Consider a body executes periodic motion given by $2 \sin (2 t + \frac{π}{4}) m .$ Then we can concluded that

Moderate

A

It represents SHM with time period of $π$ s, amplitude of 2 m and phase of $\frac{π}{4}$
B

motion is periodic but not SHM, With time period of $π$ s, amplitude of 2 m and phase of $\frac{π}{4}$
C

Motion represents SHM with time period of 2 s, amplitude 1m and phase of $\frac{π}{4}$
D

Represents SHM with time period of 2s, amplitude 2m and phase of $\frac{π}{4}$

Solution

The given equation $2 \sin (2 t + \frac{π}{4})$ can be compared with $y = A \sin (ω t + ϕ) .$ This shows that the body executes a simple harmonic motion. Time period can be determined from $' ω'$ of the given equation. On comparison of $2 \sin (2 t + \frac{π}{4})$ with $A \sin (ω t + ϕ) .$ , We get $ω = 2 r a d s^{- 1}, A = 2 m$
We know that
$\begin{array}{l} ω = 2 π t = \frac{2 π}{T} & ϕ = \frac{π}{4} \\ ∴ 2 = \frac{2 π}{T} \\ T = π s \end{array}$
Hence, We condude that time period $T = π s$ amplitude $A = 2 m$ and phase $ϕ = \frac{π}{4}$

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