Two simple harmonic motions are represented by the equations $x_{1}=5 \sin \left(2 \pi t+\frac{\pi}{4}\right)$ and $x_{2}=5 \sqrt{2}(\sin 2 \pi t+\cos 2 \pi t)$. The amplitude of second motion is _____ times the amplitude in first motion.

JEE Main · 2021 · 26 Aug 2021 · Shift 2 · Elasticity & SHM

Two simple harmonic motions are represented by the equations

x_{1} = 5 sin (2 π t + \frac{π}{4})

and

x_{2} = 52 (sin 2 π t + cos 2 π t)

. The amplitude of second motion is _____ times the amplitude in first motion.

Answer: 2

The first motion has amplitude

A_{1} = 5

.
In the second,

sin 2 π t + cos 2 π t = 2 sin (2 π t + \frac{π}{4})

, so

x_{2} = 52 \cdot 2 sin (2 π t + \frac{π}{4}) = 10 sin (2 π t + \frac{π}{4})

and

A_{2} = 10

. The amplitude of the second motion is

2

times that of the first.

JEE Main · 2021 · 26 Aug 2021 · Shift 2 · Integer answer