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## Homework Statement

An accelerometer is made of a piece of wire with a bead on it that can slide on the wire with no friction. The wire is formed as a parabola y = kx

^{2}, as shown in the drawing. The bead rests at the lowest point of the parabola when it is at rest. When accelerated parallel to the x-axis the bead will climb up some on the wire. Find the relationship between the acceleration a of the wire and the bead’s maximum horizontal displacement x relative to the wire.

## Homework Equations

F=ma

## The Attempt at a Solution

Suppose the wire is accelerating and the bead is at rest at some point P. Then in the frame of the bead, we have a gravitational force downward, a fictitious force to the left, and a normal force perpendicular to the wire.

Consider the tangent line to the wire at the point where the bead is resting. The slope of this is given by

[tex] \frac{d}{dx} (kx^2) = 2kx [/tex]

Then the angle between a line parallel to the point where the bead is at rest, and the horizontal satisfies:

[tex] tan \theta = 2kx [/tex]

Look at the components of the forces parallel to this line, and they must be equal. Then we have

[tex] ma cos \theta = mg sin \theta [/tex]

then dividing by sin theta and plugging in tan theta = 2kx we can get

[tex] x = - \frac{a}{2kg} [/tex]

However the answer should be only

[tex] x = - \frac{a}{kg} [/tex]

Do you see where I have gone wrong?

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