A skier is pulled up a slope at a constant velocity by a tow bar. The slope is inclined at 25.9° 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 55.3 kg, and the coefficient of kinetic friction between the skis and the snow is 0.141. Calculate the magnitude of the force that the tow bar exerts on the skier.
To calculate the magnitude of the force that the tow bar exerts on the skier, we need to consider the forces acting on the skier.
1. Start by drawing a force diagram for the skier:
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/ | F_applied
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(skier)
2. Identify the relevant forces:
- The weight of the skier, W, acts straight down.
- The normal force, N, acts perpendicular to the slope.
- The force applied by the tow bar, F_applied, acts parallel to the slope.
- The force of kinetic friction, f_kinetic, acts parallel to the slope and opposes the motion.
3. Break the weight vector into its components:
The weight vector can be resolved into two components, one perpendicular to the slope (W_perpendicular) and one parallel to the slope (W_parallel).
- W_parallel = W * sin(θ), where θ is the angle of the slope (25.9°).
- W_perpendicular = W * cos(θ)
4. Calculate the magnitude of the force of kinetic friction, f_kinetic:
- f_kinetic = coefficient_kinetic_friction * N
The coefficient of kinetic friction is given as 0.141. To find N, we need to consider the equilibrium of forces perpendicular to the slope. Since the skier is moving at a constant velocity, the normal force is equal in magnitude to the weight perpendicular to the slope:
- N = W_perpendicular = W * cos(θ)
5. Now, we can find the magnitude of the force applied by the tow bar, F_applied:
Since the skier is moving at a constant velocity, the net force in the direction of motion is zero:
- F_applied - f_kinetic - W_parallel = 0
Substitute the known values into the equation:
- F_applied - (coefficient_kinetic_friction * N) - (W * sin(θ)) = 0
Simplify and solve for F_applied:
- F_applied = coefficient_kinetic_friction * N + W * sin(θ)
Substitute the values:
- F_applied = (0.141) * (W * cos(θ)) + (W * sin(θ))
6. Substitute the value of W into the equation:
- W = mass * gravitational_acceleration
Mass (m) of the skier is given as 55.3 kg, and the gravitational acceleration (g) is approximately 9.8 m/s².
- W = (55.3 kg) * (9.8 m/s²)
7. Calculate the magnitude of the force applied by the tow bar, F_applied:
Using the found values, substitute the values into the equation from step 5 and solve:
- F_applied = (0.141) * [(55.3 kg) * (9.8 m/s²) * cos(25.9°)] + [(55.3 kg) * (9.8 m/s²) * sin(25.9°)]
Calculating this equation will give you the magnitude of the force that the tow bar exerts on the skier.