A skier is pulled up a slope at a constant velocity by a tow bar. The slope is inclined at 22.2° with respect to the horizontal. The force applied to the skier by the tow bar is parallel to the slope. The skier's mass is 56.0 kg, and the coefficient of kinetic friction between the skis and the snow is 0.138. Calculate the magnitude of the force that the tow bar exerts on the skier.
To find the magnitude of the force that the tow bar exerts on the skier, we need to consider the forces acting on the skier.
1. Gravitational force (Fg):
The force due to gravity acts straight downward and can be calculated using the formula Fg = mg, where m is the mass of the skier and g is the acceleration due to gravity. In this case, the mass of the skier is given as 56.0 kg, and the acceleration due to gravity is approximately 9.8 m/s^2.
Fg = (56.0 kg) * (9.8 m/s^2) = 548.8 N
2. Normal force (Fn):
The normal force acts perpendicular to the surface of contact and balances the gravitational force. On an inclined plane, the normal force reduces due to the angle of inclination. The normal force can be calculated as Fn = mg * cos(θ), where θ is the angle of inclination. In this case, θ is given as 22.2°.
Fn = (56.0 kg) * (9.8 m/s^2) * cos(22.2°) = 518.8 N
3. Force of kinetic friction (Fk):
The force of kinetic friction opposes the motion and can be calculated using the formula Fk = μk * Fn, where μk is the coefficient of kinetic friction. In this case, μk is given as 0.138.
Fk = (0.138) * (518.8 N) = 71.5 N
4. Force exerted by the tow bar (Ft):
The force exerted by the tow bar pulls the skier up the slope. Since the skier is moving up the slope at a constant velocity, the force exerted by the tow bar must be equal in magnitude but opposite in direction to the sum of the gravitational force and the force of kinetic friction.
Ft = Fg + Fk = 548.8 N + 71.5 N = 620.3 N
Therefore, the magnitude of the force that the tow bar exerts on the skier is 620.3 N.