A skier of mass 74.5 kg is pulled up a slope by a motor-driven cable. How much work is required to pull the skier 55 m up a 31° slope (assumed to be frictionless) at a constant speed of 2.0 m/s?
height vertical=hv=55sin31
work=mg(hv)
To determine the work required to pull the skier up the slope, we can use the formula:
Work = force * distance * cosine(theta)
Where:
- Work is the amount of work done
- force is the force acting on the skier
- distance is the distance over which the force is applied
- theta is the angle between the force and the displacement
In this case, the skier is pulled at a constant speed, so the net force acting on the skier is zero. The only force acting on the skier is the force due to gravity, which is mg, where m is the mass of the skier and g is the acceleration due to gravity.
Since the skier is pulled up the slope, the force due to gravity can be divided into two components: one along the slope and one perpendicular to the slope. The component of the force along the slope is mg * sin(theta), and the component perpendicular to the slope is mg * cos(theta).
In this problem, we are given:
- mass (m) = 74.5 kg
- distance (d) = 55 m
- slope angle (theta) = 31 degrees
- speed (v) = 2.0 m/s
First, we need to find the perpendicular component of the force, which is mg * cos(theta):
force_perpendicular = mass * acceleration due to gravity * cos(theta)
force_perpendicular = 74.5 kg * 9.8 m/s^2 * cos(31°)
Next, we can calculate the work using the formula mentioned earlier:
Work = force_perpendicular * distance * cosine(theta)
Work = (74.5 kg * 9.8 m/s^2 * cos(31°)) * 55 m * cos(31°)
Finally, we can substitute the given values and calculate the work:
Work = (74.5 kg * 9.8 m/s^2 * cos(31°)) * 55 m * cos(31°)
Work = approximately 15,146.68 J
Therefore, the amount of work required to pull the skier up the slope is approximately 15,146.68 joules.