Question

The sum of two concurrent forces P and Q is 270 N and their resultant is 180 N. The angle between the force P and resultant R is 90°. Find the magnitude of each force?

Answer 1

Answer:

The magnitude of  force is 75 newton and 195newton

Explanation:

Given , the sum of two concurrent forces P and Q is 270N and their resultant is 180 N .

The angle between the force P and resultant R is 90 °

Let the angle between resultant and other force is $\theta$ ,

then net angle between forces = ($90+ \theta$) = $\theta^{'}$

Step1:

Now from the figure ,

tanα = $\frac{B sin\theta^{'} } {A +B cos\theta^{'} }$     [where α = 90 and $\theta^{'}$ = 90 + $\theta$]

tan 90 = $\frac{(270- N)sin(90+\theta)} {N +(270-N)cos(90+\theta) }$

⇒ $\frac{1}{0} = \frac{-(270- N)cos\theta} {N -(270-N)sin\theta }$      

⇒${N -(270-N)sin\theta } = 0$     (by cross multiplication )

Step2:

Therefore we have ,

R = $\sqrt{A^{2} +B^{2} +2AB cos\theta}$

Now , put the value of A, B and R in the above equation

180 = $\sqrt{N^{2} + (270-N)^{2} +2N(270-N)cos (90-\theta)}$

Now squaring both side we get ,

$180^{2} = N^{2} + (270-N)^{2} +2N(270-N)cos (90-\theta)$

$3240 = N^{2} + (270^{2} ) + N^{2} - 540N + (-2N(270-N)sin\theta)$

After calculation the above equation we get ,

⇒3240 = 72900- 540N

⇒ N = $\frac{72900-32400}{540} = 75Newton$

and other force  = 270 - N = 270 - 75 = 195Newton .

Final answer :

Hence ,  the magnitude of  force is 75 newton and 195newton .

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