Show that $x(t) = A\sin(\omega t + \phi)$ is a periodic function of time, and find its period in terms of the angular frequency $\omega$.

Question

Accepted Answer

A function $x(t)$ is periodic with period $T$ if $x(t+T)=x(t)$ for every value of $t$. Consider $x(t)=A\sin(\omega t+\phi)$ and evaluate it at time $t+T$ for $T=2\pi/\omega$:

$$x(t+T) = A\sin\!\left(\omega\left(t+\frac{2\pi}{\omega}ight)+\phiight) = A\sin(\omega t + 2\pi + \phi)$$

Since the sine function itself is periodic with period $2\pi$, i.e. $\sin(	heta+2\pi)=\sin	heta$ for any angle $	heta$, this simplifies to

$$x(t+T) = A\sin(\omega t + \phi) = x(t)$$

so $x(t+T)=x(t)$ holds exactly, confirming that $x(t)=A\sin(\omega t+\phi)$ is indeed periodic, with period $T=2\pi/\omega$ (and no smaller positive value of $T$ satisfies the same condition, so this is the true, smallest period).

> [!ANSWER]
> Period $T=2\pi/\omega$.

Q.Show that $x(t) = A\sin(\omega t + \phi)$ is a periodic function of time, and find its period in terms of the angular frequency $\omega$ .

Concept understanding — Simple Harmonic Motion

Simple Harmonic Motion: The Natural Rhythm of Things

The Intuition: A Restoring Force That Fights Displacement

The Precise Statement

What Does This Motion Look Like?

The Mathematical Description (Derived from $F = -kx$ )

Real-World Examples

The Bottom Line

Q.Show that x(t)=Asin⁡(ωt+ϕ)x(t) = A\sin(\omega t + \phi)x(t)=Asin(ωt+ϕ) is a periodic function of time, and find its period in terms of the angular frequency ω\omegaω.

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Q.Show that $x(t) = A\sin(\omega t + \phi)$ is a periodic function of time, and find its period in terms of the angular frequency $\omega$ .

Unlock everything free for 14 days