https://file.aiquickdraw.com/mathbot/file/17268339956413t2ozfmi.jpg

To solve this problem of finding the tensions \( T_1 \) and \( T_2 \) in the strings, we can use the concept of equilibrium. Since the system is in equilibrium, both the horizontal and vertical forces must sum to zero.

### Step 1: Break down the forces

- The weight of the object is 100 lb, acting downward. This is balanced by the vertical components of the tensions \( T_1 \) and \( T_2 \).
- The tension forces act at angles of 50° and 30° from the horizontal for \( T_1 \) and \( T_2 \), respectively.

We can resolve the tensions into their horizontal and vertical components.

For \( T_1 \) (tension at 50°):
- Horizontal component: \( T_1 \cos(50^\circ) \)
- Vertical component: \( T_1 \sin(50^\circ) \)

For \( T_2 \) (tension at 30°):
- Horizontal component: \( T_2 \cos(30^\circ) \)
- Vertical component: \( T_2 \sin(30^\circ) \)

### Step 2: Set up the equilibrium equations

#### Vertical equilibrium:
The sum of the vertical components of \( T_1 \) and \( T_2 \) must equal the weight of the object:

\[
T_1 \sin(50^\circ) + T_2 \sin(30^\circ) = 100
\]

#### Horizontal equilibrium:
The horizontal components must cancel each other out, since there is no horizontal motion:

\[
T_1 \cos(50^\circ) = T_2 \cos(30^\circ)
\]

### Step 3: Solve the system of equations

Now we have two equations:

1. \( T_1 \sin(50^\circ) + T_2 \sin(30^\circ) = 100 \)
2. \( T_1 \cos(50^\circ) = T_2 \cos(30^\circ) \)

We can solve this system for \( T_1 \) and \( T_2 \). I'll compute the values.The tensions in the strings are:

- \( T_1 \approx 87.94 \, \text{lb} \)
- \( T_2 \approx 65.27 \, \text{lb} \)

These tensions satisfy both the vertical and horizontal equilibrium conditions for the system.

Would you like a detailed step-by-step solution for solving these equations

A 100-lb weight hangs from a string as shown in the figure. Find the tensions T1 and T2 in the string.

Solve the equilibrium problem where a 100-lb weight hangs from a string, finding the tensions T1 and T2. This involves using vector components and force equilibrium equations for vertical and horizontal forces. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation