Problem and Solution

Given two forces, (P) and (Q), their resultant force (R sqrt{3}Q) makes an angle of (30^circ) with (P). We need to determine the ratio of (P) to (Q).

Using the Law of Cosines

The magnitude of the resultant force (R) of two forces (P) and (Q) can be calculated using the Law of Cosines:

$R^2 P^2 Q^2 - 2PQ costheta$

where (theta) is the angle between forces (P) and (Q).

Given that (R sqrt{3}Q) and the angle between (R) and (P) is (30^circ), we can find (theta):

(theta 180^circ - 30^circ 150^circ)

Using the calculated angle, substitute into the Law of Cosines formula:

$(sqrt{3}Q)^2 P^2 Q^2 - 2PQ cos(150^circ)$

Simplify using (cos(150^circ) -frac{sqrt{3}}{2}):

$3Q^2 P^2 Q^2 sqrt{3}PQ$

Rearrange the equation to:

$P^2 - sqrt{3}PQ - 2Q^2 0$

This is a quadratic equation in (P). Using the quadratic formula (P frac{-b pm sqrt{b^2 - 4ac}}{2a}):

Here (a 1), (b -sqrt{3}Q), and (c -2Q^2). Calculate the discriminant: (b^2 - 4ac 3Q^2 8Q^2 11Q^2).

$P frac{sqrt{3}Q pm sqrt{11}Q}{2}$

Eliminate the negative solution since force cannot be negative:

$P frac{sqrt{3} sqrt{11}Q}{2}$

$frac{P}{Q} frac{sqrt{3} sqrt{11}}{2}$

The ratio of (P) to (Q) is (frac{sqrt{3} sqrt{11}}{2}).