Two children hang by their hands from the same tree branch. The branch is straight, and grows out from the tree trunk at an angle of 27.0° above the horizontal. One child, with a mass of 42.0 kg, is hanging 1.70 m along the branch from the tree trunk. The other child, with a mass of 27.0 kg, is hanging 2.40 m from the tree trunk. What is the magnitude of the net torque exerted on the branch by the children? Assume that the axis is located where the branch joins the tree trunk and is perpendicular to the plane formed by the branch and the trunk.

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**Topic: Understanding Torque in a Physical System**

**Example Problem: Calculating Net Torque on a Tree Branch**

**Problem Statement:**
Two children hang by their hands from the same tree branch. The branch is straight and grows out from the tree trunk at an angle of 27.0° above the horizontal. One child, with a mass of 42.0 kg, is hanging 1.70 m along the branch from the tree trunk. The other child, with a mass of 27.0 kg, is hanging 2.40 m from the tree trunk. What is the magnitude of the net torque exerted on the branch by the children? Assume that the axis is located where the branch joins the tree trunk and is perpendicular to the plane formed by the branch and the trunk.

**Detailed Solution:**
1. **Identify Forces and Distances:**
    - Child 1: Mass = 42.0 kg, Distance from trunk = 1.70 m
    - Child 2: Mass = 27.0 kg, Distance from trunk = 2.40 m

2. **Convert Mass to Force:**
    - Force due to gravity = mass × acceleration due to gravity (g = 9.8 m/s²)
    - Force Child 1 (F₁) = 42.0 kg × 9.8 m/s² = 411.6 N
    - Force Child 2 (F₂) = 27.0 kg × 9.8 m/s² = 264.6 N

3. **Calculate Torque for Each Child:**
    - Torque (τ) = Force (F) × Lever Arm (r) × sin(θ)
    - Lever Arm is the distance from the axis to the point where force is applied
    - θ = 27.0°

    For Child 1:
    - τ₁ = 411.6 N × 1.70 m × sin(27.0°)

    For Child 2:
    - τ₂ = 264.6 N × 2.40 m × sin(27.0°)

4. **Compute Sine of 27.0°:**
    - sin(27.0°) ≈ 0.454

5. **Calculate the Torques:**
    - τ₁ = 411.6 N × 1.70 m × 0
Transcribed Image Text:**Topic: Understanding Torque in a Physical System** **Example Problem: Calculating Net Torque on a Tree Branch** **Problem Statement:** Two children hang by their hands from the same tree branch. The branch is straight and grows out from the tree trunk at an angle of 27.0° above the horizontal. One child, with a mass of 42.0 kg, is hanging 1.70 m along the branch from the tree trunk. The other child, with a mass of 27.0 kg, is hanging 2.40 m from the tree trunk. What is the magnitude of the net torque exerted on the branch by the children? Assume that the axis is located where the branch joins the tree trunk and is perpendicular to the plane formed by the branch and the trunk. **Detailed Solution:** 1. **Identify Forces and Distances:** - Child 1: Mass = 42.0 kg, Distance from trunk = 1.70 m - Child 2: Mass = 27.0 kg, Distance from trunk = 2.40 m 2. **Convert Mass to Force:** - Force due to gravity = mass × acceleration due to gravity (g = 9.8 m/s²) - Force Child 1 (F₁) = 42.0 kg × 9.8 m/s² = 411.6 N - Force Child 2 (F₂) = 27.0 kg × 9.8 m/s² = 264.6 N 3. **Calculate Torque for Each Child:** - Torque (τ) = Force (F) × Lever Arm (r) × sin(θ) - Lever Arm is the distance from the axis to the point where force is applied - θ = 27.0° For Child 1: - τ₁ = 411.6 N × 1.70 m × sin(27.0°) For Child 2: - τ₂ = 264.6 N × 2.40 m × sin(27.0°) 4. **Compute Sine of 27.0°:** - sin(27.0°) ≈ 0.454 5. **Calculate the Torques:** - τ₁ = 411.6 N × 1.70 m × 0
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