Question

`Question: $x_(1)=(5)/(2)[\sin (2 \pi t)+\cos (2 \pi t)], x_(2)=5\left[\sin \left(2 \pi t+(\pi)/(4)\right)\right]$`

Find the ratio of the amplitude of the given motion?
Options:

`(a) $√(2): 1$\\(b) $2: 1$\\(c) $1: √(2)$\\(d) $1: 2$`

Answer 1

Condense x₁ to get

sin(2πt) + cos(2πt) = √2 • (sin(2πt) + cos(2πt))/√2

… = √2 (cos(2πt) cos(π/4) + sin(2πt) sin(π/4))

… = √2 cos(2πt - π/4)

So the amplitude of x₁ is (5/2) • √2 = 5/√2, while the amplitude of x₂ is 5. The ratio between them is then 5/√2 : 5, or equivalently 1/√2 : 1 or 1 : √2.