A skateboarder with mass ms = 42 kg is standing at the top of a ramp which is hy = 3.7 m above the ground. The skateboarder then jumps on his skateboard and descends down the ramp. His speed at the bottom of the ramp is vf = 6.3 m/s.
Write an expression for the work, Wf, done by the friction force between the ramp and the skateboarder in terms of the variables given in the problem statement.
The ramp makes an angle θ with the ground, where θ = 30 degrees. Write an expression for the magnitude of the friction force, fr, between the ramp and the skateboarder.
To find the work done by the friction force, we first need to determine the distance over which the friction force acts. In this case, the distance is the length of the ramp, which can be calculated using the height of the ramp and the angle it makes with the ground.
We can use trigonometry to find the length of the ramp:
length of ramp = hy / sin(θ)
Given that hy = 3.7 m and θ = 30 degrees, we can calculate the length of the ramp:
length of ramp = 3.7 / sin(30°)
Now, to find the work done by the friction force, we use the formula:
work = force x distance
The friction force is parallel to the ramp, i.e., in the direction of motion, and opposes the skateboarder's motion. Therefore, the work done by the friction force is negative.
The magnitude of the friction force, fr, can be calculated using the formula:
fr = μ * N
Here, μ is the coefficient of friction and N is the normal force exerted by the ramp on the skateboarder. Since the skateboarder is on a ramp, the normal force can be calculated using the equation:
N = m * g * cos(θ)
Given that m = 42 kg and g = 9.8 m/s², we can calculate the normal force:
N = 42 * 9.8 * cos(30°)
Finally, we need to substitute the values into the formulas.
Expression for the length of the ramp:
length of ramp = 3.7 / sin(30°)
Magnitude of the friction force:
fr = μ * N
fr = μ * (42 * 9.8 * cos(30°))
Therefore, the expression for the magnitude of the friction force, fr, is μ * (42 * 9.8 * cos(30°)), and the expression for the work, Wf, done by the friction force is -fr * length of ramp, i.e., -μ * (42 * 9.8 * cos(30°)) * (3.7 / sin(30°)).