Question

#. The greatest and least resultant of two forces acting at a point are 29kg wi, and 5kg wt. respectively. If each force h Increased by 3kg wc, then the magnitude of the resultant of new forces acting at right angles to each other is (in kg wt) 1) 45 2)35 3) 25 4) 13​

Answer 1

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》 The greatest and least resultant of two forces acting at a point are 29kg wi, and 5kg wt. respectively. If each force h Increased by 3kg wc, then the magnitude of the resultant of new forces acting at right angles to each other is 35 kg wt.

Answer 2

Therefore the resultant of new forces is 25 N. ( Option-3 )

Given:

The maximum value of the resultant of 2 forces = 29 N

The minimum value of the resultant of 2 forces = 5 N

The forces are increased by 3 N individually.

To Find:

The value of the resultant of new forces acting at the right angle to each other.

Solution:

The given question can be solved as shown below.

Let F₁ and F₂ be the 2 forces.

The resultant of 2 forces is given by,

⇒ F = √ ( F₁² + F₂² + 2F₁F₂ cosθ )

Where θ is the angle between 2 forces.

Case-1: Maximum value of resultant:

The maximum value of the resultant is obtained when both forces are parallel and acting in the same direction.

Then θ = 0 ⇒ Cos θ = 1

⇒ Then resultant of 2 forces = F =√ ( F₁² + F₂² + 2F₁F₂ ) = 29

⇒ √( F₁ + F₂ )² = 29

⇒ F₁ + F₂ = 29      (i.)

Case-2: Minimum value of resultant:

The minimum value of the resultant is obtained when both forces are parallel and acting in the opposite direction.

Then θ = 180° ⇒ Cos θ = -1

⇒ Then resultant of 2 forces = F =√ ( F₁² + F₂² - 2F₁F₂ ) = 5

⇒ √( F₁ - F₂ )² = 5

⇒ F₁ - F₂ = 5      (ii.)

Solving equation-(i.) and equation-(ii.),

⇒ ( F₁ + F₂ ) + ( F₁ - F₂ ) = 29 + 5 = 34

⇒ 2F₁ = 34 ⇒ F₁ = 17 N

Then F₁ + F₂ = 29

⇒ F₂ = 29 - F₁ = 29 - 17 = 12 N

⇒ F₁ = 17 N and F₂ = 12 N

So 2 forces are 17 N and 12 N.
To get new forces each of the obtained forces is added by 3 N.

⇒  F₁ = 17 N + 3 N = 20 N and F₂ = 12 N + 3 N = 15 N

⇒  New forces, F₁= 20 N and F₂ = 15 N

If both the forces are at the right angle to each other, then Cos 90 = 0

⇒ So the resultant = F = √( F₁² + F₂² + 0 )

⇒ F = √( F₁² + F₂² )

⇒ F = √ ( 20² + 15² )

⇒ F = √ ( 400 + 225 )

⇒ F = √625 = 25

Therefore the resultant of new forces is 25 N.

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