Question

A helicopter of mass \(M\) is lowering a truck of mass \(m\) onto the deck of a ship. (a) At first, the helicopter and the truck move downward together (the length of the cable doesn't change). If their downward speed is decreasing at a rate of \(0.10 g,\) what is the tension in the cable? (b) As the truck gets close to the deck, the helicopter stops moving downward. While it hovers, it lets out the cable so that the truck is still moving downward. If the truck's downward speed is decreasing at a rate of \(0.10 g,\) while the helicopter is at rest, what is the tension in the cable?

Short answer

Question: Calculate the tension in the cable for case (a) and case (b), given that the mass of the truck (m) is 2000 kg, mass of the helicopter (M) is 8000 kg, and the acceleration due to gravity (g) is 9.81 m/s². Solution: For case (a): Using the formula, \(T = (M + m)g - (M + m)(0.10g)\): T = (8000 + 2000)(9.81) - (8000 + 2000)(0.10)(9.81) T = (10,000)(9.81) - (10,000)(0.981) T = 98,100 - 9,810 T = 88,290 N The tension in the cable for case (a) is 88,290 N. For case (b): Using the formula, \(T = mg - m(0.10g)\): T = (2000)(9.81) - (2000)(0.10)(9.81) T = 19,620 - 1,962 T = 17,658 N The tension in the cable for case (b) is 17,658 N.

Step-by-step solution

Analyzing case (a) - Helicopter and truck moving downward together

When the helicopter and truck are moving downward together, their combined mass can be represented as \((M + m)\). The downward speed decreases at a rate of \(0.10 g\), which represents the acceleration. Since the acceleration is acting upward, the net force acting on the combined system can be written as \(F_{net} = (M + m)(-0.10g)\). This net force is due to the tension force in the cable and the gravitational force acting on the system.

Calculate the tension for case (a)

To calculate the tension in the cable during case (a), we'll use Newton's second law of motion. The net force acting on the combined system is given by \(F_{net} = T - (M + m)g\). From Step 1, we have \(F_{net} = (M + m)(-0.10g)\). Therefore, we get the equation: \(T - (M + m)g = (M + m)(-0.10g)\). Solve for tension, \(T\), to get: \(T = (M + m)g - (M + m)(0.10g)\)

Analyzing case (b) — Helicopter hovering and the truck moving downward

In this case, the helicopter is stationary while the truck is moving downward. The forces acting on the truck include the tension in the cable and the gravitational force acting on the truck. The acceleration of the system remains the same as in the previous case with \(-0.10g\).

Calculate the tension for case (b)

To calculate the tension in the cable during case (b), we'll use Newton's second law of motion for the truck only. The net force acting on the truck is given by \(F_{net} = T - mg\). From Step 3, we have \(F_{net} = m(-0.10g)\). Therefore, we get the equation: \(T - mg = m(-0.10g)\). Solve for tension, \(T\), to get: \(T = mg - m(0.10g)\)