If I have a signal that is time limited, say a sinusoid that only lasts for T seconds, and I take the FFT of that signal, I see the frequency response. In the example this would be a spike at the sinusoid's main frequency.

Now, say I take that same time signal and delay it by some time constant and then take the FFT, how do things change? Is the FFT able to represent that time delay?

I recognize that a time delay represents a $exp(-j\omega t)$ change in the frequency domain, but I'm having a hard time determining what that actually means.

Practically speaking, is the frquency domain an appropriate place to determine the time delay between various signals?

If I have a signal that is time limited, say a sinusoid that only lasts for $T$ seconds, and I take the FFT of that signal, I see the frequency response. In the example this would be a spike at the sinusoid's main frequency.

Now, say I take that same time signal and delay it by some time constant and then take the FFT, how do things change? Is the FFT able to represent that time delay?

I recognize that a time delay represents a $\exp(-j\omega t)$ change in the frequency domain, but I'm having a hard time determining what that actually means.

Practically speaking, is the frequency domain an appropriate place to determine the time delay between various signals?