The concepts in this problem are similar to those in Multiple-Concept Example 4, except that the force doing the work in this problem is the tension in the cable. A rescue helicopter lifts a 72.4-kg person straight up by means of a cable. The person has an upward acceleration of 0.830 m/s2 and is lifted from rest through a distance of 11.9 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.
To solve this problem, we can use concepts from mechanics and the work-energy theorem. Let's break it down step by step:
(a) To find the tension in the cable, we can use Newton's second law of motion. The force equation for the person being lifted is:
F = m * a
Where:
F is the total force (tension in the cable + weight of the person),
m is the mass of the person (72.4 kg),
a is the upward acceleration (0.830 m/s²).
Rearranging the equation, we get:
F = m * a
F = 72.4 kg * 0.830 m/s²
F = 60.092 N
Therefore, the tension in the cable is 60.092 N.
(b) To calculate the work done by the tension in the cable, we use the formula:
Work = Force * Distance * cos(θ)
In this case, the angle (θ) is 0 degrees because the force and the distance are in the same direction (vertical). Therefore:
Work = 60.092 N * 11.9 m * cos(0°)
Work = 60.092 N * 11.9 m * 1
Work = 713.728 Joules
So, the tension in the cable does 713.728 Joules of work.
(c) The work done by the person's weight can be calculated using the formula:
Work = Force * Distance * cos(θ)
The angle (θ) between the weight force and the distance is 180 degrees because the person is lifted straight up against gravity. Therefore:
Work = (m * g) * distance * cos(180°)
Where:
m is the mass of the person (72.4 kg),
g is the acceleration due to gravity (9.8 m/s²),
distance is the distance lifted (11.9 m).
Work = (72.4 kg * 9.8 m/s²) * 11.9 m * cos(180°)
Work = -8426.512 Joules
Note that the negative sign indicates that the work done by the person's weight is negative since it opposes the direction of motion.
(d) Now, using the work-energy theorem, we can find the final speed of the person. The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy.
The initial kinetic energy of the person is zero because they start from rest. Therefore:
Work done by tension + Work done by weight = Change in kinetic energy
(713.728 J) + (-8426.512 J) = (1/2) * m * vf² - (1/2) * m * vi²
Since the person starts from rest, the initial velocity (vi) is zero. We can assume vf as the final velocity.
713.728 J - 8426.512 J = (1/2) * 72.4 kg * vf² - (1/2) * 72.4 kg * 0²
-7712.784 J = (1/2) * 72.4 kg * vf²
Simplifying further:
vf² = -7712.784 J * 2 / 72.4 kg
vf² = -212.259
vf ≈ √(-212.259)
vf ≈ 14.57 m/s
Therefore, the final speed of the person is approximately 14.57 m/s.