A 110 kg object is suspended below the xy-plane by a short cable attached to a steel ring located at the origin (0, 0, 0) as shown below in the figure. An object, 5 cables, 4 vectors, and a square on the x y z coordinate space. The square has its corners at (5, "-5," 5), (5, 5, 5), (-5, 5, 5), and (-5, "-5," 5). 4 of the cables connect the corners of the square to the origin. The fifth cable extends from where the other 4 meet down to the 110 kg object. The four vectors all begin at the origin and extend up along one of the cables. The vector on the cable connecting the origin to (5, "-5," 5) is labeled T_1. The vector on the cable connecting the origin to (5, 5, 5) is labeled T_2. The vector on the cable connecting the origin to (-5, 5, 5) is labeled T_3. The vector on the cable connecting the origin to (-5, "-5," 5) is labeled T_4.110 kg The ring itself is attached to 4 ropes which are attached to the ceiling, which is 5 meters above and parallel to the xy-plane. The ropes are attached to the ceiling at the points (−5, −5, 5), (5, −5, 5), (5, 5, 5), and (−5, 5, 5). Use symmetry to find the exact magnitudes T1, T2, T3, and T4 of the tensions in the cables, and use g = 9.8 m/s2 for the acceleration of gravity.

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 \( 466.92 \, \text{N} \).

---

Would you like further details or have any questions?

### 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.

Learn how to calculate tensions (T_1, T_2, T_3, T_4) in cables supporting a 110 kg object suspended in space. This problem involves statics, vector analysis, and trigonometry, with detailed step-by-step calculations. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation