A snowboarder of mass 87.1 kg (including all of the gear and clothing), starting with a downhill speed of 6.1 m/s, slides down a slope at an angle $\theta$ of 36.7° with the horizontal. The coefficient of kinetic friction is 0.106. What is the net work done on the snowboarder in the first 8.02 s of descent?
Fs = m*g = 87.1kg * 9.8N/kg = 853.6 N. =
Force of snowboarder.
Fp = 853.6*sin36.7 = 510.1 N. = Force parallel to hill.
Fn = 853.6*cos36.7 = 684.4 N = Normal =
Force perpendicular to hill.
Fk = u*Fn = 0.106 * 684.4 = 72.55 N. =
Force of kinetic friction.
a=(Fp-Fk)/m=(510.1-72.55)/87.1=5.02 m/s^2.
d = Vo*t + 0.5a*t^2
d = 6.1*8.02 + 2.51*8.02^2 = 210.4 m.
Down the slope.
Work = F*d = (510.1-72.55) * 210.4 =
92,061 Joules.
To find the net work done on the snowboarder, we need to calculate the work done by the gravitational force and the work done by the frictional force. The net work is the sum of these two:
Net Work = Work by Gravity + Work by Friction
1. Work by Gravity:
The gravitational force can be broken down into two components: one parallel to the slope (mg sinθ) and one perpendicular to the slope (mg cosθ). Since the snowboarder is moving downhill, only the component parallel to the slope is in the direction of motion. So, we'll calculate the work done by the parallel component of the gravitational force.
Work by Gravity = Force × Distance
= (mg sinθ) × Distance
2. Work by Friction:
The frictional force is the product of the coefficient of kinetic friction (μk) and the normal force (mg cosθ). The normal force can be calculated as mg cosθ, where θ is the angle of the slope.
Work by Friction = Force × Distance
= (μk × mg cosθ) × Distance
Now, let's calculate the net work done on the snowboarder in the first 8.02 s of descent using the given values:
Given:
Mass of the snowboarder, m = 87.1 kg
Initial speed, v = 6.1 m/s
Angle of the slope, θ = 36.7°
Coefficient of kinetic friction, μk = 0.106
Time, t = 8.02 s
1. Work by Gravity:
First, let's find the distance traveled by the snowboarder in the given time.
Distance = Initial Speed × Time
= 6.1 m/s × 8.02 s
Next, we can calculate the work done by the parallel component of the gravitational force:
Work by Gravity = (m × g × sinθ) × Distance
2. Work by Friction:
The normal force can be calculated as mg cosθ.
Then, we can calculate the work done by friction:
Work by Friction = (μk × m × g × cosθ) × Distance
Finally, we can find the net work done by adding the work done by gravity and the work done by friction:
Net Work = Work by Gravity + Work by Friction
You can substitute the given values into the formulas and calculate the net work done on the snowboarder in the first 8.02 s of descent.