Question

the resultant of two forces at right angle is 17 n if the maximum resultant is 23n magnitude of the greater force is 3x N what is the value of x​

Answer 1

Answer:-

Given:-

Resultant of two forces at right angle = 17 N

Maximum resultant = 23 N

Greater force = 3x N

We know that,

Resultant force (F) = √(P² + Q² + 2PQ cos θ)

where P & Q are the magnitudes of the forces and θ is the angle between them.

Let the forces be P & Q. (P > Q)

⟹ 17 = √(P² + Q² + 2PQ cos 90°)

[ ∵ P & Q form a right angle ]

⟹ 17 = √(P² + Q²). [ ∵ cos 90° = 0 ]

On squaring both sides we get,

⟹ 289 = P² + Q² -- equation (1).

Also given that,

Maximum resultant = 23 N

Maximum resultant is the sum of the magnitudes of the forces.

⟹ P + Q = 23 N

⟹ Q = (23 - P) N -- equation (2)

Substitute Q = 23 - P in equation (1).

⟹ 289 = P² + (23 - P)²

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

⟹ 289 = P² + (23)² + P² - 46P

⟹ 0 = 2P² - 46P + 529 - 289

⟹ 0 = 2P² - 46P + 240

⟹ 2(P² - 23P + 120) = 0

⟹ P² - 23P + 120 = 0

⟹ P² - 8P - 15P + 120 = 0

⟹ P(P - 8) - 15(P - 8) = 0

⟹ (P - 8)(P - 15) = 0

⟹ P = 8 (or) 15 N

Since, P is a greater force, maximum value i.e., 15 N is considered.

It is given that,

⟹ P (greater force) = 3x N

So,

⟹ 15 = 3x

⟹ 15/3 = x

⟹ x = 5

∴ Value of x is 5.