a 40 kg skier skis down a frictionless slope angled at 10 degrees to the horizontal. Assume the skier moves in the negative direction of an x axis along the slope. A wind force with component Fx acts on the skier. What is Fx is the magnitude of the skier's velocity is (a) constant, (b) increasing at a rate of 1.0 m/s^2, and (c) increasing at a rate of 2.0 m/s^2?
To find the value of Fx in each scenario, we need to analyze the forces acting on the skier along the slope.
Let's break down the forces acting on the skier:
1. Gravitational force (mg): The weight of the skier acts vertically downward and can be expressed as mg, where m is the mass of the skier and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Normal force (N): The normal force acts perpendicular to the slope and cancels out the component of the gravitational force that is perpendicular to the slope. It can be calculated as N = mg * cos θ, where θ is the angle of the slope (10 degrees).
3. Force due to the wind (Fx): This is the force we need to find.
In all scenarios, the skier is moving downward along the slope, so the acceleration due to gravity (g) is in the negative y-direction.
(a) When the magnitude of the skier's velocity is constant, it means the skier is moving with a constant speed. Since there is no acceleration in the x-direction (ax = 0), all the forces must balance out. Therefore, Fx = 0.
(b) When the skier's velocity is increasing at a rate of 1.0 m/s^2, we have the acceleration in the x-direction (ax = 1.0 m/s^2). In this case, the net force in the x-direction can be calculated using Newton's second law: Fx = m * ax.
Fx = (40 kg) * (1.0 m/s^2)
Fx = 40 N
So, when the skier's velocity is increasing at a rate of 1.0 m/s^2, the magnitude of the force due to the wind (Fx) is 40 N.
(c) Similar to scenario (b), when the skier's velocity is increasing at a rate of 2.0 m/s^2, we have ax = 2.0 m/s^2. Again, using Newton's second law:
Fx = m * ax
Fx = (40 kg) * (2.0 m/s^2)
Fx = 80 N
Therefore, when the skier's velocity is increasing at a rate of 2.0 m/s^2, the magnitude of the force due to the wind (Fx) is 80 N.