bead parabola accelerometer

An accelerometer is made of a piece of wire in the shape of a parabola  y = kx²  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) = mv²/2,    or    v² = –2 (ax + gy).

Substituting  y = kx² we get

v² = – 2x(a + gkx).

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