Example 4.6 Solution: Equilibrium of a Suspended Mass Step by Step

  How to Find the Angle of a Rope in Equilibrium | Example 4.6

- Dr.Sanjaykumar Pawar  

Free body diagram and equilibrium analysis of a 6 kg mass suspended by a rope with a 50 N horizontal force applied at the midpoint.

INTERNAL LINKS

1. What is Equilibrium of Forces in Physics?

2. Introduction to Free Body Diagrams (FBD)

3. Types of Equilibrium: Static and Dynamic

4. Tension in Strings and Ropes Explained

5. Resolution of Forces into Components

6. Laws of Motion and Force Analysis

7. Vector Addition and Resolution of Vectors

8. NCERT Class 11 Physics Chapter 4 Notes

9. Solved Numerical Problems on Equilibrium

10. Engineering Mechanics Basics for Beginners

11. Common Mistakes in Free Body Diagrams

12. Practice Questions on Equilibrium and Tension

Example 4.6 See Fig. 4.8. A mass of 6 kg is suspended by a rope of length 2 m from the ceiling. A force of 50 N in the horizontal direction is applied at the mid- point P of the rope, as shown. What is the angle the rope makes with the vertical in equilibrium ? (Take g = 10 m s-2). Neglect the mass of the rope.

Example 4.6 - Equilibrium of a Suspended Mass

Question

A mass of 6 kg is suspended by a rope of length 2 m. A horizontal force of 50 N is applied at the midpoint P of the rope. Find the angle made by the rope with the vertical in equilibrium.

Given:

• Mass (m) = 6 kg
• Horizontal Force = 50 N
• g = 10 m/s²
• Rope is massless

Step 1: Identify Forces on the Mass

The mass is hanging at rest. Therefore, it is in equilibrium. The forces acting on the mass are:

• Weight (W) acting downward
• Tension (T₂) acting upward

Since the mass is in equilibrium:

T₂ = mg

T₂ = 6 × 10

T₂ = 60 N

Therefore,

T₂ = 60 N

Step 2: Consider Equilibrium at Point P

Three forces act at point P:

• Tension T₁ in the upper rope
• Tension T₂ = 60 N in the lower rope
• Horizontal force = 50 N

Since point P is also in equilibrium:

• Net horizontal force = 0
• Net vertical force = 0

Step 3: Resolve Tension T₁ into Components

If the rope makes an angle θ with the vertical:

• Vertical component = T₁ cos θ
• Horizontal component = T₁ sin θ

Step 4: Apply Vertical Equilibrium

Upward force = Downward force

T₁ cos θ = T₂

T₁ cos θ = 60

Step 5: Apply Horizontal Equilibrium

Horizontal component balances the applied force.

T₁ sin θ = 50

Step 6: Divide the Equations

(T₁ sin θ) / (T₁ cos θ) = 50 / 60

tan θ = 5 / 6

Step 7: Calculate the Angle

θ = tan⁻¹ (5/6)

θ ≈ 39.8°

Final Answer: θ ≈ 40°

Important Observation

• The answer does not depend on the rope length (2 m).
• The answer does not depend on where the 50 N force is applied.
• Only the force balance conditions determine the angle.

Exam Shortcut

T₂ = mg = 6 × 10 = 60 N

T₁ cos θ = 60

T₁ sin θ = 50

tan θ = 50/60 = 5/6

θ = tan⁻¹(5/6) ≈ 40°

Answer = 40°