A skier of mass 70kg is pulled up a slope by a motor-driven cable. The acceleration of gravity is 9.8m/s^2. How much work is required to pull him 60m up a 30.7º slope (frictionless) at a constant speed of 3 m/s? answer in units of kJ.
In this question, I think I gotta use mgh and 1/2mv^2 to calculate the energy but i don't know how. can anyone help me? thanks in advance!
Forget about kinetic energy in this problem. Since the pulling is being done at constant speed, the kinetic energy does not change and the speed does not matter when computing the work required.
The work energy required equals the change in potential energy,
W = M g H sin 30.7.
(H sin 30.7 is the change in altitude)
M = 70 kg
H = 60 m.
no friction
net force = ma (up) = F (applied, up) - mg sin 30
ma = F - mg sin 30
it moves with constant speed >> v = constant, a = dv/dt =0
0 = F - mg sin 30
F = mg sin 30 = 343 N (applied force up)
work done W = Force *displacement cos (theta)
W = F (up) * d (up) cos (0)
W = 343 * 60 = 20580 Joule >>work done by motor
-------------
Power (motor) = F (average) * v (constant)
P = 343 *2 = 686 watt
To solve this problem, you are correct in using the conservation of energy principle. The work done to pull the skier up the slope is equal to the change in potential energy.
The formula for potential energy is given by mgh, where m is the mass of the skier, g is the acceleration due to gravity, and h is the change in altitude (the vertical distance traveled).
In this case, the change in altitude is given by H sin θ, where H is the total vertical distance traveled (60 m) and θ is the angle of the slope (30.7 degrees).
So, the formula for the work done is:
W = mgh sin θ
Substituting the given values:
W = (70 kg)(9.8 m/s^2)(60 m)(sin 30.7)
Now, you can use a calculator to evaluate this expression:
W ≈ 19,445 J
Since the question asks for the answer in units of kJ (kilojoules), you can convert the answer by dividing by 1000:
W ≈ 19.45 kJ
Therefore, the work required to pull the skier up the slope is approximately 19.45 kJ.
To calculate the work required to pull the skier up the slope, you can use the equation:
Work = (mass of the skier) * (acceleration due to gravity) * (change in height) * (sinθ),
where:
- (mass of the skier) is 70 kg,
- (acceleration due to gravity) is 9.8 m/s^2,
- (change in height) is 60 m,
- (sinθ) is the sine of the angle of the slope, which is sin(30.7º).
Let's plug in the values and calculate the work:
Work = (70 kg) * (9.8 m/s^2) * (60 m) * (sin(30.7º))
To calculate the sine of 30.7º, we need to convert the angle to radians:
sin(30.7º) = sin(30.7º * π / 180) ≈ 0.508.
Now, let's substitute the values and calculate the work:
Work = (70 kg) * (9.8 m/s^2) * (60 m) * (0.508)
Work ≈ 204,384 J.
To convert the work from joules (J) to kilojoules (kJ), divide by 1000:
Work = 204,384 J / 1000 ≈ 204.384 kJ.
Therefore, the work required to pull the skier up the slope is approximately 204.384 kJ.