skier of mass 79 kg is pulled up a slope by a motor-driven cable.

(a) How much work is required to pull him 80 m up a 30° slope (assumed frictionless) at a constant speed of 3.4 m/s?
1 J

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

a.=Wt. of skier=m*g=79kg9.8N/kg=774.2 N.

Fp = 774.2*sin30 = 387.1 N. = Force
parallel to the slope.

Work = F*d = 387.1 * 80 = 30,968 J.

b. Power=F*V = 387.1 * 3.4=1316 J./s.
= 1316 Watts.

1316W * 1hp/746W. = 1.76 hp.

(a) Well, let's calculate the work required to pull this skier up the slope. First, we need to calculate the gravitational potential energy gained by the skier.

We can use the formula for gravitational potential energy:
Potential Energy = mass x gravity x height

Given that the mass of the skier is 79 kg, the height is 80 m, and gravity is approximately 9.8 m/s^2, we can calculate:
Potential Energy = 79 kg x 9.8 m/s^2 x 80 m

Now, the work done is equal to the potential energy gained:
Work = 79 kg x 9.8 m/s^2 x 80 m

Calculating this, we get:
Work = 62,432 J

(b) To calculate the power, we need to use the formula:
Power = Work / Time

Since the skier is moving at a constant speed of 3.4 m/s, we can calculate the time required to cover 80 m:
Time = Distance / Speed
Time = 80 m / 3.4 m/s

Now, plug in the values to calculate the power:
Power = 62,432 J / (80 m / 3.4 m/s)

Calculating this, we get:
Power ≈ 2,086 W

Now, to convert this to horsepower, we have to divide by 746 (since 1 horsepower is equal to 746 watts):
Power(hp) ≈ 2,086 W / 746

Calculating this, we get:
Power(hp) ≈ 2.8 hp

So, the motor must have approximately 2.8 horsepower to perform this task.

To find the work required to pull the skier up the slope, we will use the formula:

Work = force x distance

(a) First, let's calculate the force required to pull the skier up the slope. The force can be broken down into two components: the parallel force and the perpendicular force. The parallel force is responsible for overcoming the gravitational force acting down the slope, and the perpendicular force is responsible for moving the skier up the slope.

The parallel force is given by:

Parallel force = mass x acceleration due to gravity x sin(angle of the slope)

Parallel force = 79 kg x 9.8 m/s^2 x sin(30°)
Parallel force ≈ 381.49 N

The perpendicular force is given by:

Perpendicular force = mass x acceleration due to gravity x cos(angle of the slope)

Perpendicular force = 79 kg x 9.8 m/s^2 x cos(30°)
Perpendicular force ≈ 679.89 N

Since the skier is moving at a constant speed, the work done should counterbalance the sum of both forces. Therefore, the work required to pull the skier up the slope is equal to the perpendicular force multiplied by the distance:

Work = Perpendicular force x distance
Work = 679.89 N x 80 m
Work ≈ 54391.2 J

So, the work required to pull the skier up the slope is approximately 54391.2 J.

(b) To find the power required, we can use the formula:

Power = work / time

However, the time taken is not given in the problem. But we know that velocity is distance / time, so we can rearrange the formula to solve for time:

time = distance / velocity
time = 80 m / 3.4 m/s
time ≈ 23.53 s

Now, we can calculate the power in watts:

Power = work / time
Power = 54391.2 J / 23.53 s
Power ≈ 2311.09 W

Finally, let's convert the power to horsepower (hp):

1 hp = 745.7 W

Power in hp ≈ 2311.09 W / 745.7 W/hp
Power in hp ≈ 3.10 hp

Therefore, the motor must have approximately 3.10 hp to perform this task.

To find the work required to pull the skier up the slope and the power required for the motor, we can use the following formulas:

Work (W) = force (F) * distance (d) * cosθ
Power (P) = work (W) / time (t)

First, let's calculate the work required to pull the skier up the slope:

(a) Given information:
Mass of the skier (m) = 79 kg
Distance traveled (d) = 80 m
Angle of the slope (θ) = 30°
Speed (v) = 3.4 m/s

To find the force, we can use the equation:
Force (F) = mass (m) * acceleration (a)

Since the skier is moving at a constant speed up the slope, the net force is zero. The gravitational force (mg) is balanced by the force exerted by the motor-driven cable (F).

Gravitational force (mg) = mass (m) * acceleration due to gravity (g)

Acceleration due to gravity (g) = 9.8 m/s^2 (approx.)

So, the force exerted by the motor-driven cable (F) is equal to the gravitational force (mg). Hence, F = mg.

The angle of the slope (θ) is given, and we can use trigonometry to find the gravitational force in the direction of the slope:

Force (mg) = mass (m) * acceleration due to gravity (g) * sinθ

Now, we can calculate the force exerted by the motor-driven cable (F), which is equal to the gravitational force:

F = mg = m * g * sinθ

Plugging in the given values:

F = 79 kg * 9.8 m/s^2 * sin(30°)

Next, we can calculate the work required to pull the skier up the slope using:

Work (W) = F * d * cosθ

Plugging in the values:

W = F * d * cosθ = (79 kg * 9.8 m/s^2 * sin(30°)) * 80 m * cos(30°)

Calculate the numerical value of these expressions, and the answer will be in Joules (J).

(b) To find the power of the motor required to perform this task, we can use the formula:

Power (P) = Work (W) / time (t)

However, time (t) is not given in the question. We can use the speed (v) to find the time taken for the skier to travel the given distance:

t = d / v

Plugging in the values:

t = 80 m / 3.4 m/s

Calculate the numerical value of this expression to find the time taken.

Finally, we can calculate the power (P) using:

P = W / t

Plugging in the calculated values for work (W) and time (t), and convert the answer to horsepower (hp).

Remember to use the appropriate conversion factor: 1 horsepower (hp) = 746 Watts (W).