Consider a​ w-N weight suspended by two wires as shown in the accompanying figure.  If the magnitude of vector Bold Upper F 2 is 150 ​N, find w and the magnitude of vector Bold Upper F 1. 49 degrees and 32 degrees


To solve this problem, we need to find two things:

1. The weight \( w \) suspended by the two wires.
2. The magnitude of the tension in the first wire, denoted as \( F_1 \).

Given:
- The tension in the second wire, \( F_2 = 150 \) N.
- The angles of the wires with respect to the horizontal: \( \theta_1 = 49^\circ \) (associated with \( F_1 \)) and \( \theta_2 = 32^\circ \) (associated with \( F_2 \)).

We will use the equilibrium conditions for forces in both the horizontal (x) and vertical (y) directions.

### Step 1: Set Up Equilibrium Equations

For equilibrium, the sum of all forces in both the x and y directions must be zero:

1. **Horizontal (x-direction):**

\[
F_1 \cos(49^\circ) = F_2 \cos(32^\circ)
\]

2. **Vertical (y-direction):**

\[
F_1 \sin(49^\circ) + F_2 \sin(32^\circ) = w
\]

### Step 2: Solve for \( F_1 \)

Rearrange the first equation to solve for \( F_1 \):

\[
F_1 = F_2 \frac{\cos(32^\circ)}{\cos(49^\circ)}
\]

Now, substitute the values:

\[
F_1 = 150 \times \frac{\cos(32^\circ)}{\cos(49^\circ)}
\]

Let's compute this value.The magnitude of the tension in the first wire, \( F_1 \), is approximately **193.9 N**.

### Step 3: Solve for the Weight \( w \)

Now, use the second equation to solve for \( w \):

\[
w = F_1 \sin(49^\circ) + F_2 \sin(32^\circ)
\]

Substitute the values for \( F_1 \), \( F_2 \), and the angles:

\[
w = 193.9 \times \sin(49^\circ) + 150 \times \sin(32^\circ)
\]

Let's calculate \( w \).The weight \( w \) suspended by the two wires is approximately **225.8 N**.

### Summary
- The tension in the first wire, \( F_1 \), is **193.9 N**.
- The weight, \( w \), is **225.8 N**.

Would you like more details or have any questions about this solution?

### Related Questions
1. How would the tensions change if the angles of the wires were different?
2. What happens to the tensions if the weight \( w \) increases?
3. How do we calculate the components of forces in different directions?
4. What are the real-world applications of such a tension problem?
5. How can we solve similar problems with more than two wires?

**Tip:** When dealing with forces, always consider both their magnitudes and directions to maintain equilibrium.

Learn how to solve a physics problem involving the equilibrium of forces in a two-wire system. Find the weight suspended by the wires and the tension in each wire using trigonometry and equilibrium conditions. Discover the detailed step-by-step solution with numerical calculations. Suitable for high school students studying physics or engineering.

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