A rescue helicopter lifts a 85.2-kg person straight up by means of a cable. The person has an upward acceleration of 0.764 m/s2 and is lifted from rest through a distance of 12.5 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 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 multiplied by its acceleration. In this case, the person is being lifted with an upward acceleration of 0.764 m/s².
(a) The equation for the tension in the cable can be written as:
Tension = mass * acceleration
Tension = 85.2 kg * 0.764 m/s²
Tension = 65.0928 N
So, the tension in the cable is 65.0928 N.
Now, let's move on to finding the work done by the tension in the cable and the person's weight.
(b) To find the work done by the tension in the cable, we use the formula:
Work = force * distance * cos(θ)
Since the cable is being lifted straight up, the angle (θ) between the force and displacement is 0 degrees, and the cosine of 0 degrees is 1.
Work done by the tension in the cable = Tension * distance * cos(0)
Work done by the tension in the cable = 65.0928 N * 12.5 m * 1
Work done by the tension in the cable = 813.66 J
Therefore, the work done by the tension in the cable is 813.66 Joules.
(c) Now, let's calculate the work done by the person's weight. Since the person is being lifted vertically, the angle (θ) between the weight and displacement is 180 degrees, and the cosine of 180 degrees is -1.
Work done by the person's weight = Weight * distance * cos(180)
The weight can be calculated using the formula:
Weight = mass * acceleration due to gravity
Weight = 85.2 kg * 9.8 m/s²
Weight = 835.96 N
Work done by the person's weight = 835.96 N * 12.5 m * -1
Work done by the person's weight = -10449.5 J
Therefore, the work done by the person's weight is -10449.5 Joules.
(d) Finally, we can use the work-energy theorem to 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.
Total work done on the person = Work done by the tension in the cable + Work done by the person's weight
Total work done = 813.66 J + (-10449.5 J)
Total work done = -9626.84 J
The work done is negative because the person is being lifted against gravity.
By the work-energy theorem, the change in kinetic energy is equal to the total work done:
Change in kinetic energy = Total work done
Initial kinetic energy + Final kinetic energy = Total work done
Since the person starts from rest, the initial kinetic energy is zero:
0 + Final kinetic energy = -9626.84 J
Final kinetic energy = -9626.84 J
However, kinetic energy cannot be negative, so the final kinetic energy is zero.
Since kinetic energy is given by the formula:
Kinetic energy = 0.5 * mass * velocity^2
0.5 * mass * velocity^2 = 0
Since the mass is non-zero, the only way for the equation to hold true is if the final velocity is zero.
Hence, the final speed of the person is 0 m/s.