  We determine the Sato-Tate groups and prove the generalized Sato-Tate
conjecture for the Jacobians of curves of the form $$ y^2=x^p-1 \text{ and }
y^2=x^{2p}-1,$$ where $p$ is an odd prime. Our results rely on the fact the
Jacobians of these curves are nondegenerate, a fact that we prove in the paper.
Furthermore, we compute moment statistics associated to the Sato-Tate groups.
These moment statistics can be used to verify the equidistribution statement of
the generalized Sato-Tate conjecture by comparing them to moment statistics
obtained for the traces in the normalized $L$-polynomials of the curves.
