To solve for the tensions $T_1$, $T_2$, $T_3$, and $T_4$ in the cables, we'll use symmetry and basic principles of mechanics.

Step 1: Analyze the Forces

The object has a mass of 110 kg, which means its weight $W$ is: $W = mg = 110 \times 9.8 = 1078 \, \text{N}$ This weight is balanced by the vertical components of the tensions in the four cables. The setup is symmetric, so each cable should contribute equally to supporting the weight.

Step 2: Symmetry Consideration

The four cables are attached symmetrically at the points $(5, -5, 5)$, $(5, 5, 5)$, $(-5, 5, 5)$, and $(-5, -5, 5)$. Since the cables are symmetrical, the magnitudes of the tensions $T_1 = T_2 = T_3 = T_4 = T$.

Step 3: Determine the Vertical Component of the Tension

Each cable forms a triangle with the vertical line from the origin to the object, the horizontal line along the xy-plane, and the cable itself.

The length of each cable can be calculated using the distance formula. Since each cable connects the origin (0, 0, 0) to a point on the square, the length $L$ of each cable is: $L = \sqrt{5^2 + 5^2 + 5^2} = \sqrt{75} = 5\sqrt{3} \, \text{meters}$

Step 4: Calculate the Vertical Component of the Tension

The vertical component $T_z$ of the tension $T$ in each cable is given by: $T_z = T \times \cos(\theta)$ where $\theta$ is the angle between the cable and the vertical axis. The cosine of the angle $\theta$ can be found by: $\cos(\theta) = \frac{\text{vertical height}}{\text{length of the cable}} = \frac{5}{5\sqrt{3}} = \frac{1}{\sqrt{3}}$

Step 5: Equilibrium Condition

The sum of the vertical components of the tension must equal the weight of the object: $4 \times T_z = W$ $4 \times T \times \frac{1}{\sqrt{3}} = 1078$

Step 6: Solve for $T$

$T \times \frac{4}{\sqrt{3}} = 1078$ $T = \frac{1078 \times \sqrt{3}}{4} = \frac{1078 \times 1.732}{4} \approx 466.92 \, \text{N}$

Thus, the magnitude of each tension $T_1 = T_2 = T_3 = T_4 = T \approx 466.92 \, \text{N}$.

Final Answer

The magnitude of the tension in each cable is approximately $466.92 \, \text{N}$.

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Related Questions

1. How would the tensions change if the object’s mass were increased?
2. What would happen to the tensions if one of the cables were longer?
3. How does the angle of the cables affect the tension in them?
4. What role does the acceleration due to gravity play in determining the tensions?
5. How could we use vectors to represent the forces in this system?

Tip:

Always check if symmetry can simplify a problem before diving into complex calculations. It can save time and reduce the risk of errors.