravity, which acts vertically downwards, can be broken down into two components: the perpendicular component (mg*cosθ) and the parallel component (mg*sinθ). Here, θ represents the angle of incline of the hill.
2. The force of air resistance opposes the motion of the bicycle and acts in the opposite direction of its velocity. Since the bicycle is coasting at a constant speed, the force of air resistance is equal in magnitude but opposite in direction to the force of friction when climbing the hill.
3. The force of friction opposes the motion of the bicycle, acts parallel to the hill, and is responsible for providing the necessary force to move up the hill.
Since the bicyclist is climbing the hill at the same constant speed, the force of air resistance is equal in magnitude to the force of friction. Therefore, the force of air resistance is equal to the force of friction required to climb the hill.
Now, to find the force of friction, we need to consider the parallel component of the force of gravity (mg*sinθ) and the force of air resistance, which are equal:
force of friction = mg*sinθ
To calculate the force of friction, we need to know the mass (m) of the rider and bicycle, and the angle of incline (θ) of the hill.
Using the values you provided:
mass of rider and bicycle (m) = 80 kg
angle of incline (θ) = 15 degrees
force of friction = (80 kg) * (9.8 m/s^2) * sin(15 degrees)
Calculating this equation will give us the force of friction required for the bicyclist to climb the hill.