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At what angle do the forces (P + Q ) and...

At what angle do the forces (P + Q ) and (P-Q) act so that the resultant is `sqrt ( 3 P ^(2) + Q ^(2))` ?

Text Solution

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We are given `F _(1) = (P + Q ) , F _(2) = vec P - Q`
We are to calculate, `theta`
Using relation : `R = sqrt (F_(1) ^(2) + F _(2) ^(2) + 2 F _(1 ) F _(2) cos theta)`
We have,
`sqrt ( 3 P^(2) + Q ^(2)) = sqrt (( P + Q) ^(2) + (P+Q) ^(2) + 2 ( P+Q) (P+Q) cos theta)`
Squaring on both sides, we get
or `3 P ^(2) + Q ^(2) = P ^(2) + Q ^(2) + 2 PQ + P ^(2) + Q ^(2) - 2 PQ + 2 ( P ^(2) - Q ^(2)) cos theta `
or `3 P ^(2) + Q ^(2) = 2 P ^(2) + 2 Q ^(2) + 2 ( P ^(2) - Q ^(2) ) cos theta `
or `P ^(2) - Q ^(2) = 2 ( P ^(2) - Q ^(2)) cos theta or cos theta = (1)/(2)`
`therefore theta = 60^(@)`
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