**bead parabola accelerometer**

An accelerometer is made
of a piece of wire in the shape of a parabola * y = kx*^{2 } with a bead on it that can slide
without friction, as shown in the drawing. The bead is initially attached to the
wire at the lowest point of the parabola. The wire is accelerated
with a constant acceleration* *parallel to the* x*-axis, and then the
bead is released. Find the
relationship between the acceleration *a *of the wire and the bead’s
maximum horizontal
displacement *x *relative to the wire.

__Solution by
Sukumar Chandra__

Consider the motion of the bead relative to the wire in a reference frame fixed
to the wire, which is a non-inertial reference frame moving with acceleration *
a* horizontally to the right. Call the horizontal axis ‘*x*’ and the
vertical axis ‘*y*’, and choose the origin to be the starting position of
the bead. Thus at any given time the bead is at (*x, y*) moving with some
speed *v* along the wire. The forces on the bead in this frame and the work
done by them are:

1) Gravity, *mg* vertically downward; the work done by it is –*mgy*.

2) Normal reaction force, which is always perpendicular to the displacement so it does no work.

3) Pseudo-force *ma* horizontally leftward; the work done by it is –*max*.

Initially the bead is at rest so its change of kinetic energy equals *mv*^{
2}/2. The work done by all the forces on a particle is equal to the
change in its kinetic energy. Therefore

*(–mgy) + (–max) =* *mv*^{2}/2, or *v*^{2}
= *–*2 (*ax + gy*).

Substituting *y = kx ^{2}* we get

*v*^{2} = *– 2x*(*a + gkx*).

When the bead comes to rest *v* = 0, and this occurs when *x = *0*
*or* x = -a /(gk)*.