A skier of mass 79.1 kg, starting from rest, slides down a slope at an angle q of 38 with the horizontal. The coefficient of kinetic friction, m, is 0.09. What is the net work (the net gain in kinetic energy) done on the skier in the first 8.9 s of descent?
m•a= μ•m•g•cosα
a= μ•g•cosα
v=at = μ•g•cosα•t
W= ΔKE=KE2-KE1 =mv²/2 -0= mv²/2
To find the net work done on the skier, we need to consider the forces acting on the skier and calculate the work done by each of them. The net work is the sum of the work done by all the forces.
First, let's draw a free-body diagram to visualize the forces acting on the skier:
|\
| \
Fg | \
| \
| \
_______|_____\_______
|| S ||
|| || Ff || ||
|| || || ||
|| || || ||
In the diagram:
- Fg represents the gravitational force acting vertically downward.
- Ff represents the friction force acting parallel to the slope.
- S represents the component of the weight force acting parallel to the slope.
The net work done on the skier can be calculated by the formula:
Net work = Work done by Fg + Work done by Ff + Work done by S
Work done by Fg:
The work done by the gravitational force is given by the formula:
Work = force x distance x cos(theta)
In this case, the force is the weight (W = m * g), the distance is the vertical displacement, and theta is the angle between the force and the displacement (which is 0 degrees, as the displacement is vertical).
So, the work done by Fg is zero, since the vertical displacement is perpendicular to the force.
Work done by Ff:
The work done by the friction force is given by the formula:
Work = force x distance x cos(theta)
In this case, the force is the friction force (Ff = m * g * m), the distance is the length of the slope, and theta is the angle between the force and the displacement (which is the angle of the slope, q).
So, the work done by Ff is equal to:
Work = Ff x d x cos(q)
Work done by S:
The work done by the component of the weight force parallel to the slope is also given by the formula:
Work = force x distance x cos(theta)
In this case, the force is the component of the weight force parallel to the slope (S = m * g * sin(q)), the distance is the length of the slope, and theta is the angle between the force and the displacement (which is the angle of the slope, q).
So, the work done by S is equal to:
Work = S x d x cos(q)
Now, let's calculate the net work done by adding up the work done by each force.
Net work = Work done by Fg + Work done by Ff + Work done by S
= 0 + Ff x d x cos(q) + S x d x cos(q)
= m * g * m * d * cos(q) + m * g * sin(q) * d * cos(q)
Substituting the given values:
mass (m) = 79.1 kg
angle (q) = 38 degrees
coefficient of kinetic friction (m) = 0.09
length of the slope (d) = distance = speed * time = 0 (since the skier is starting from rest)
Substituting these values into the equation, we get:
Net work = 79.1 kg * 9.8 m/s^2 * 0.09 * 0 * cos(38 degrees) + 79.1 kg * 9.8 m/s^2 * sin(38 degrees) * 0 * cos(38 degrees)
Since the displacement is zero (the skier starts from rest and the distance traveled in 8.9 s is zero), the net work done on the skier in the first 8.9 s of descent is zero.