How will the period of revolution of a charged particle in the cyclotron change when its velocity is doubled?

When a particle rotates in a circle, the Lorentz force F = qBv makes a particle with mass m move with centripetal acceleration v² / r:
mv² / r = qBv,
where m is the mass of the particle,
v – speed,
r is the radius of the orbit,
q – charge,
B – magnetic field induction.

We cancel the equality by v.
mv = qBr
v = qBr / m
Orbit length – 2 * pi * r.
The orbital period is the quotient of dividing the length of the orbit by the speed:
T = (2 * pi * r) / (qBr / m) = (2 * pi * r * m) / qBr = 2 * pi * m / qB.
The formula shows that the period does not depend on the speed.

Answer: When you change the speed, the period will not change.