Pete slides a crate up a ramp at an angle of
27◦ by exerting a 242 N force parallel to the
ramp. The crate moves at a constant speed.
The coefficient of friction is 0.4.
How much work did Pete do when the crate
was raised a vertical distance of 2.22 m?
Answer in units of J
To find the work done by Pete when the crate was raised a vertical distance, we need to calculate the gravitational potential energy gained by the crate.
The gravitational potential energy (PE) gained by an object of mass (m) when it is raised a vertical distance (h) can be calculated using the formula:
PE = m * g * h
Where:
m is the mass of the object
g is the acceleration due to gravity (approximately 9.8 m/s^2)
h is the vertical distance
However, we need to consider the work done against friction as well.
The work done against friction is given by the equation:
Work_friction = force_friction * distance
To calculate the force of friction:
force_friction = coefficient_of_friction * normal_force
The normal force is the force exerted perpendicular to the ramp due to the weight of the crate:
normal_force = m * g * cos(angle_of_ramp)
So, the total work done by Pete can be calculated as:
Work_total = Work_against_gravity + Work_against_friction
Now, let's calculate each part of the equation:
Mass of the crate (m) is not provided in the question. If you have the mass, substitute it into the formula. Let's assume the mass is 10 kg for this example.
m = 10 kg
Acceleration due to gravity (g) is approximately 9.8 m/s^2.
g = 9.8 m/s^2
The vertical distance (h) raised is given as 2.22 m.
h = 2.22 m
The coefficient of friction (μ) is given as 0.4.
μ = 0.4
The angle of the ramp is given as 27 degrees.
angle_of_ramp = 27 degrees = 27°
First, calculate the normal force:
normal_force = m * g * cos(angle_of_ramp)
normal_force = 10 kg * 9.8 m/s^2 * cos(27°)
Next, calculate the force of friction:
force_friction = μ * normal_force
Then, calculate the work done against gravity:
Work_against_gravity = m * g * h
Finally, calculate the work done against friction:
Work_against_friction = force_friction * distance
Since the crate moves at a constant speed, the distance is the same as the vertical distance raised, which is 2.22 m.
Now, substitute the values into the equations and calculate the answers:
normal_force = 10 kg * 9.8 m/s^2 * cos(27°)
force_friction = 0.4 * normal_force
Work_against_gravity = 10 kg * 9.8 m/s^2 * 2.22 m
Work_against_friction = force_friction * 2.22 m
Finally, calculate the total work done by Pete:
Work_total = Work_against_gravity + Work_against_friction
Substitute the calculated values into the equation and find the answer in joules (J).