A bicyclist is coasting straight down a hill at a constant speed. The mass of the rider and bicycle is 55.0 kg, and the hill is inclined at 16.0° 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. How much force (directed parallel to the hill) must be applied to the bicycle in order for the bicyclist to climb the hill?

force friction= forcegravity down hill= mg*SinTheta figure that out.

then, going up the hill, the rider has to over come gravity (same amount) and friction, or force= 2mgSinTheta

56

To determine the force required for the bicyclist to climb the hill, we need to consider the forces acting on the bicycle.

First, let's analyze the forces when the bicyclist is coasting down the hill:

1. Gravitational Force (Weight): This force is directed downward and can be calculated as the product of the mass (55.0 kg) and the acceleration due to gravity (9.8 m/s^2):
Weight = mass × acceleration due to gravity
= 55.0 kg × 9.8 m/s^2

2. Normal Force: The normal force is the force exerted by the hill on the bicycle perpendicular to the surface. Since the bicycle is on an inclined hill, the normal force can be calculated as:
Normal Force = Weight × cos(angle of incline)
= Weight × cos(16.0°)

3. Air Resistance: This opposes the motion of the cyclist, but since the problem states that the cyclist is coasting down the hill at a constant speed, we can assume that the air resistance is balanced by the gravitational force.

Now, let's consider the forces when the bicyclist climbs the hill at the same constant speed:

4. Force Applied (Parallel to the Hill): This is the force that needs to be applied to the bicycle to overcome the gravitational force (which is directed downward) and propel the bicyclist up the hill at a constant speed. This force can be calculated as:
Force Applied = Weight × sin(angle of incline)
= Weight × sin(16.0°)

Therefore, to climb the hill at the same constant speed, the bicyclist must apply a force parallel to the hill equal to Weight × sin(16.0°).

To determine the force required for the bicyclist to climb the hill, we can break down the problem into different forces acting on the bicycle.

1. Gravity force (Fg): This is the force pulling the bicyclist downward, calculated by multiplying the mass of the rider and bicycle (55.0 kg) by the acceleration due to gravity (9.8 m/s^2).
Fg = (mass) * (acceleration due to gravity)
Fg = 55.0 kg * 9.8 m/s^2

2. Normal force (Fn): Since the hill is inclined, the ground exerts a perpendicular force to support the weight of the bicyclist. The normal force is equal in magnitude but opposite in direction to the vertical component of the weight.
Fn = (mass) * (acceleration due to gravity) * (cosθ)
θ = 16.0°

3. Friction force (Ff): This is the force opposing the forward motion of the bicycle. The friction force can be calculated using the equation Ff = μ * Fn, where μ is the coefficient of friction between the bicycle tires and the surface of the hill. However, since the bicyclist is moving at a constant speed, the friction force is equal to zero.

4. Force applied to the bicycle (Fapp): This is the force that needs to be exerted parallel to the hill for the bicyclist to climb it at a constant speed.

Since the bicycle is moving at a constant speed, the net force in the horizontal direction is zero. Therefore, the force applied to the bicycle (Fapp) must equal the force of gravity component parallel to the hill.

The force of gravity component parallel to the hill can be calculated using the equation Fg_parallel = (mass) * (acceleration due to gravity) * (sinθ), where θ is the angle of the hill.

Now, we can substitute the values into the equations to find the force needed to climb the hill:

Fn = (mass) * (acceleration due to gravity) * (cosθ)
Fg_parallel = (mass) * (acceleration due to gravity) * (sinθ)
Fapp = Fg_parallel = Fn

Note: Since the hill is inclined at the same angle both while coasting down and climbing up, the force required to climb the hill at a constant speed is equal to the force of gravity component parallel to the hill while coasting down.

Substitute the values:
Fapp = Fg_parallel = Fn = (mass) * (acceleration due to gravity) * (sinθ)
Fapp = 55.0 kg * 9.8 m/s^2 * sin(16.0°)

Calculating this value using a calculator, the force required to climb the hill at a constant speed is approximately:

Fapp ≈ 161.5 N