A rescue helicopter is lifting a man (weight = 797 N) from a capsized boat by means of a cable and harness. (a) What is the tension in the cable when the man is given an initial upward acceleration of 0.711 m/s2? (b) What is the tension during the remainder of the rescue when he is pulled upward at a constant velocity?

To find the tension in the cable, we can use Newton's second law of motion, which states that the force on an object is equal to its mass multiplied by its acceleration. In this case, the force is the tension in the cable.

(a) When the man is given an initial upward acceleration of 0.711 m/s^2, we can start by calculating the force required to accelerate the man upward. The force required is equal to the mass of the man multiplied by the acceleration.

Given:
Weight of the man (force due to gravity) = 797 N
Acceleration = 0.711 m/s^2

Using Newton's second law, we have:
Force = Mass x Acceleration

Since force equals weight, we can rearrange the equation to find the mass of the man:

Mass = Weight / Acceleration

Mass = 797 N / 9.8 m/s^2 (Note: we use the acceleration due to gravity, 9.8 m/s^2)

Mass ≈ 81.33 kg

Now, we can calculate the tension in the cable using Newton's second law:

Tension = Mass x Acceleration

Tension = 81.33 kg x 0.711 m/s^2

Tension ≈ 57.83 N

Therefore, the tension in the cable when the man is given an initial upward acceleration of 0.711 m/s^2 is approximately 57.83 N.

(b) During the remainder of the rescue when the man is pulled upward at a constant velocity, there is no net acceleration acting on the man. This means that the upward force provided by the tension in the cable is equal to the downward force of gravity.

Therefore, the tension in the cable during the remainder of the rescue when the man is pulled upward at a constant velocity is equal to his weight.

Tension = Weight = 797 N

To calculate the tension in the cable, we need to consider the forces acting on the man at different stages of rescue.

(a) When the man is given an initial upward acceleration of 0.711 m/s^2:
In this case, the net force acting on the man is the force due to his weight (mg) minus the tension in the cable (T). We can use Newton's second law of motion (F = ma) to find the tension.
The equation will be: T - mg = ma

Given:
Weight of the man (mg) = 797 N
Acceleration (a) = 0.711 m/s^2

Substituting the values into the equation:
T - 797 N = (797 N) * (0.711 m/s^2)

Now, we can solve for T:
T = 797 N + (797 N * 0.711 m/s^2)

(b) When the man is pulled upward at a constant velocity:
When the man is pulled upward at a constant velocity, the net force acting on him is zero. This means that the tension in the cable is equal to his weight. So, the tension will be the same as the weight of the man.

Therefore, the tension during the remainder of the rescue (when the man is pulled upward at a constant velocity) is 797 N.