In a daring rescue by helicopter, two men with a combined mass of 172-kg are lifted to safety. (a) If the helicopter lifts the men straight up with constant acceleration, is the tension in the rescue cable greater than, less than, or equal to the combined weight of the men? Explain. (b) Determine the tension in the cable if the men are lifted with a constant acceleration of 1.10 m/s2.

F = M (g + a)

The acceleration, a, is positive if upward. That makes the tension greater than the weight in that case.

(a) In order to determine whether the tension in the rescue cable is greater than, less than, or equal to the combined weight of the men, we need to consider the forces acting on the system.

When the helicopter lifts the men straight up with constant acceleration, there are two forces acting on the men: their weight (mg) and the tension in the rescue cable (T). According to Newton's second law, the net force acting on the men is equal to the product of their mass and acceleration, so we have:

Net force = mass × acceleration

The net force is the difference between the tension in the cable and the weight of the men, so we can rewrite the equation as:

T - mg = m × acceleration

Now, let's consider the possible cases:

1. Tension in the cable is greater than the combined weight: If the tension in the cable is greater than the combined weight (mg), the net force would be positive and greater than zero. This would result in an upward acceleration greater than zero, suggesting that the helicopter is lifting the men. Therefore, the tension in the cable is greater than the combined weight.

2. Tension in the cable is equal to the combined weight: If the tension in the cable is equal to the combined weight, the net force would be zero. This would result in zero acceleration, meaning the men would remain stationary. However, in this scenario, the men are being lifted, indicating that the tension in the cable must be greater than the combined weight. So, the tension in the cable is not equal to the combined weight.

3. Tension in the cable is less than the combined weight: If the tension in the cable is less than the combined weight, the net force would be negative. This would result in a downward acceleration, implying that the men are not being lifted. Again, this contradicts our given scenario, so the tension in the cable cannot be less than the combined weight.

Therefore, the tension in the rescue cable is greater than the combined weight of the men.

(b) To determine the tension in the cable when the men are lifted with a constant acceleration of 1.10 m/s², we can use the same formula as before:

T - mg = m × acceleration

Plugging in the values, we have:

T - (m1g + m2g) = (m1 + m2) × acceleration

Where m1 and m2 are the masses of the two men, g is the acceleration due to gravity (approximately 9.8 m/s²), and acceleration is 1.10 m/s².

Given that the combined mass of the men is 172 kg, let's assume one man has a mass of m1 kg, and the other man has a mass of m2 kg.

m1 + m2 = 172 kg

Since the problem doesn't specify the mass of each individual, we cannot find the exact values for m1 and m2. However, we can solve the equation using this assumption.

T - (m1g + m2g) = (m1 + m2) × acceleration

T - (m1g + m2g) = 172 kg × 1.10 m/s²

At this point, without further information, it is not possible to determine the exact tension in the cable (T).

To answer part (a) of the question, we need to consider the forces acting on the men and the cable when they are lifted straight up with constant acceleration. The forces at play here are the gravitational force (weight) acting downward on the men and the tension in the rescue cable acting upward.

According to Newton's second law of motion, the net force acting on an object is equal to the mass of the object times its acceleration. In this case, since the helicopter is lifting the men with constant acceleration, the net force acting on the men will be the tension in the cable.

If the tension in the cable is greater than the combined weight of the men, then the net force will be upward and the men will be lifted. If the tension in the cable is less than the combined weight of the men, then the net force will be downward and the men will not be lifted. Finally, if the tension in the cable is equal to the combined weight of the men, then the net force will be zero and the men will remain stationary.

In this scenario, the tension in the cable needs to be greater than the combined weight of the men in order to lift them. This is because the tension in the cable should provide a net upward force that is greater than the combined gravitational force acting downward on the men. Only then will the men be lifted off the ground.

Moving on to part (b) of the question, we now know that the tension in the cable needs to be greater than the combined weight of the men to lift them. To determine the tension in the cable when the men are lifted with a constant acceleration of 1.10 m/s^2, we can use the equation:

Tension = (Mass of the men) × (Acceleration)

Given that the combined mass of the men is 172 kg and the acceleration is 1.10 m/s^2, we can substitute these values into the equation:

Tension = 172 kg × 1.10 m/s^2

Tension = 189.2 N

Therefore, the tension in the cable when the men are lifted with a constant acceleration of 1.10 m/s^2 is 189.2 N.