the torsional pendulum
A torsional pendulum is something that twists back and forth instead of swinging side to side. Hang a disk from a thin wire, give it a twist, and let go: it rotates one way, unwinds, overshoots the other way, and repeats. The balance wheel in a mechanical watch and the delicate torsion balance used to weigh tiny forces are torsional pendulums.
Precisely, a body is suspended by a wire or fiber, and twisting it by an angle theta produces a restoring torque proportional to the twist: tau = -kappa theta, where kappa (kappa) is the torsion constant of the wire and the minus sign means the torque winds it back. This is the rotational version of Hooke's law. The result is angular simple harmonic motion with period T = 2 pi sqrt(I / kappa), where I is the moment of inertia of the body about the twist axis.
Torsional pendulums power precise instruments: Henry Cavendish used a torsion balance to measure the gravitational constant G, and sensitive galvanometers use the same principle. One honest point worth noticing: here the restoring agent is the elastic stiffness of the wire in twisting, not gravity. Because gravity plays no role in the restoring torque, the period does not depend on g at all.
A disk with moment of inertia I = 0.02 kg m^2 hangs from a wire with torsion constant kappa = 0.08 N m/rad. It twists back and forth with period T = 2 pi sqrt(0.02 / 0.08) = 2 pi sqrt(0.25) = about 3.1 s.
A torsional pendulum's rhythm is set by the wire's stiffness and the body's moment of inertia, never by gravity.
The restoring effect comes from twisting the wire, not from gravity, so a torsional pendulum's period is independent of g.