A father pushes horizontally on his daughter's sled to move it up a snowy incline, as illustrated in the figure, with h = 3.4 m and θ = 14°. The total mass of the sled and the girl is 35 kg and the coefficient of kinetic friction between the sled runners and the snow is 0.20. If the sled moves up the hill with a constant velocity, how much work is done by the father in moving it from the bottom to the top of the hill?
To determine the work done by the father in moving the sled from the bottom to the top of the hill, we need to calculate the total work done against gravity and the work done against friction.
1. Work done against gravity:
The force of gravity acts vertically downward. However, since the sled is moving horizontally up the incline, we need to find the component of the force of gravity in that direction. The formula for the force of gravity is given by:
F_gravity = mass * acceleration_due_to_gravity
In this case, the mass is the total mass of the sled and the girl (35 kg) and the acceleration due to gravity is approximately 9.8 m/s².
F_gravity = 35 kg * 9.8 m/s²
Next, we calculate the component of the force of gravity in the direction of motion. This can be found using the following formula:
F_gravity_parallel = F_gravity * sin(θ)
Where θ is the angle of inclination.
F_gravity_parallel = (35 kg * 9.8 m/s²) * sin(14°)
Now, we can calculate the work done against gravity using the formula:
Work_gravity = F_gravity_parallel * distance
In this case, the distance is the height of the incline (3.4 meters).
Work_gravity = (35 kg * 9.8 m/s² * sin(14°)) * 3.4 meters
2. Work done against friction:
The force of friction can be calculated using the formula:
F_friction = coefficient_of_friction * normal_force
The normal force can be calculated as the component of the force of gravity perpendicular to the incline. This can be found using the following formula:
F_normal = F_gravity * cos(θ)
F_normal = (35 kg * 9.8 m/s²) * cos(14°)
Now, we can calculate the force of friction using the formula:
F_friction = 0.20 * F_normal
Next, we calculate the work done against friction:
Work_friction = F_friction * distance
In this case, the distance is the height of the incline (3.4 meters).
Work_friction = (0.20 * (35 kg * 9.8 m/s² * cos(14°))) * 3.4 meters
3. Total work done:
The total work done by the father is the sum of the work done against gravity and the work done against friction.
Total_work = Work_gravity + Work_friction
Substituting the calculated values, we can find the total work done by the father in moving the sled from the bottom to the top of the hill.