Question

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Answer 1

Question:-

The sum of magnitudes of two forces acting at a point is 18 N and the magnitude of their resultant is 12 N. If the resultant makes an angle of 90° with the force of smaller magnitude , what are the magnitude of the two forces ?

Answer:-

Let the magnitudes of two force be A N and B N (A > B)

Given:-

Sum of these two forces = 18 N

⟹ A + B = 18 N

⟹ B = (18 - A) N -- equation (1).

Also given that,

Magnitude of resultant (R) = 12 N.

And,

Resultant forms an angle of 90° with the force which has smaller magnitude (B).

We know that,

Magnitude of resultant (R) = √(A² + B² + 2AB cos θ)

where θ is the angle between A , B.

⟹ A = √(R² + B² + 2RB cos 90°)

[ θ = 90° ]

⟹ A = √(12)² + (18 - A)² + 2(12)(18 - A)(0)

[ From equation (1) ]

• cos 90° = 0

On squaring both sides we get;

⟹ A² = [ √{144 + (18 - A)² + 0} ]²

⟹ A² = 144 + (18 - A)²

using (a - b)² = a² + b² - 2ab in RHS we get,

⟹ A² = 144 + (18)² + A² - 36A

⟹ A² - A² - 144 - 324 = - 36A

⟹ - 468 = - 36A

⟹ - 468/ - 36 = A

⟹ 13 N = A

Substitute the value of A in equation (1).

⟹ B = 18 - A

⟹ B = 18 - 13

⟹ B = 5 N

∴

• Magnitude of larger force (A) = 13 N

• Magnitude of smaller force (B) = 5 N

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Answer 2

${\large{\bold{\rm{\underline{Question}}}}}\; \; \; \; \green \bigstar$

★ The sum of the magnitude of two forces acting at a point is 18N and the magnitude of their resultant is 12N. If the resultant makes an angle of 90° with the force of smaller magnitude what are the magnitude of the two forces ?

${\large{\bold{\rm{\underline{Given \; that}}}}}\; \; \; \; \green \bigstar$

★ The sum of the magnitude of two forces acting at a point is 18N.

★ The magnitude of their resultant is 12N.

★ The resultant makes an angle of 90° with the force of smaller magnitude.

${\large{\bold{\rm{\underline{To \; find}}}}}\; \; \; \; \green \bigstar$

★ The magnitude of the two forces.

${\large{\bold{\rm{\underline{Solution}}}}}\; \; \; \; \green \bigstar$

★ The magnitude of the two forces = (13 and 5) Newton's.

${\large{\bold{\rm{\underline{Full \; Solution}}}}}\; \; \; \; \green \bigstar$

~ Let ${\bf{\red{S}}}$ as the magnitude of the smaller force, let ${\bf{\red{L}}}$ as the magnitude of the larger force and let ${\bf{\red{R}}}$ as the resultant of the force..!

~ As it's already given that the resultant makes an angle of 90° with the force of smaller magnitude..! Henceforth, it's cleared that it form of a right angled triangle. So, let us use formula of Phythagoras Theorm..!

• ➝ L² - S² = R²

• ➝ L² - S² = 12²

• ➝ L² - S² = 144 (Equation 1)

~ The sum of the two magnitude is given as

• ➝ S + L = 18

• ➝ L = 18 - S (Equation 2)

~ Substituting the values from (Equation 2) in (Equation 1) we get the following,

• ➝ (18-S)² - S² = 144

• ➝ S = 5N

${\frak{Henceforth, \: S \: is \: 5N}}$

~ Now substituting the value of S in (Equation 1) we get the following results,

• ➝ L = 18 - 5

• ➝ L = 13N

${\frak{Henceforth, \: L \: is \: 13N}}$

Henceforth, 13N and 5N is the magnitude of the two forces.