Find the magnitude of the tension in each supporting cable shown below. In each case, the weight of the suspended body

To solve the problem of finding the magnitude of the tension in each supporting cable, we will proceed with the following steps:

1. Breakdown the Problem

1. Identify the weights and angles involved with the cables.
2. Set up the equilibrium equations for the system.

2. Relevant Concepts

1. For a suspended body in static equilibrium, the sum of the vertical forces must equal zero, as must the sum of the horizontal forces. This can be expressed as:

   • $\Sigma F_x = 0$
   • $\Sigma F_y = 0$

2. Use trigonometric identities to resolve tensions into their components:

   • $T_i \cos(\theta_i)$ for horizontal components
   • $T_i \sin(\theta_i)$ for vertical components

3. Analysis and Detail

1. Let $T_1$ and $T_2$ be the tensions in the cables at angles $\theta_1$ and $\theta_2$ respectively.

2. We can express the equations based on equilibrium:

   • Vertical forces: $T_1 \sin(\theta_1) + T_2 \sin(\theta_2) = W$
   • Horizontal forces: $T_1 \cos(\theta_1) = T_2 \cos(\theta_2)$

3. Rearranging the horizontal forces equation gives: $T_2 = T_1 \frac{\cos(\theta_1)}{\cos(\theta_2)}$

4. Substitute $T_2$ into the vertical forces equation: $T_1 \sin(\theta_1) + \left( T_1 \frac{\cos(\theta_1)}{\cos(\theta_2)} \right) \sin(\theta_2) = W$

4. Verify and Summarize

1. Rearrange to isolate $T_1$: $T_1 \left( \sin(\theta_1) + \frac{\cos(\theta_1)}{\cos(\theta_2)} \sin(\theta_2) \right) = W$ Hence, $T_1 = \frac{W}{\sin(\theta_1) + \frac{\cos(\theta_1)}{\cos(\theta_2)} \sin(\theta_2)}$

2. Substitute $T_1$ back into the equation for $T_2$: $T_2 = T_1 \frac{\cos(\theta_1)}{\cos(\theta_2)}$

Final Answer

The magnitudes of the tensions in the cables can be expressed as:

• $T_1 = \frac{W}{\sin(\theta_1) + \frac{\cos(\theta_1)}{\cos(\theta_2)} \sin(\theta_2)}$
• $T_2 = \frac{W \cos(\theta_1)}{\cos(\theta_2) \left( \sin(\theta_1) + \frac{\cos(\theta_1)}{\cos(\theta_2)} \sin(\theta_2) \right)}$

Make sure to substitute the appropriate values for $W$, $\theta_1$, and $\theta_2$ to find the numerical values of $T_1$ and $T_2$.