A skier of mass 74 kg is pulled up a slope by a motor-driven cable.

(a) How much work is required to pull him 90 m up a 30° slope (assumed frictionless) at a constant speed of 3.0 m/s?
J

(b) What power (expressed in hp) must a motor have to perform this task?
hp

a) Work=Force x distance

To find force take your weight (74 kg) and multiply it by gravity (9.8) to get a force of 725 N
Next, find your distance by doing sin(30) times how many meters up the slope (90 m)
So then Work=(725)(45)= 32,625 J

(a) The speed does not matter, since the kinetic energy does not change.

Multiply the weight M*g = 725 N by the vertical rise 90 sin30 = 45 m. You should get 32,630 Joules.

(b) Power = (Work done)/(Time required)
The time required is (90 m)/(3 m/s) = 30 s.

Power = 32,630/30 = 1088 Watts
= 1.45 hp

(a) Well, let's calculate the work done in pulling this skier up the slope. The work done is given by the equation W = F * d * cosθ, where F is the force applied, d is the displacement, and θ is the angle between the force direction and the displacement direction.

First, let's find the magnitude of the force applied. The force can be given by F = m * g * sinθ, where m is the mass of the skier and g is the acceleration due to gravity.

F = 74 kg * 9.8 m/s^2 * sin(30°) = 362.97 N

Next, let's find the displacement in the direction of the force. In this case, it's equal to the distance traveled up the slope, which is 90 m.

Now, let's calculate the work done:

W = 362.97 N * 90 m * cos(30°)

Using a calculator, we get:

W ≈ 27499.25 J

Therefore, the work required to pull the skier up the slope is approximately 27499.25 Joules.

(b) To find the power, we can use the equation P = W / t, where P is power, W is work, and t is time. In this case, since the skier is traveling at a constant speed, we can use the formula P = F * v, where F is force and v is velocity.

First, let's find the velocity: v = 3.0 m/s

Now, let's calculate the power:

P = F * v = 362.97 N * 3.0 m/s

Using a calculator, we get:

P ≈ 1088.91 Watts

To convert watts to horsepower, we can use the fact that 1 hp is approximately equal to 746 Watts:

P ≈ 1088.91 W / 746 W/hp

Using a calculator, we get:

P ≈ 1.46 hp

Therefore, the motor must have approximately 1.46 horsepower to perform this task.

To find the work required to pull the skier up the slope, we need to consider the forces involved and the displacement of the skier.

(a) Firstly, let's analyze the forces acting on the skier. The main force is the force of gravity, which can be resolved into two components - one parallel to the slope and one perpendicular to the slope. The perpendicular component doesn't contribute to the work since it's perpendicular to the displacement.

The net force acting on the skier along the slope is the force required to overcome gravity, given by:
Fnet = mg * sinθ

where m is the mass of the skier (74 kg) and θ is the angle of the slope (30°). The work done can be calculated by multiplying the net force by the displacement:
Work = Fnet * d * cosθ

where d is the distance traveled (90 m) and cosθ accounts for the direction of the displacement.

Substituting the values into the equation:
Work = (mg * sinθ) * d * cosθ

Now, let's calculate the work:
Work = (74 kg * 9.8 m/s^2 * sin(30°)) * 90 m * cos(30°)
Work ≈ 11548.44 J

Therefore, the work required to pull the skier up the slope is approximately 11548.44 Joules (J).

(b) Power is defined as the rate at which work is done. It can be calculated by dividing the work done by the time taken to do it:
Power = Work / time

Since the speed of the skier is constant at 3.0 m/s, we can find the time taken to travel the given distance using the equation:
time = distance / speed

Substituting the values:
time = 90 m / 3.0 m/s
time = 30 s

Now, let's calculate the power:
Power = 11548.44 J / 30 s

To express power in horsepower (hp), we use the conversion:
1 hp = 745.7 W

So, the power required can be calculated as:
Power(hp) = Power(W) / 745.7

Substituting the values:
Power(hp) = (11548.44 J / 30 s) / 745.7
Power(hp) ≈ 5.168 hp

Therefore, a motor with approximately 5.168 horsepower is needed to perform this task.