**Simple Harmonic Motion and Uniform**

**Circular Motion**

In Galileo, using his newly constructed telescope, discovered the four principal moons of Jupiter. Over weeks of observation, each moon seemed to him to be moving back and forth relative to the planet in what today we would call simple harmonic motion; the disk of the planet was the midpoint of the motion. The record of Galileo’s observations, written in his own hand, is still available. A. P. French 0 MIT used Galileo’s data to work out the position of the moon Callisto relative to The angle between Jupiter and its moon Callisto as seen from Earth. The circles are based on Galileo’s

measurements.The curve is a best fit, strongly suggesting simple harmonic motion. At Jupiter’s mean distance minutes of arc corresponds to about 2 X 106 km. (Adapted from A. P. French, Newtonian Mechanics, W. W. Norton & Company,New York.

Jupiter. In the results shown in the circles are based on Galileo’s observations and the curve is a best fit to the data. The curve strongly suggests the displacement function for SHM. A period of about days can be measured Actually, Callisto moves with essentially constant speed in an essentially circular orbit around Jupiter. Its true motion–far from being simple harmonic–is uniform circular motion. What Galileo saw–and what you can see with a good pair of binoculars and a little patience–is the projection of this uniform circular motion on a line in the plane of the motion. We are led by Galileo’s remarkable observations to the conclusion that simple harmonic motion is uniform circular motion viewed edge-on. In more formal language. Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion occurs . gives an example. It shows a reference particle P’ moving in uniform circular motion with (constant) angular speed w in a reference circle. The radius Xm of the circle is the magnitude of the particle’s position vector. At any time t, the angular position of the particle is wt + 4>, where 4> is its angular position at t = O. The projection of particle P I onto the x axis is a point P, which we take to be a second particle. The projection of the position vector of particle P I onto the x axis gives the location x(t) of P. Thus, we find

x(t) = xm cos(wt + 4»,

which is precisely Our conclusion is correct. reference particle P I moves

in uniform circular motion, its projection particle P moves in simple harmonic motion along a diameter of the circle. b shows the velocity of the reference particle. From (v = wr), the magnitude of the velocity vector is wXm; its projection on the x axis is

v(t) = -wXm sin(wt + 4»,

A reference particle pi moving with uniform circular motion in a reference

circle of radius x”,. Its projection P on the x axis executes simple harmonic motion. (b) The projection of the velocity of the reference particle is the velocity of SHM. (c) The projection of the radial acceleration a of the reference particle is the acceleration of SHM. which is exactly The minus sign appears because the velocity component of P in b is directed to the left, in the negative direction of x. shows the radial acceleration a of the reference particle. From (a. = w2r), ‘the magnitude of the radial acceleration vector is w2xm.

aCt) = -w2xm cos(wt + r/J),

which is exactly Thus, whether we look at the displacement, the velocity

or the acceleration, the projection of uniform circular motion is indeed simple harmonic motion projection on the x axis is