A mobile at the art museum has a 2.0 kg steel cat and a 4.0 kg steel dog suspended from a lightweight cable. It is found that θ1 = 29 ∘ when the center rope is adjusted to be perfectly horizontal. What is the tension in the rope 3?

To solve this problem, we'll analyze the forces involved and apply equilibrium conditions, since the mobile is in static equilibrium. The goal is to find the tension in rope 3, which we'll denote as \( T_3 \).

### Step-by-step approach:

1. **Free-Body Diagram Analysis**:
   - There are three ropes involved:
     - Rope 1: Supporting the 2.0 kg steel cat, at an angle \( \theta_1 = 29^\circ \).
     - Rope 2: Supporting the 4.0 kg steel dog, at an unknown angle \( \theta_2 \).
     - Rope 3: The center rope, which is perfectly horizontal, has the tension \( T_3 \) we need to find.

2. **Forces Acting on Each Object**:
   - The force due to gravity on the steel cat is:
     \[
     F_{\text{cat}} = m_{\text{cat}} \cdot g = 2.0 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 19.6 \, \text{N}
     \]
   - The force due to gravity on the steel dog is:
     \[
     F_{\text{dog}} = m_{\text{dog}} \cdot g = 4.0 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 39.2 \, \text{N}
     \]
   These weights act vertically downward.

3. **Equilibrium Conditions**:
   - In the horizontal direction, the horizontal components of the tensions must balance out:
     \[
     T_1 \cdot \cos(\theta_1) = T_2 \cdot \cos(\theta_2)
     \]
   - In the vertical direction, the sum of the vertical components of the tensions must balance the total weight:
     \[
     T_1 \cdot \sin(\theta_1) + T_2 \cdot \sin(\theta_2) = F_{\text{cat}} + F_{\text{dog}}
     \]

4. **Solve for Tension \( T_3 \)**:
   Since rope 3 is horizontal and the system is in equilibrium, \( T_3 \) must be equal to the horizontal components of either \( T_1 \) or \( T_2 \).

Thus, \( T_3 = T_1 \cdot \cos(\theta_1) \).

Let me calculate the exact value of \( T_3 \).The tension in rope 3 (\( T_3 \)) is approximately **35.36 N**.

Would you like further details or clarification on any part of the solution? 

Here are some related questions to deepen your understanding:
1. How would the tension in rope 3 change if the steel dog were heavier?
2. What happens to the angles of the ropes if rope 3 is not perfectly horizontal?
3. How does changing the angle \( \theta_1 \) affect the tension in rope 1?
4. What would be the tension in rope 2 if we know the angle \( \theta_2 \)?
5. How does static equilibrium apply to other structures like bridges or cranes?

**Tip:** In problems involving static equilibrium, always break the forces into horizontal and vertical components and apply the equilibrium conditions for each direction separately.

A mobile at the art museum has a 2.0 kg steel cat and a 4.0 kg steel dog suspended from a lightweight cable. It is found that θ1 = 29 ∘ when the center rope is adjusted to be perfectly horizontal. What is the tension in the rope 3?

This problem involves finding the tension in a horizontal rope of a mobile with a 2.0 kg steel cat and a 4.0 kg steel dog. Learn to apply concepts of static equilibrium and trigonometry to solve for the tension using equilibrium conditions. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation