Question

Question #f1ade

Answer 1

see below

Explanation:

A linear restorative force is a key ingredient of shm. So in a spring system, the spring forces the mass back toward the centre. Thus `F = ma = -kx`, the acceleration being directed back toward the centre of the harmonic oscillator and proportional to the extension.

The solution to the related differential equations are sinusoids, typically stated as:

`x(t) = A cos omega t + B sinomega t qquad triangle`

Uniform circular motion is a different thing. There is no restorative force in the sense that the mass is always a distance/radius `R` from the centre of rotation.

And that force is constant in magnitude, but changes direction as it is always centre-seeking force ... in the sense that there is a centrifugal force that causes the mass to accelerate towards the centre of the circle.

You can connect them up by projecting uniform circular motion onto a straight line. For convenience, the x or y axis

Uniform circular motion of fixed radius `R` at constant angular velocity `vec omega` in rectangular coordinates is

`vec r = R ((cos omega t),(sin omega t))`

Projecting this onto the x-axis using the dot product....
`x(t) = vec r * hat y = R ((cos omega t),(sin omega t))*((1),(0)) = R cos omega t qquad square`

So the motion of the projection is sinusoidal.

You can compare this directly to `triangle`, you even plug in initial values to see that they are identical.