The function $g$ is defined by $g(x)=|x-1|-2$. If $g(a)=g(0)$ and $a>0$, what is the value of $a$?

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Find $g(a)$ and $g(0)$.

$$g(x)=|x-1|-2$$ $$g(a)=|a-1|-2$$
$$g(0)=|0-1|-2$$ $$= |-1|-2$$ $$= 1-2$$ $$= -1$$

Setting $g(a)=g(0)$:

$$|a-1|-2=-1$$ $$|a-1|=1$$ $$a-1 = 1 \text{ or } -1$$ $$a= 2 \text{ or } 0$$

Since $a>0, a=\boxed{2}$