Seconds pendulum
This activity belongs to the GeoGebra book The Domain of the Time.
This animation simulates the motion of a seconds pendulum in real time, neglecting the weight of the rod and friction. The animation does not use formulas (neither trigonometry nor equations nor differential calculus); it only makes the necessary variations in the vectors that direct the motion.
In the previous activity about the simple pendulum, we observed that for small amplitudes (less than approximately 10º), the period T is practically constant and equal to the period T0 of simple harmonic motion (SHM).
Since the period of SHM is:
we can calculate the length of the rod so that this period is exactly 2 seconds (one second for the forward swing and one second for the return swing). This value is:
Using a pendulum of this type, it was possible for the first time in history to measure a quantity of time as small as a second with considerable accuracy (a day has 86,400 seconds). As you can see, the length of the rod is almost exactly one meter. This is not a coincidence. This measurement was originally proposed in the late 17th century as a standard measure of length, eventually evolving into our current meter, which differs by just over half a centimeter.
- Note: We will see later, in the activity on the cycloidal pendulum, how Huygens ingeniously avoided the small time difference between the period of the pendulum and the period of SHM, successfully constructing accurate pendulum clocks.
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
SetValue(aux, vt)
SetValue(v, vt + dt gt)
SetValue(M, M + dt v)
# Record the period time and the number of complete oscillations
SetValue(reg, If(x(aux) < 0 ∧ x(vt) > 0, Append(t, reg), reg))
SetValue(osci, If(x(aux) < 0 ∧ x(vt) > 0, osci + 1, osci))
Author of the activity and GeoGebra construction: Rafael Losada.