Two simple harmonic motion, are represented by the equations y₁ = 10 sin ( 3π t + π/3 ) y₂ = 5(sin 3π t + √3 cos 3π t) Ratio of amplitude of y₁ to y₂ = x : 1. The value of x is ______________.

Solution

Physics sir · 4 steps

Write y₂ as one sine wave first. Its amplitude is not 5, it is hidden in the bracket.

1. Amplitude of y1. y₁ = 10 sin(3π t + π/3) is already a single sine wave, so its amplitude is 10.
2. Expand y2. y₂ = 5 sin 3π t + 5√3 cos 3π t. A sum a sin θ + b cos θ is one SHM with amplitude √(a² + b²).

   A₂ = √(5² + (5√3)²) = √(25 + 75) = 10

3. Check. In fact y₂ = 10(1/2 sin 3π t + √3/2 cos 3π t) = 10 sin(3π t + π/3), which is the same motion as y₁.
4. So. The ratio of amplitudes is 10 : 10 = 1 : 1, so x = 1.

The common slip: 2. Reading the factor 5 in front of the bracket as the amplitude of y₂ gives 10 : 5, so x = 2.

The idea to keep. Whenever you see a sin θ + b cos θ, combine it into one sine of amplitude √(a²+b²) before comparing amplitudes. The number in front is not the amplitude.

Answer from the NTA final answer key. Solution by PrepWiser.