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 of the two cables, we can use Newton's second law of motion. The sum of the forces acting on an object is equal to the mass of the object multiplied by its acceleration.
First, let's find the tension in Cable A.
1. Determine the force acting on the first crate (attached to Cable A):
F1 = m1 * a
where m1 is the mass of the first crate and a is the acceleration of the helicopter.
Plug in the values:
m1 = 151 kg
a = 2.0 m/s^2
F1 = (151 kg) * (2.0 m/s^2)
F1 = 302 kg*m/s^2
However, since we need to find the tension in Cable A, which is measured in kN (kilonewtons), we need to convert the force from Newtons to kilonewtons.
1 kN = 1000 N
F1 = (302 kg*m/s^2) / (1000 N/kN)
F1 = 0.302 kN
The tension in Cable A is 0.302 kN.
Next, let's find the tension in Cable B.
2. Determine the net force acting on the second crate (attached to Cable B):
The net force is equal to the force acting on the second crate minus the weight of the second crate.
Force acting on the second crate:
F2 = m2 * a
where m2 is the mass of the second crate and a is the acceleration of the helicopter.
Plug in the values:
m2 = 79 kg
a = 2.0 m/s^2
F2 = (79 kg) * (2.0 m/s^2)
F2 = 158 kg*m/s^2
Weight of the second crate:
W2 = m2 * g
where g is the acceleration due to gravity.
We assume g to be approximately 9.8 m/s^2.
W2 = (79 kg) * (9.8 m/s^2)
W2 = 774.2 kg*m/s^2
Net force on the second crate:
Net force = F2 - W2
Net force = (158 kg*m/s^2) - (774.2 kg*m/s^2)
Net force = -616.2 kg*m/s^2
As you can see, the net force is negative because the upward force (F2) is smaller than the downward force (W2). This negative net force represents the tension in Cable B.
Convert the net force to kilonewtons:
Net force = (-616.2 kg*m/s^2) / (1000 N/kN)
Net force = -0.6162 kN
The tension in Cable B is -0.6162 kN. Although it has a negative sign, it indicates that Cable B is pulling downward on the second crate, counteracting the upward force.
To find the tension in each of the two cables, we need to consider the forces acting on the crates.
First, let's consider the crate attached to cable B. The tension in cable B is equal to the weight of both crates combined minus the force required to accelerate them upward. The weight of an object is given by the mass multiplied by the acceleration due to gravity.
For the first crate (151 kg):
Weight of the first crate = mass of the first crate * acceleration due to gravity
Next, let's consider the tension in cable A. Since both crates are attached to the helicopter, the tension in cable A will be the total weight of both crates plus the force required to accelerate them.
Now let's calculate the values step-by-step:
1. Calculate the weight of the first crate:
Weight of the first crate = mass of the first crate * acceleration due to gravity
Given:
mass of the first crate = 151 kg
acceleration due to gravity = 9.8 m/s^2
Weight of the first crate = 151 kg * 9.8 m/s^2
2. Calculate the total weight of both crates:
Total weight of both crates = weight of the first crate + weight of the second crate
Given:
mass of the second crate = 79 kg
Weight of the second crate = mass of the second crate * acceleration due to gravity
Total weight of both crates = (mass of the first crate + mass of the second crate) * acceleration due to gravity
3. Calculate the tension in cable A:
Tension in cable A = total weight of both crates + force required to accelerate them
Given:
acceleration = 2.0 m/s^2
Force required to accelerate the crates = total mass of both crates * acceleration
Tension in cable A = total weight of both crates + force required to accelerate the crates
Now, let's calculate the values:
1. Calculate the weight of the first crate:
Weight of the first crate = 151 kg * 9.8 m/s^2
2. Calculate the total weight of both crates:
Total weight of both crates = (151 kg + 79 kg) * 9.8 m/s^2
3. Calculate the force required to accelerate the crates:
Force required to accelerate the crates = (151 kg + 79 kg) * 2.0 m/s^2
4. Calculate the tension in cable A:
Tension in cable A = total weight of both crates + force required to accelerate the crates
Finally, convert the tension from Newtons (N) to kilonewtons (kN) by dividing by 1000:
Tension in cable A (in kN) = (Tension in cable A in N) / 1000
Repeat the same calculations for the tension in cable B.