Question

Two forces P and Q acting at a point have a
resultant R. If Q is doubled R is
doubled. Again if Q is reversed in direction,
then also R is doubled. Show that
p² :Q²:R²= 2:3:2
Slove question​

Answer 1

Step-by-step explanation:

The resultant of two forces P and Q is equal to Q√3 and it makes an angle of π/6 with the direction of P. How do you show that P=Q or 2Q? The resultant of two forces P and Q is R. If Q is doubled, R is doubled. If Q is reversed, R is again doubled.

Answer 2

Solution :

Let, θ be the angle between $\vec{P}$ and $\vec{Q}$

Given that, $\vec{R}$ is resultant

Then, R² = P² + Q² + 2PQcosθ ...(i)

When $\vec{Q}$ is doubled, $\vec{R}$ gets doubled. Then,

(2R)² = P² + (2Q)² + 2P (2Q) cosθ

⇒ 4R² = P² + 4Q² + 4PQcosθ ...(ii)

When $\vec{Q}$ is reversed, $\vec{R}$ gets doubled. Then

(2R)² = P² + Q² + 2PQ cos(π - θ)

⇒ 4R² = P² + Q² - 2PQcosθ ...(iii)

Adding (i) and (iii), we get

R² + 4R² = P² + Q² + 2PQcosθ + P² + Q² - 2PQcosθ

⇒ 5R² = 2P² + 2Q² ...(iv)

Adding (ii) and {(iii) × 2}, we get

4R² + 8R² = P² + 4Q² + 4PQcosθ + 2P² + 2Q² - 4PQcosθ

⇒ 12R² = 3P² + 6Q²

⇒ 4R² = P² + 2Q² ...(v)

Now, (iv) - (v) gives

R² = P²

⇒ R²/P² = 1/1

⇒ R/P = 1/1

⇒ R : P = 1 : 1

⇒ P² : R² = 1 : 1

⇒ P² : R² = 2 : 2

Putting R² = P² in (v), we get

4R² = R² + 2Q²

⇒ 3R² = 2Q²

⇒ R²/Q² = 2/3

⇒ R² : Q² = 2 : 3

⇒ Q² : R² = 3 : 2

∴ P² : Q² : R² = 2 : 3 : 2 ( proved )