a rope pulls a 82.5 kg skier at a constant speed up a 18.7 degree slope with uk equals 0.150. How much force does the rope exert
To calculate the force the rope exerts on the skier, we need to consider the forces acting on the skier along the incline.
1. Determine the weight of the skier:
The weight, due to gravity, can be calculated using the equation:
Weight = mass * gravity
Given that the mass of the skier is 82.5 kg and the acceleration due to gravity is approximately 9.8 m/s², the weight is:
Weight = 82.5 kg * 9.8 m/s² = 808.5 N
2. Calculate the component of the weight parallel to the incline:
The component of the weight parallel to the incline is:
Weight_parallel = Weight * sin(θ)
θ is the angle of the slope, which is 18.7 degrees in this case.
Weight_parallel = 808.5 N * sin(18.7°) = 270.2 N
3. Calculate the force of friction:
The force of friction can be calculated using the equation:
Force_friction = coefficient_of_friction * Weight_normal
The normal force (perpendicular to the incline) can be determined as:
Weight_normal = Weight * cos(θ)
Weight_normal = 808.5 N * cos(18.7°) = 769.1 N
Given that the coefficient of friction (μk) is 0.150, the force of friction is:
Force_friction = 0.150 * 769.1 N = 115.4 N
4. Determine the net force along the incline:
Since the skier is moving at a constant speed, the net force is zero.
Net_force = Force_applied - Force_friction - Weight_parallel
0 = Force_applied - 115.4 N - 270.2 N
Force_applied = 115.4 N + 270.2 N
Force_applied = 385.6 N
Therefore, the force exerted by the rope on the skier is approximately 385.6 Newtons.
To calculate the force that the rope exerts on the skier, we need to consider the forces acting on the skier.
There are two main forces at play here: the force of gravity and the force exerted by the rope. The skier is being pulled up the slope at a constant speed, which means the net force acting on the skier is zero (since acceleration is zero).
Let's break down these forces:
1. Force of gravity (downward direction): The force of gravity can be calculated using the formula: Fg = m * g, where m is the mass of the skier and g is the acceleration due to gravity (approximately 9.8 m/s²).
Fg = (82.5 kg) * (9.8 m/s²) = 808.5 N
2. Force exerted by the rope (upward direction): The force exerted by the rope can be calculated using the formula: Fr = N - Ff, where N is the normal force and Ff is the force of friction.
To find the normal force N, we need to decompose the weight of the skier into components perpendicular and parallel to the slope. The perpendicular component is N, and it can be calculated as: N = mg * cos(θ), where θ is the angle of the slope.
N = (82.5 kg) * (9.8 m/s²) * cos(18.7°) = 760.2 N
The force of friction Ff can be calculated as: Ff = uk * N, where uk is the coefficient of friction.
Ff = (0.150) * (760.2 N) = 114.0 N
Finally, the force exerted by the rope can be determined:
Fr = N - Ff = 760.2 N - 114.0 N = 646.2 N
Therefore, the rope exerts a force of approximately 646.2 N on the skier.
weight parallel to the slope ... m * g * sin(18.7º)
frictional force ... m * g * cos(18.7º) * 0.150
the rope has to overcome the sum of the forces