Does the mass of a particle increase with increasing velocity In the chapter of magnetic effect of current I came across cyclotron limitations.
Statement:
The cyclotron cannot accelerate the particles to velocities as high as comparable to the speed of light.
Reason:
The reason is that at these velocities the mass of a particle increases with increasing velocity.
MyDoubt:
The reason for the statement is difficult for me to digest since I have never seen a particle or read about such a particle that follows this kind of reason as mentioned.
Can you provide some answers related to this?
 A: The inertia of the electron, as measured by the change in velocity wrought by a given impulse and as seen in the cyclotron rest frame, does indeed increase with the speed of the electron.
In the early days of relativity (i.e. more than 80 to 90 years ago now) people did indeed describe this phenomenon as an increase in the electron's relativistic mass: one thought of the kinetic energy as being stored up as extra mass of the electron. This idea, although not wrong, is awkward, as I explain here. In particular, there's no way to make an increasing mass convention fit smoothly with other Newtonian concepts. Do we measure it through inertia? Well, OK, but now our definition of mass depends on the angle between the 3-force and the 3-velocity vector: a particle resists a shove more if the shove is along its direction of motion! Do we measure it through increase in total energy? That's possible, and that's what was done in the past, but, as described, it doesn't mean the same thing as inertia.
The modern description simply uses four vectors, and then Newton's second law behaves itself as it ought; it reads $\vec{F} = m_0\,\vec{A}$ where now $\vec{F}$ and $\vec{A}$ are the force and acceleration four vectors and the electron mass $m_0$ is the Lorentz invariant electron rest mass.
So now, what should our description of the cyclotron phenomenon be? Simply that the Lorentz factor $\gamma$ that relates the four acceleration to the more readily measured rates in our frame changes. Indeed:
$$\vec{A} =\frac{d\vec{U} }{d \tau} = \gamma(\vec{u}) \left(\frac{d{\gamma}(\mathbf{u})}{dt} c, \frac{d{\gamma}(\vec{u})}{dt} \vec{u} + \gamma(\vec{u})\, \vec{a} \right)$$
where $\vec{a}$ and $\vec{u}$ are the co-ordinate accelerations and velocities (as we'd measure them by our clocks and distances in the cyclotron rest frame).
The description you cite is an old-time attempt to absorb the speed dependent $\gamma$ factor into the mass and describe the phenomenon as a variation in that mass rather than leaving everything in a covariant description.
