Question

. find the magnitude of the two forces such that it they are at right angles, their resultant is 30N, But if they act an 60 degree ,their resultant is 35N

Answer 1

Answer:

The magnitude of the forces are 27.59 N and 11.96 N

Explanation:

Resultant of the two forces P and Q is given by

 $R=\sqrt{P^2+Q^2+2PQcos\theta}$

For first condition resultant is 30 N with right angle is

$30=\sqrt{P^2+Q^2+2PQcos\theta} \\ R=\sqrt{P^2+Q^2+2PQcos90\\\\\\ 30=\sqrt{P^2+Q^2+2} \\\\P^2+Q^2=900........(1)$

For second condition, resultant is 35 N and they act at 60 degree

$R=\sqrt{P^2+Q^2+2PQcos\theta} \\ 35=\sqrt{P^2+Q^2+2PQcos60} \\ 35=\sqrt{P^2+Q^2+PQ} \\{P^2+Q^2+PQ}=1225................(2)$

from equation 1 and 2, we get

${P^2+Q^2+PQ}=1225\\ PQ=1225-900\\PQ=325$

From this we can write  in formula (a+b)^2, we get

$(P+Q)^2=P^2+Q^2+2PQ\\(P+Q)^2=900+2\times 325\\(P+Q)^2=1550\\P+Q=\sqrt{1550} \\P+Q=39.37$

Now

$P=39.37-Q\\\\(39.37-Q)\times Q=325\\39.37Q-Q^2=325\\Q^2-39.37Q+ 325 =0$

Solving this equation , we get

Q=27.59 and

P=11. 96

The magnitude of the forces are 27.59 N and 11.96 N