A light, rigid horizontal bar AB of length 1.5\text{ m} is smoothly pivoted at end A. A ve… — JAMB UTME

A light, rigid horizontal bar $AB$ of length $1.5\text{ m}$ is smoothly pivoted at end $A$ . A vertical downward load of $40\text{ N}$ is hung from end $B$ . The bar is kept in horizontal equilibrium by a light string attached at point $C$ , located $1.0\text{ m}$ from $A$ . The string exerts a tension force $T$ pulling upwards at an angle of $30^\circ$ relative to the horizontal bar. What is the magnitude of the tension $T$ in newtons?

Answer: 120 N

Answer

The magnitude of the tension

T

in the string is

120\text{ N}

For the bar to maintain rotational equilibrium, the clockwise moment about pivot

A

must equal the counterclockwise moment about

A

. The

40\text{ N}

load exerts a clockwise moment of

40\text{ N} \times 1.5\text{ m} = 60\text{ N}\cdot\text{m}

. The string tension

T

exerts a counterclockwise moment given by its vertical component multiplied by the distance from the pivot:

(T \sin 30^\circ) \times 1.0\text{ m} = 0.5T \text{ N}\cdot\text{m}

. Equating the two moments gives

0.5T = 60\text{ N}\cdot\text{m}

, yielding

T = 120\text{ N}

Step-by-Step Solution

Calculate the clockwise moment about the pivot at end

A

τ_{\text{clockwise}} = 40\text{ N} \times 1.5\text{ m} = 60\text{ N}\cdot\text{m}

The weight at

B

acts vertically downward at a perpendicular distance of

1.5\text{ m}

from pivot

A

Determine the perpendicular component of tension

T

relative to the bar

F_{\perp} = T \sin 30^\circ = 0.5T

Only the component of force perpendicular to the bar produces a moment about the pivot.

Set up the counterclockwise moment about pivot

A

τ_{\text{counterclockwise}} = (0.5T) \times 1.0\text{ m} = 0.5T \text{ N}\cdot\text{m}

The string is attached at point

C

, which is

1.0\text{ m}

away from pivot

A

Apply the principle of moments for rotational equilibrium and solve for

T

0.5T = 60 \implies T = 120\text{ N}

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

Key Concept

Principle of moments and rotational equilibrium for forces acting at non-perpendicular angles.

Estimated Time:1m 30s

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