Three coplanar forces act simultaneously at a point O. The first force of magnitude 10\tex… — JAMB UTME

Three coplanar forces act simultaneously at a point $O$ . The first force of magnitude $10\text{ N}$ acts due East, and the second force of magnitude $10\text{ N}$ acts at an angle of $60^\circ$ North of East. If a third force keeps the system in static equilibrium, what is the magnitude of this third force?

$10\sqrt{3}\text{ N}$ Answer
B
$20\text{ N}$
C
$10\sqrt{2}\text{ N}$
D
$10\text{ N}$

Answer

The magnitude of the third force required for equilibrium is

10\sqrt{3}\text{ N}

Resolving the forces along the horizontal (East) and vertical (North) axes gives components of

15\text{ N}

and

5\sqrt{3}\text{ N}

respectively. The magnitude of the resultant force is

\sqrt{15^2 + (5\sqrt{3})^2} = \sqrt{300} = 10\sqrt{3}\text{ N}

. Since the third force balances the system, its magnitude must equal that of the resultant, which is

10\sqrt{3}\text{ N}

Step-by-Step Solution

Resolve the two given forces into horizontal (

x

) and vertical (

y

) Cartesian components.

F_{1x} = 10\text{ N}

F_{1y} = 0\text{ N}

;

F_{2x} = 10 \cos(60^\circ) = 5\text{ N}

F_{2y} = 10 \sin(60^\circ) = 5\sqrt{3}\text{ N}

Vector addition requires breaking non-orthogonal vectors into perpendicular component directions.

Calculate the total horizontal (

R_x

) and vertical (

R_y

) components of the resultant of the first two forces.

R_x = 10 + 5 = 15\text{ N}

R_y = 0 + 5\sqrt{3} = 5\sqrt{3}\text{ N}

Adding aligned components yields the net component along each axis.

Compute the magnitude of the resultant force

R

R = \sqrt{R_x^2 + R_y^2} = \sqrt{15^2 + (5\sqrt{3})^2} = \sqrt{225 + 75} = \sqrt{300} = 10\sqrt{3}\text{ N}

Applying Pythagoras' theorem to orthogonal components determines the net magnitude.

Determine the magnitude of the equilibrant (third force).

Equibrant magnitude

= R = 10\sqrt{3}\text{ N}

For static equilibrium, the third force must be equal in magnitude and opposite in direction to the resultant of the first two forces.

Key Concept

Vector Addition and Equilibrium of Forces

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