A bead slides along a frictionless wire and starts with an initial downward velocity v₀. Find the shape of the wire that maintains a constant downward component of velocity. In other words, find the shape that converts all the gravitational energy into new horizontal kinetic energy. Rather than an energy argument, we will begin by finding equations of motion using Newton's law, F=mA, or total applied force equals mass times the second derivative of position. Since the wire is frictionless, it can only produce a force perpendicular to the bead. Since the vertical speed cannot increase, the weight of the bead, mg must equal the upward portion of the wire force. This is shown in the rough sketch Figure 12.1.

Figure 12.1: Isochrone Forces

The fact that the force from the wire is perpendicular to the wire means that a tiny incremental triangle along the wire is similar to the triangle of forces, yielding

We may use the Chain Rule to express the geometric slope in terms of the horizontal and vertical speeds,

The vertical speed, , is to remain constant (since we balanced the force of gravity with the upward component of wire force), so

The horizontal motion is goverened by Newton's F=mA law,

Now use the phase variable trick; let so the equation above becomes

Separate variables in the horizontal speed equation above and show that

if we suppose that the initial horizontal velocity is zero, u(0)=0.

Use the equation for to show that the horizontal position is given by

for x(0)=0.

We also know (if we measure y downward, so its velocity is positive) that .

Prove that the vertical position is given by

if we start at y(0)=0.

We can solve the y equation fot t=y/v₀ and substitute into the horizontal position equation, obtaining .

Solve this equation for y=y(x)=kx^2/3 showing that .

Figure 12.2: The Isochrone

We have found several equations for the isochrone.

12.1 Conservation of Energy

The equations for the isochrone can also be found from an energy argument. You need to know that the increase in kinetic energy changing from to a higher horizontal speed u is

You also need to know that the decrease in gravitational potential energy in moving down a distance y is

The physical principle of conservation of energy (which we showed for a linear oscillator in Chapter 17 of the main text) says these two quantities are equal, so

Use the Chain Rule to express the differential equation above in the form

Then separate variables and show that

as before.

How could you find the equation for x(t) from this equation?

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