Question

If two equal forces of magnitude P act on angle of θ° their resultant will be-
1. \(P\cos \frac{\theta }{2}\)
2. \(2P\sin \frac{\theta }{2}\)
3. \(P\tan \frac{\theta }{2}\)
4. \(2P\cos \frac{\theta }{2}\)

Answer 1

Correct Answer - Option 4 : \(2P\cos \frac{\theta }{2}\)

Explanation:

When two force making an angle θ, the resultant (R) of the two force is given by:

\(R=\sqrt{F_1^2+F_2^2+2F_1F_2\cosθ}\;\)

Since the two forces are equal;

\(R=\sqrt{F_1^2+F_2^2+2F_1F_2\cosθ}\;\)

\(R=\sqrt{F^2+F^2+2F^2\cosθ}\;\)

\(R=\sqrt{2F^2+2F^2\cosθ}\;\)

\(R=\sqrt{2F^2(1\;+\;\cosθ)}\;\)

We know that;

1 + cos θ = 2 cos2(θ/2)

\(R=\sqrt{2F^2[2\cos^2(θ/2)]}\;\)

R = 2F cos(θ/2)

Here, F = P, so

R = 2P cos(θ/2)