A rescue helicopter lifts a 85.8-kg person straight up by means of a cable. The person has an upward acceleration of 0.643 m/s2 and is lifted from rest through a distance of 13.1 m. (a) What is the tension in the cable? How much work is done by (b) the tension in the cable and (c) the person's weight? (d) Use the work-energy theorem and find the final speed of the person.
tension=m(g+a)
b. work=tension*distance
c. work=-weight*height
final PE+finalKE=workdone
final KE=workdone-finalpe
1/2 m v^2=m(g+a)h -mgh
solve for v
To find the tension in the cable, we can use Newton's second law, which states that the net force acting on an object is equal to its mass times its acceleration. In this case, the net force is the tension in the cable and the acceleration is the upward acceleration of the person.
(a) We can calculate the tension in the cable using the following formula:
Tension = Mass * Acceleration
Given:
Mass (m) = 85.8 kg
Acceleration (a) = 0.643 m/s^2
Tension = 85.8 kg * 0.643 m/s^2
Tension ≈ 55.19 N
Therefore, the tension in the cable is approximately 55.19 N.
Now, let's calculate the work done by the tension in the cable and the person's weight.
(b) The work done by the tension in the cable can be calculated using the formula:
Work = Force * Distance * cos(theta)
In this case, the force is the tension in the cable, the distance is the vertical distance the person is lifted, and theta is the angle between the force and the displacement (which is 0 degrees since the force is acting in the same direction as the displacement).
Given:
Force (F) = Tension ≈ 55.19 N
Distance (d) = 13.1 m
Theta (θ) = 0 degrees
Work = 55.19 N * 13.1 m * cos(0)
Work = 723.269 J
The work done by the tension in the cable is approximately 723.269 J.
(c) The work done by the person's weight can be calculated using the formula:
Work = Force * Distance * cos(theta)
In this case, the force is the person's weight, which is equal to their mass multiplied by the acceleration due to gravity (9.8 m/s^2), the distance is the vertical distance the person is lifted (13.1 m), and theta is again 0 degrees.
Given:
Force (F) = mass (m) * acceleration (g) = 85.8 kg * 9.8 m/s^2
Distance (d) = 13.1 m
Theta (θ) = 0 degrees
Work = (85.8 kg * 9.8 m/s^2) * 13.1 m * cos(0)
Work = 11228.016 J
The work done by the person's weight is approximately 11228.016 J.
(d) According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy. Since the person starts from rest and is lifted, all the work done on them is converted into kinetic energy.
We can use the work-energy theorem to find the final speed (vf) of the person:
Work done on the person = Change in Kinetic Energy
Work done on the person = Work done by the tension in the cable + Work done by the person's weight
Work done on the person = 723.269 J + 11228.016 J
Change in Kinetic Energy = 723.269 J + 11228.016 J
Since the initial kinetic energy of the person is zero (starting from rest), the final kinetic energy is equal to the change in kinetic energy.
Final Kinetic Energy = 723.269 J + 11228.016 J
Finally, we can use the formula for kinetic energy to find the final speed:
Kinetic Energy = (1/2) * mass * velocity^2
Kinetic Energy = (1/2) * mass * final velocity^2
723.269 J + 11228.016 J = (1/2) * 85.8 kg * final velocity^2
11951.285 J = 0.5 * 85.8 kg * final velocity^2
final velocity^2 = (11951.285 J) / (0.5 * 85.8 kg)
final velocity^2 ≈ 278.086 m^2/s^2
Taking the square root of both sides, we get:
final velocity ≈ √(278.086 m^2/s^2)
final velocity ≈ 16.676 m/s
Therefore, the final speed of the person is approximately 16.676 m/s.