A helicopter is lifting two crates simultaneously. One crate with a mass of 151 kg is attached to the helicopter by cable A. The second crate with a mass of 79 kg is hanging below the first crate and attached to the first crate by cable B. As the helicopter accelerates upward at a rate of 2.0 m/s2, what is the tension in each of the two cables?
Cable A .....in kN
Cable B .....in kN
To find the tension in each cable, we can use the concepts of Newton's second law and free-body diagrams.
First, let's consider the forces acting on the crates. The weight of each crate can be calculated using the formula:
Weight (W) = mass (m) * acceleration due to gravity (g)
For the first crate with a mass of 151 kg:
Weight of crate 1 = 151 kg * 9.8 m/s^2 (acceleration due to gravity)
Weight of crate 1 = 1479.8 N
For the second crate with a mass of 79 kg:
Weight of crate 2 = 79 kg * 9.8 m/s^2
Weight of crate 2 = 774.2 N
Since the crates are accelerating upward, we need to consider the net force acting on each crate. The net force is given by:
Net force (F_net) = mass (m) * acceleration (a)
For the first crate:
F_net of crate 1 = 151 kg * 2.0 m/s^2 (acceleration of the helicopter)
F_net of crate 1 = 302 N
For the second crate:
F_net of crate 2 = 79 kg * 2.0 m/s^2
F_net of crate 2 = 158 N
Now, let's consider the forces acting on each crate individually. For the first crate, we have two forces:
1. Tension in cable A (T_A): This force is directed upward and equal to the net force acting on the crate when considering upward motion.
T_A - Weight of crate 1 = F_net of crate 1
T_A - 1479.8 N = 302 N
T_A = 1781.8 N
Thus, the tension in cable A is 1781.8 N or 1.78 kN (kilonewtons).
For the second crate, we have three forces:
1. Tension in cable B (T_B): This force is directed upward and equal to the net force acting on the crate when considering upward motion.
T_B - Weight of crate 2 = F_net of crate 2
T_B - 774.2 N = 158 N
T_B = 932.2 N
Thus, the tension in cable B is 932.2 N or 0.932 kN (kilonewtons).
So, the tension in cable A is 1.78 kN and the tension in cable B is 0.932 kN.
To find the tension in each cable, we need to consider the forces acting on each crate separately.
For crate A:
- The weight of crate A is equal to its mass multiplied by the acceleration due to gravity: Fg = m * g
where m = 151 kg and g = 9.8 m/s^2 (acceleration due to gravity).
- The tension in cable A is equal to the net force acting on crate A: Tension A = Fg - Fnet
where Fnet = m * a (mass multiplied by acceleration).
- Substituting the values into the equation, we have:
Tension A = (m * g) - (m * a)
Tension A = (151 kg) * (9.8 m/s^2) - (151 kg) * (2.0 m/s^2)
Tension A = 1481.8 N - 302 N
Tension A = 1179.8 N
The tension in Cable A is 1179.8 N (or approximately 1.18 kN).
For crate B:
- The weight of crate B is equal to its mass multiplied by the acceleration due to gravity: Fg = m * g
where m = 79 kg and g = 9.8 m/s^2.
- The tension in cable B is equal to the net force acting on crate B: Tension B = Fg - Fnet
- The net force acting on crate B is the same as crate A because they are connected by the same cable.
- Substituting the values into the equation, we have:
Tension B = (m * g) - (m * a)
Tension B = (79 kg) * (9.8 m/s^2) - (79 kg) * (2.0 m/s^2)
Tension B = 774.2 N - 158 N
Tension B = 616.2 N
The tension in Cable B is 616.2 N (or approximately 0.62 kN).
Therefore, the tension in Cable A is approximately 1.18 kN and the tension in Cable B is approximately 0.62 kN.