Two simple harmonic motions are represented by the equations
x₁ = 5 sin ( 2π t + π/4 ) and x₂ = 5√2 (sin 2π t + cos 2π t). The amplitude of second motion is ................ times the amplitude in first motion.

Solution

Physics sir · 3 steps

Fold sin plus cos into a single sine wave first. Only then can you read off the amplitude.

1. First motion. x₁ = 5 sin(2π t + π/4) is already a single sine, so its amplitude is A₁ = 5.
2. Combine sin and cos. For any angle θ, sin θ + cos θ = √2(1/√2 sin θ + 1/√2 cos θ) = √2 sin(θ + π/4). Put θ = 2π t.

   x₂ = 5√2×√2 sin(2π t + π/4) = 10 sin(2π t + π/4)

3. So. A₂ = 10, and A₂/A₁ = 10/5 = 2.

The common slip: 1.41. Taking the factor 5√2 outside the bracket as the amplitude gives a ratio of √2 ≈ 1.41. The bracket sin + cos adds another factor of √2.

The idea to keep. The amplitude is the number in front of a single sine or cosine. For a sin ω t + b cos ω t, it is √(a² + b²), not the factor written outside the bracket.

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