SHM and UCM: isochronism
This activity belongs to the GeoGebra book The Domain of the Time.
Simple Harmonic Motion (SHM) can be seen as a projection of Uniform Circular Motion (UCM). Let's see how.
The following construction is designed so that the green point MM follows a UCM around O, but always remains vertically aligned with the blue point M. This implies that the radius of the circle must be equal to the amplitude A of the SHM. The centripetal acceleration of this UCM is c (green dashed vector), whose magnitude we have seen equals ω2 A.
To keep MM always vertically aligned with M, the horizontal component of c must be precisely the acceleration a of the SHM. This is because the triangle with a as a leg and c as the hypotenuse must be similar to the triangle with x as a leg and A as the hypotenuse, as shown in the construction.
Due to this similarity of triangles, we have that |a|/|c| = x/A, meaning |a| = |c| x/A = ω2 x.
But we also knew that |a| = k/m x, so it must hold that ω2 = k/m, meaning the angular velocity of the UCM must be ω = and the period must be T = 2π/ω = 2π. Since every time MM makes one full revolution, M completes one full oscillation, the period of both motions must be the same.
We then deduce that the period of the SHM does not depend on the amplitude A; it only depends on the mass m and the elasticity k. For a spring with elasticity k, the mass m will always take the same amount of time to complete a full oscillation. This property is known as isochronism.
SCRIPT FOR SLIDER anima
# Calculate the elapsed seconds dt; add one second if t1(1) < tt
SetValue(tt, t1(1))
SetValue(t1, First(GetTime(), 3))
SetValue(dt, (t1(1) < tt) + (t1(1) − tt)/1000)
# Move M and MM
SetValue(MM, O + (r; t ω))
SetValue(aux, v)
SetValue(v, v + dt a)
SetValue(M, M + dt v)
# Record the period time and the number of complete oscillations
SetValue(reg, If(x(aux) > 0 ∧ x(v) < 0, Append(t, reg), reg))
SetValue(osci, If(x(aux) > 0 ∧ x(v) < 0, osci + 1, osci))
Author of the activity and GeoGebra construction: Rafael Losada.