# Two forces, while acting on a particle in opposite; directions, have the resultant of 10N. If they act at right angles to each other, the resultant is found to be 50N. Find the two forces.

The resultant force is described as the total amount of force acting on the object or body along with the direction of the body. The resultant force is zero when the object is at rest or it is travelling with the same velocity as the object. The resultant force should be equal for all the force since all the force is acting in the same direction.

## The formula of Resultant Force

If one force is acting perpendicular to another, the resultant force is determined by using the Pythagorean theorem. The Resultant force formula is given by,

FR = F1 + F2 + F3

Where

F1, F2, F3 are the three forces acting in the same direction on an object.

### Solution

Step 1: Resultant force when the forces are opposite

Let F1 and F2 are the magnitudes of two forces R1 = √F12 +F22 + 2F1F2 cosθ

⇒ 10N = √F12 +F22 + 2F1F2 cosθ1800

⇒ 10 = √(F1)2 + F22 −2F1F2

⇒ 10 = F1−F2 …..(1)

Step 2: Resultant force when the forces are at right angle

R1 = √F12 +F22 + 2F1F2 cosθ 900

⇒ 50 = √(F1)2 + F22

⇒ 50 = F12 +F22

Step 3: Solving equations

From eq (1) substitute the value of F1 in eq (2)

2500 = (F2 +10)2 +(F22)

⇒F22 +10F2 −1200 = 0

⇒(F+ 40)(F2 −30) = 0

F2 =30N [Since the magnitude of force can’t be negative , so we will not consider negative value]

Substituting this value of F2 in eq (1)

10 = F1 −30

⇒ F1 =40N

Hence, the Magnitudes of two forces are 40N and 30N.

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