Answer :
To solve this problem, we'll first analyze the forces acting on each crate and apply Newton's Second Law of Motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration:
[tex]F_{net} = m \cdot a[/tex]
Analyzing the forces on the bottom crate (103 kg):
Weight of the bottom crate (W2):
[tex]W_2 = m_2 \cdot g = 103 \text{ kg} \cdot 9.81 \text{ m/s}^2 = 1010.43 \text{ N}[/tex]
where [tex]g = 9.81 \text{ m/s}^2[/tex] is the acceleration due to gravity.
Tension in the cable between the crates (T2):
The net force acting on the bottom crate is the tension in the cable (T2) minus the weight of the bottom crate, and it accelerates upward at 1.41 m/s². Therefore, we can use Newton's Second Law:
[tex]T_2 - W_2 = m_2 \cdot a[/tex]
Plug in the known values:
[tex]T_2 - 1010.43 = 103 \cdot 1.41[/tex]
[tex]T_2 - 1010.43 = 145.23[/tex]
[tex]T_2 = 145.23 + 1010.43[/tex]
[tex]T_2 = 1155.66 \text{ N}[/tex]
Analyzing the forces on the upper crate (289 kg):
Weight of the upper crate (W1):
[tex]W_1 = m_1 \cdot g = 289 \text{ kg} \cdot 9.81 \text{ m/s}^2 = 2833.29 \text{ N}[/tex]
Tension in the cable between the helicopter and the upper crate (T1):
The tension (T1) must overcome both the weight of the upper crate and the tension (T2) from the bottom crate as this crate also accelerates upward at 1.41 m/s².
Using Newton's Second Law:
[tex]T_1 - W_1 - T_2 = m_1 \cdot a[/tex]
Plug in the known values:
[tex]T_1 - 2833.29 - 1155.66 = 289 \cdot 1.41[/tex]
[tex]T_1 - 3988.95 = 407.49[/tex]
[tex]T_1 = 407.49 + 3988.95[/tex]
[tex]T_1 = 4396.44 \text{ N}[/tex]
Therefore, the tension in the upper cable (T1) is 4396.44 N, and the tension in the cable between the crates (T2) is 1155.66 N.
