4.1.2 Using Moments - Equilibrium

Problem: Ladder in Equilibrium

infoNote

Given:

A uniform ladder $AB$ is leaning against a smooth vertical wall on rough horizontal ground at an angle of 70° to the horizontal.
The ladder has a length of 8 m.
It is held in equilibrium by a frictional force of magnitude 60 N acting horizontally at $B$ .
The ladder's weight acts at its centre of mass. Find:

$R_W = :success[60 N]$ . Result: The normal reaction of the wall on the ladder at $A$ is 60 N.

\Rightarrow mg \cos(70^\circ) \times 4 - 60 \times 8 \times \sin(70^\circ)=0\\ \Rightarrow mg \cos(70^\circ) \times 4 = 60 \times 8 \times \sin(70^\circ)

\Rightarrow m = \frac{60 \times 8 \times \sin(70^\circ)}{4 \times \cos(70^\circ)}

Simplifying:

m = \frac{60 \times \sin(70^\circ)}{\cos(70^\circ) \times 4} \approx :success[33.64 kg]

Result: The mass of the ladder is approximately 33.64 kg.

infoNote

Given:

A uniform rod $AB$ has weight 20 N and length 3 m.
The end $A$ is freely hinged to a point on a vertical wall.
The rod is held horizontally and in equilibrium by a light inextensible string.
One end of the string is attached to the rod at $B$ .
The other end of the string is attached to a point $C$ , which is 1 m directly above $A$ . Find: Calculate the tension in the string.

\theta = \tan^{-1} \left(\frac{1}{3}\right) \approx :highlight[18.435°]

The rod's weight acts at its midpoint, so the distance from $A$ is 1.5 m.
The tension $T$ in the string is at an angle $\theta$ to the horizontal, giving components $T \cos \theta$ and $T \sin \theta$ .

1.5 \times 20 = T \sin(18.435^\circ) \times 3

T \sin(18.435^\circ) \times 3 = 30

T = \frac{30}{3 \sin(18.435^\circ)} \approx :success[31.62 N]

Result: The tension in the string is approximately 31.62 N.

infoNote

Identify the pivot point: Choose a point where one or more unknown forces act, as forces through the pivot have no moment. This simplifies the equation by eliminating unknowns.
Apply the principle of moments: In equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any point. Set up an equation for these moments to solve for unknown forces or distances.
Use equilibrium conditions: In addition to moments, apply the conditions for translational equilibrium: