A horizontal rod with a mass of 10 kg and length 12 m is hinged to a wall at one end and supported by a cable which makes an angle of 30º with the rod at its other end. Calculate the tension in the cable and the force exerted by the hinge.

12 hours ago

Introduction

This is a problem involving static equilibrium. The horizontal rod is supported at one end by a hinge and at the other end by a cable inclined at an angle. To solve this, we apply the conditions of equilibrium:

Sum of all horizontal forces = 0
Sum of all vertical forces = 0
Sum of all torques (moments) = 0

Given Data

Mass of rod, m = 10 kg
Length of rod, L = 12 m
Angle between cable and rod = 30º
Acceleration due to gravity, g = 9.8 m/s²
Weight of rod, W = mg = 10 × 9.8 = 98 N

Step 1: Torque Equation (About the Hinge)

Let T be the tension in the cable. The weight of the rod acts at its center (L/2 = 6 m from hinge). The tension provides a vertical component (Tsinθ) which produces torque in the opposite direction to the weight.

∑τ = 0 ⇒ T × L × sin(30º) = W × (L/2)

T × 12 × 0.5 = 98 × 6

T × 6 = 588

T = 588 / 6 = 98 N

Step 2: Vertical Force Balance

Let the vertical reaction at the hinge be R_v.

∑F_vertical = 0 ⇒ R_v + Tsin(30º) = W

R_v + 98 × 0.5 = 98

R_v + 49 = 98 ⇒ R_v = 49 N

Step 3: Horizontal Force Balance

Let the horizontal reaction at the hinge be R_h.

∑F_horizontal = 0 ⇒ R_h = Tcos(30º)

R_h = 98 × cos(30º) = 98 × 0.866 ≈ 84.87 N

Step 4: Total Reaction Force at the Hinge

R = √(R_h² + R_v²) = √(84.87² + 49²) ≈ √(7201.6 + 2401) ≈ √9602.6 ≈ 98 N

Final Answers

Tension in the cable: 98 N
Horizontal force by hinge: 84.87 N
Vertical force by hinge: 49 N
Total hinge force: ~98 N

Conclusion

By applying the principles of static equilibrium, we determined that the tension in the supporting cable is 98 N and the hinge provides both horizontal and vertical support to maintain the system in balance. This problem illustrates the practical application of torque and force balance in real-world structures.