A skier of mass 70.4 kg, starting from rest, slides down a slope at an angle $\theta$ of 36.8° with the horizontal. The coefficient of kinetic friction is 0.14. What is the net work, i.e. net gain in kinetic energy, (in J) done on the skier in the first 8.4 s of descent?
To find the net work done on the skier, we need to consider the forces acting on the skier and the distance over which those forces act.
1. Determine the gravitational force acting on the skier:
The gravitational force acting on the skier is given by the formula F_gravity = m * g, where m is the mass of the skier and g is the acceleration due to gravity.
F_gravity = 70.4 kg * 9.8 m/s^2
F_gravity = 689.92 N
2. Determine the component of gravitational force parallel to the slope:
The component of gravitational force parallel to the slope is given by the formula F_parallel = F_gravity * sin(theta), where theta is the angle of the slope with respect to the horizontal.
F_parallel = 689.92 N * sin(36.8°)
F_parallel = 404.85 N
3. Determine the force of kinetic friction:
The force of kinetic friction can be calculated using the formula F_friction = coefficient_of_friction * F_normal, where F_normal is the normal force acting on the skier. On an inclined plane, the normal force is given by F_normal = m * g * cos(theta), where theta is the angle of the slope with respect to the horizontal.
F_normal = 70.4 kg * 9.8 m/s^2 * cos(36.8°)
F_normal = 581.09 N
F_friction = 0.14 * 581.09 N
F_friction = 81.35 N
4. Determine the net force acting on the skier:
The net force acting on the skier is given by the formula F_net = F_parallel - F_friction.
F_net = 404.85 N - 81.35 N
F_net = 323.5 N
5. Determine the distance traveled by the skier:
The distance traveled by the skier can be calculated using the formula d = v_0 * t + (1/2) * a * t^2, where v_0 is the initial velocity, t is the time, and a is the acceleration.
Since the skier starts from rest, the initial velocity is 0.
d = 0 * 8.4 s + (1/2) * 0 * (8.4 s)^2
d = 0 m
6. Determine the net work done:
The net work done on the skier is given by the formula W_net = F_net * d, where W_net is the net work done, F_net is the net force, and d is the distance traveled.
W_net = 323.5 N * 0 m
W_net = 0 J
Therefore, the net work done on the skier in the first 8.4 s of descent is 0 J.
To find the net work done on the skier, we need to determine the change in kinetic energy.
First, let's find the gravitational force acting on the skier parallel to the slope. We can use the formula:
\[
F_{\text{{gravity}}} = m \cdot g \cdot \sin(\theta)
\]
where
- $m$ is the mass of the skier (70.4 kg),
- $g$ is the acceleration due to gravity (9.8 m/s^2), and
- $\theta$ is the angle of the slope (36.8°).
Substituting these values into the equation, we get:
\[
F_{\text{{gravity}}} = 70.4 \, \text{{kg}} \cdot 9.8 \, \text{{m/s}}^2 \cdot \sin(36.8°)
\]
Next, let's calculate the frictional force acting on the skier. We can use the formula:
\[
F_{\text{{friction}}} = \mu \cdot F_{\text{{normal}}}
\]
where
- $\mu$ is the coefficient of kinetic friction (0.14), and
- $F_{\text{{normal}}}$ is the normal force acting on the skier.
Since the skier is on a slope, the normal force is given by:
\[
F_{\text{{normal}}} = m \cdot g \cdot \cos(\theta)
\]
Substituting the values, we have:
\[
F_{\text{{normal}}} = 70.4 \, \text{{kg}} \cdot 9.8 \, \text{{m/s}}^2 \cdot \cos(36.8°)
\]
Finally, we can calculate the net work done on the skier by subtracting the work done by friction from the work done by gravity:
\[
\text{{Net work}} = (F_{\text{{gravity}}} - F_{\text{{friction}}}) \cdot d
\]
where
- $d$ is the distance covered by the skier (which we haven't been given directly).
Unfortunately, we don't have the exact value for the distance covered by the skier in the first 8.4 seconds of descent. Without this information, we cannot calculate the net work done on the skier.