A cable is lifting a construction worker and a crate, as the drawing shows. The weights of the worker and crate are 930 and 1640 N, respectively. The acceleration of the cable is 0.620 m/s2, upward.
(a) What is the tension in the cable below the worker?
(b) What is the tension in the cable above the worker?
To find the tension in the cable below the worker, we need to consider the forces acting on the system. The forces involved are the weight of the worker and the tension in the cable.
Let's start by finding the tension in the cable below the worker.
(a) Tension in the cable below the worker:
Step 1: Identify the forces acting on the worker:
- Weight of the worker (930 N) acting downward
- Tension in the cable acting upward
Step 2: Apply Newton's second law of motion:
The sum of all the forces in the vertical direction equals the mass of the worker times the acceleration:
ΣF = m · a
The net force (ΣF) in the vertical direction is the difference between the tension in the cable and the weight of the worker:
Tension - Weight = m · a
Step 3: Substitute the known values into the equation:
Tension - 930 N = (mass of the worker / g) · a
Step 4: Solve for the tension:
Tension = (mass of the worker / g) · a + 930 N
Here, "g" represents the acceleration due to gravity (approximately 9.8 m/s^2).
Now that we have the formula, let's plug in the values:
Tension = (930 N / 9.8 m/s^2) · 0.620 m/s^2 + 930 N
Therefore, the tension in the cable below the worker is the sum of the weight and the additional tension due to the acceleration:
Tension = 930 N + 601.86 N ≈ 1531.86 N
So, the tension in the cable below the worker is approximately 1531.86 N.
(b) To find the tension in the cable above the worker, we can use a similar approach.
Step 1: Identify the forces acting on the crate:
- Weight of the crate (1640 N) acting downward
- Tension in the cable acting upward
Step 2: Apply Newton's second law of motion:
The sum of all the forces in the vertical direction equals the mass of the crate times the acceleration:
ΣF = m · a
The net force (ΣF) in the vertical direction is the sum of the tension in the cable and the weight of the crate:
Tension + Weight = m · a
Step 3: Substitute the known values into the equation:
Tension + 1640 N = (mass of the crate / g) · a
Step 4: Solve for the tension:
Tension = (mass of the crate / g) · a - 1640 N
Now, let's plug in the values:
Tension = (1640 N / 9.8 m/s^2) · 0.620 m/s^2 - 1640 N
Therefore, the tension in the cable above the worker is the difference between the additional tension due to the acceleration and the weight:
Tension = -441.86 N
The negative sign indicates that the tension in the cable above the worker is directed downward. Therefore, the tension in the cable above the worker is approximately -441.86 N.
Note: The tension in a cable or rope cannot be negative in reality. In this case, the negative sign indicates that the cable is in compression, meaning it is being pushed together rather than pulled apart. However, since cables are generally not compressible, the tension would effectively be zero.