Suppose $\sin x=0$ and $\sin(x+z)=\frac{\sqrt3}2$. What is the smallest possible positive value of $z$?

1. $\sin x=0$ means $x=k\pi$ for some integer $k$.
2. By the addition formula, $\sin(x+z)=\sin(k\pi+z)=\sin k\pi\cos z+\cos k\pi\sin z=(-1)^k\sin z$.
3. If $k$ is even, the condition is $\sin z=\dfrac{\sqrt3}2$, so $z=\dfrac\pi3$ or $\dfrac{2\pi}3$ (plus multiples of $2\pi$). If $k$ is odd, it is $\sin z=-\dfrac{\sqrt3}2$, so $z=\dfrac{4\pi}3$ or $\dfrac{5\pi}3$ (plus multiples of $2\pi$).
4. The smallest possible positive value of $z$ is therefore $\boxed{\dfrac\pi3}$.