A skier of mass 71 kg is pulled up a slope by a motor-driven cable.
The acceleration of gravity is 9.8 m/s2. How much work is required to pull him 64.6 m up a 33.8� slope (assumed to be frictionless) at a constant speed of 2.53 m/s?
Answer in units of kJ
work= force*distance
= mg*sin33.8*distance
YOu can figure power also
power= work/time= force*velocity/distance
= 71*g*Sin33.8*2.53 watts.
Well, well, well, looks like someone wants to calculate some work. Let's get to it, shall we?
To figure out the amount of work required, we need to take a few things into account. First, we need to find the gravitational force acting on the skier. We can do this by multiplying the mass (71 kg) by the acceleration due to gravity (9.8 m/s^2). So, the gravitational force is 71 kg * (9.8 m/s^2) = 696.8 N.
Next, we need to determine the vertical distance the skier is being pulled up. Now, since the slope is at an angle of 33.8 degrees, we need to calculate the vertical height by multiplying the total distance (64.6 m) by the sine of the angle (33.8 degrees). So, the vertical distance is 64.6 m * sin(33.8 degrees) = 35.1 m.
Since the skier is being pulled up at a constant speed, the work done against gravity is equal to the change in potential energy. The change in potential energy can be calculated by multiplying the force (696.8 N) by the vertical distance (35.1 m). So, the work required is 696.8 N * 35.1 m = 24,416.68 J.
But hold your horses, we're not in the mood for joules today. We want our answer in kilojoules. So, 24,416.68 J divided by 1000 gives us 24.41668 kJ.
Therefore, the amount of work required to pull the skier up the slope is approximately 24.41668 kJ.
And there you have it! Work all done, with a touch of humor for good measure. Now go on, tell your friends about our little calculation spectacle!
To find the work required to pull the skier up the slope, we need to calculate the gravitational potential energy and the work done against gravity.
1. Calculate the gravitational potential energy:
Gravitational Potential Energy = mass * acceleration due to gravity * height
Height = distance * sin(angle)
Height = 64.6 m * sin(33.8 degrees)
Height ≈ 37.03 m
Gravitational Potential Energy = 71 kg * 9.8 m/s^2 * 37.03 m
Gravitational Potential Energy ≈ 25,248.678 J
2. The work done against gravity is equal to the gravitational potential energy. Therefore:
Work = 25,248.678 J
3. Convert the work from joules to kilojoules:
Work = 25,248.678 J / 1000
Work ≈ 25.25 kJ
Therefore, the work required to pull the skier up the slope is approximately 25.25 kJ.
To find the work required to pull the skier up the slope, we can use the formula for work, which is given by:
Work = Force x Distance
In this case, the force required to pull the skier up the slope is the component of the gravitational force parallel to the slope. This force can be found using the formula:
Force = Weight x sin(theta)
Where Weight is the gravitational force acting on the skier and theta is the angle of the slope. The weight (W) is given by:
Weight = mass x acceleration due to gravity
In this case, the mass of the skier is given as 71 kg and the acceleration due to gravity is 9.8 m/s^2.
First, let's calculate the weight:
Weight = 71 kg x 9.8 m/s^2 = 696.8 N
Next, let's find the force:
Force = 696.8 N x sin(33.8 degrees)
Now, we need to calculate the work. Since the skier is moving at a constant speed, the net work done is zero since no change in kinetic energy occurs. Therefore, the work done by the cable is equal in magnitude but opposite in direction to the work done by gravity.
Work = -Force x Distance
Work = -Force x 64.6 m
Finally, we can calculate the work:
Work = -Force x Distance = -Force x 64.6 m
Note: The negative sign indicates that the work is done against the force of gravity.
Substitute the values into the equation:
Work = - (Force) x 64.6 m = - (Force) x 64.6 m = - (Force) x 64.6 m = - (696.8 N x sin(33.8 degrees)) x 64.6 m
Calculate the numerical value:
Work = - (696.8 N x sin(33.8 degrees)) x 64.6 m = - (696.8 N x 0.5586) x 64.6 m = - 415.0236 N x 64.6 m
Now, convert the work from joules to kilojoules:
Work = - 415.0236 N x 64.6 m = - 26792.57496 J
Work in kJ = - 26792.57496 J / 1000 = - 26.79257496 kJ
Round the answer to the appropriate number of significant figures:
Work = - 26.79 kJ
Therefore, the work required to pull the skier 64.6 m up a 33.8° slope at a constant speed of 2.53 m/s is approximately -26.79 kJ.