Static Equilibrium: Beam Supported by Cable and Hinge

Q4. A uniform beam, 2.20m long with a mass m=25kg, is mounted by a small hinge on a wall as shown in the figure. The beam is held in a horizontal position by a cable that makes θ = 30°. The beam supports a sign of mass M = 28kg suspended from its end. Determine the components of the force FH and the tension, FT, in the supporting cable.

Background

Topic: Static Equilibrium and Forces

This question tests your understanding of static equilibrium, force decomposition, and torque. You must analyze the forces acting on a beam held by a hinge and a cable, and calculate the horizontal and vertical components of the hinge force, as well as the tension in the cable.

Key Terms and Formulas

• Static Equilibrium: An object is in static equilibrium if the net force and net torque on it are both zero.

• Torque ($\tau$): $\tau = F \cdot r \cdot \sin\theta$

• Force Decomposition: $F_T$ (tension) can be split into horizontal ($F_{Tx}$) and vertical ($F_{Ty}$) components.

• Sum of Forces: $\sum F_x = 0$, $\sum F_y = 0$

• Sum of Torques: $\sum \tau = 0$

• Weight: $W = mg$

Step-by-Step Guidance

1. Draw a free-body diagram of the beam, labeling all forces: the weight of the beam ($mg$) at its center, the weight of the sign ($Mg$) at the end, the tension in the cable ($F_T$) at the end, and the hinge force ($F_H$) at the wall.

2. Write the equilibrium equations for forces in the x and y directions:

   $\sum F_x = 0$

   $\sum F_y = 0$

3. Decompose the tension in the cable into components:

   $F_{Tx} = F_T \cos\theta$

   $F_{Ty} = F_T \sin\theta$

4. Set up the torque equilibrium equation about the hinge (choose the hinge as the axis to eliminate unknown hinge forces):

   $\sum \tau_{hinge} = 0$

   Include torques from the beam's weight, the sign's weight, and the cable's tension.

Try solving on your own before revealing the answer!

Final Answer:

$F_H$ components and $F_T$ tension calculated using:

$F_T = \frac{(mg \cdot \frac{L}{2} + Mg \cdot L)}{L \sin\theta}$

$F_{Tx} = F_T \cos\theta$, $F_{Ty} = F_T \sin\theta$

Hinge force components found from force equilibrium equations.

The values are determined by plugging in the masses, length, angle, and $g$.