Question

Two forces equal to $P$ and $2P$ act on a particle. If the first be doubled and second be increased by $20$ newton the direction of the resultant is unaltered. Find the value of $P$.

Answer 1

If the direction of the resultant force does not change, thenthe angle between them remains same.

Let us assume that the angle was initially ${45}^0$.

$\Rightarrow \tan {45}^0=\frac{2P}{2P}=1$

$\Rightarrow$ When first force is doubled, it becomes $=4P$

Given that $20N$ is added to the second force, it becomes $=2P+20$

$\Rightarrow \tan {45}^0=\frac{4P}{(2P+20)}=1$

$\Rightarrow 4P=2P+20$

$\Rightarrow 2P=20$

$\Rightarrow P=10N$

alt method :

$tan\beta =\frac{\left(2Psin\theta \right)}{P+2Pcos\theta }⟹\left(1\right)$

$tan\beta =\frac{\left\{(2P+20)sin\theta \right\}}{2P+(2P+20)cos\theta }⟹\left(2\right)$

by equating (1) & (2)

2P=20

P=10 N