A uniform rigid bar AB of length 2.0\text{ m} and weight 120\text{ N} is hinged smoothly a… — JAMB UTME

A uniform rigid bar $AB$ of length $2.0\text{ m}$ and weight $120\text{ N}$ is hinged smoothly at end $A$ to a vertical post. The bar is held in equilibrium at an angle of $60^\circ$ above the horizontal by a force $F$ applied at end $B$ acting perpendicular to the bar. What is the magnitude of the force $F$ ?

$30\text{ N}$ Answer
B
$60\text{ N}$
C
$120\text{ N}$
D
$15\text{ N}$

Answer

The magnitude of the force required to keep the bar in equilibrium is

30\text{ N}

To maintain rotational equilibrium, the clockwise moment created by the weight of the bar about hinge

A

must equal the counterclockwise moment created by force

F

. The weight of

120\text{ N}

acts at the bar's midpoint (

1.0\text{ m}

from

A

), and its perpendicular distance to the vertical line of action is

1.0 \cos(60^\circ) = 0.5\text{ m}

. Thus, the clockwise moment is

120 \times 0.5 = 60\text{ N}\cdot\text{m}

. Since force

F

acts perpendicularly at the end of the

2.0\text{ m}

bar, its moment is

F \times 2.0

. Setting

2.0 F = 60

gives

F = 30\text{ N}

Step-by-Step Solution

Identify the center of gravity and the position of applied forces.

For a uniform bar of length

L = 2.0\text{ m}

, its weight

W = 120\text{ N}

acts vertically downward at its center of gravity, which is at a distance of

1.0\text{ m}

from hinge

A

The weight of a uniform body acts through its midpoint.

Determine the perpendicular distance for each force relative to the pivot at

A

Perpendicular distance for weight:

d_W = 1.0\text{ m} \times \cos(60^\circ) = 0.5\text{ m}

. Perpendicular distance for force

F

d_F = 2.0\text{ m}

(since

F

is perpendicular to the bar).

The moment of a force is defined as the product of the force magnitude and the perpendicular distance from the pivot to the line of action of the force.

Apply the Principle of Moments about the pivot

A

\sum M_A = 0 \implies F \times 2.0\text{ m} = 120\text{ N} \times 0.5\text{ m} \implies 2.0 F = 60 \implies F = 30\text{ N}

For rotational equilibrium, the total counterclockwise moment about any pivot must equal the total clockwise moment.

Key Concept

Principle of Moments and Rotational Equilibrium

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