An 56.0 N box of clothes is pulled 5.2 m up a 30.0° ramp by a force of 104 N that points along the ramp. If the coefficient of kinetic friction between the box and the ramp is 0.22, calculate the change in the box's kinetic energy.
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To calculate the change in the box's kinetic energy, we need to find the work done by the net force on the box.
1. Calculate the gravitational force acting on the box.
Gravitational force (Fg) = mass (m) × gravitational acceleration (g)
Given that the mass (m) = 56.0 N ÷ gravitational acceleration (g) = 9.8 m/s²
m = 5.71 kg
Fg = (5.71 kg) × (9.8 m/s²)
Fg = 55.94 N
2. Calculate the force parallel to the ramp.
Force parallel to the ramp (Fpar) = applied force (Fapplied) - component of gravity parallel to the ramp
Component of gravity parallel to the ramp = Fg × sin(theta)
Given that the applied force (Fapplied) = 104 N and the angle (theta) = 30°
Component of gravity parallel to the ramp = (55.94 N) × sin(30°)
Component of gravity parallel to the ramp = 27.97 N
Fpar = 104 N - 27.97 N
Fpar = 76.03 N
3. Calculate the work done by the force parallel to the ramp.
Work (W) = Fpar × distance (d)
Given that the distance (d) = 5.2 m
W = (76.03 N) × (5.2 m)
W = 395.72 J
4. Calculate the work done by friction.
Friction force (Ffriction) = coefficient of kinetic friction (μk) × normal force (Fn)
Normal force (Fn) = Fg × cos(theta)
Fn = (55.94 N) × cos(30°)
Fn = 48.45 N
Ffriction = (0.22) × (48.45 N)
Ffriction = 10.66 N
Work done by friction (Wfriction) = Ffriction × d
Wfriction = (10.66 N) × (5.2 m)
Wfriction = 55.43 J
5. Calculate the net work done on the box.
Net work (Wnet) = W - Wfriction
Wnet = 395.72 J - 55.43 J
Wnet = 340.29 J
The change in the box's kinetic energy is equal to the net work done on the box, so the change in the box's kinetic energy is 340.29 J.
To calculate the change in the box's kinetic energy, we first need to find the work done by the force of 104 N in pulling the box up the ramp.
The work done by a force can be calculated using the formula:
Work = Force * Distance * cos(theta)
In this case, the force is 104 N, the distance is 5.2 m, and theta is the angle between the force and the direction of motion, which is 30°. So we have:
Work = 104 N * 5.2 m * cos(30°)
Next, we need to find the work done against friction. The work done against friction is equal to the force of friction multiplied by the distance moved. The force of friction can be calculated using the formula:
Force of friction = coefficient of friction * Normal force
The normal force is the force perpendicular to the surface of the ramp and can be calculated using the formula:
Normal force = mass * gravitational acceleration * cos(theta)
The mass of the box is given as 56.0 N, and the gravitational acceleration is 9.8 m/s^2.
Therefore, the normal force is:
Normal force = 56.0 N * 9.8 m/s^2 * cos(30°)
Using the coefficient of kinetic friction given as 0.22, we can calculate the force of friction:
Force of friction = 0.22 * (56.0 N * 9.8 m/s^2 * cos(30°))
Finally, we can calculate the work done against friction:
Work against friction = Force of friction * Distance
Substituting the values into the equation, we have:
Work against friction = (0.22 * (56.0 N * 9.8 m/s^2 * cos(30°))) * 5.2 m
The change in kinetic energy of the box is equal to the work done minus the work done against friction.
Change in kinetic energy = Work - Work against friction
Substitute the values and calculate:
Change in kinetic energy = (104 N * 5.2 m * cos(30°)) - ((0.22 * (56.0 N * 9.8 m/s^2 * cos(30°))) * 5.2 m)
Now, you can plug in the values and calculate the change in the box's kinetic energy in Joules.