Question

The resultant amplitude of two waves y₁ = a sin (ωt + π/6) and y₂ = a cos ωt will be ______

Options

• a

• `"a"sqrt2`

• `"a"sqrt3`

• 2a

Answer 1

The resultant amplitude of two waves y₁ = a sin (ωt + π/6) and y₂ = a cos ωt will be `underline("a"sqrt3)`.

Explanation:

The general equation of S.H.M. is given by

y = a sin(ωt + Φ) ................(i)

Given that

y₁ = a sin (ωt + π/6) ................(ii)

Comparing equation (ii) with equation (i), we get,

a₁ = a; `Φ_1 = pi/6`

y₂ = a cos ωt = a sin(ωt - π/2) ............ (iii)

Comparing equation (iii) with equation (i),

we get

a₂ = a; `Φ_2 = pi/2`

So, phase difference `DeltaPhi = Phi_2 - Phi_1`

= `pi/2 - pi/6 = pi/3`

Now, the resultant amplitude of two waves is given by

A = `sqrt("a"_1^2 + "a"_2^2 + 2"a"_1"a"_2 cosPhi)`

Substituting a₁ = a; a₂ = a; `Phi = DeltaPhi = pi/3`, we get 

A = `"a"sqrt3`