02 Mathematical pendulum (2) Large angle

02 Mathematical pendulum (2) Large angle#

Aim#

To show that the period of motion of a simple pendulum depends on the angle the pendulum makes with the vertical.

Subjects#

  • 3A10 (Pendula)

Diagram#

../../../../../_images/figure_0124.png

Fig. 289 .#

Equipment#

  • Pendulum; brass bob attached to a threaded rod \((I=50 \mathrm{~cm})\) and connected to a support with ball bearing.

  • Large cardboard with the principal angles of deflection indicated on it (see Diagram).

  • Photogate

  • Computerinterface.

  • Computer with data-acquisition system. (we use PASCO Science Workshop)

Presentation#

The photogate is placed just offset the rest-position of the pendulum. The data-acquisition system is set up in such a way that a graph of periodtimes can be presented. The data-acquisition is started, and by hand the pendulum is given a deflection of almost \(180^{\circ}\) and released. When \(\theta\) has reached angles smaller than \(90^{\circ}\), the data-acquisition is stopped. During the data-acquisition the students observe the graph displayed (see red line in Figure 290).

../../../../../_images/figure_195.png

Fig. 290 .#

(This means that the expected range of the axes of the graph have to be prepared before the demonstration is started.)

A second run is made, giving the pendulum the smallest deflection possible. After about \(10-20\) registrations of \(T\) the data-acquisition is stopped. The complete graph can be observed and discussed now.

Explanation#

The equation that describes the motion of the mass \(m\) is given by \(a_{x}=\frac{d^{2 s}}{d t^{2}}=-g \sin \theta\) (x-direction along the tangent of the circle; see Figure 291A). This is not a simple harmonic motion since \(\sin \theta\) is not proportional to \(s\).

Only for small amplitude oscillations \(\sin \theta \approx \theta=\frac{S}{l}\) and the equation of motion reduces to \(\frac{d^{2} s}{d t^{2}}=-\frac{g}{l} s\) This is the differential equation for simple harmonic motion. Then the period is given by \(T=2 \pi \sqrt{\frac{l}{g}}\)

For large amplitudes we need \(a_{x}=-g \sin \theta\) in stead of \(a_{x}=-g \theta\). Since \(\sin \theta<\theta\), this means that \(a_{x}\) is smaller than the small-amplitude equation indicates: The mass will need more time than \(T=2 \pi \sqrt{\frac{l}{g}}\) to reach its maximum deflection. In other words: \(T\) is larger than \(2 \pi \sqrt{\frac{l}{g}}\).

(For an exact solution to the equation of motion: see SourcesXX.)

Remarks#

  • Also see the demonstration “Mathematical pendulum (1) - Simple harmonic motion”. With that demonstration the effect on the acceleration a can be observed very well.

  • When you observe the pendulum directly by eye it can be seen directly that the period of oscillation is larger at larger angles.

  • The software is setup in such a way that the period is presented after the pendulum has passed three times through the photogate. Every next period is presented after every second passage (see Figure 291B).

../../../../../_images/figure_244.png

Fig. 291 .#

  • Since the system measures the complete period the position of the photogate can be at any arbitrary point along the arc of motion.

Sources#

  • Borghouts, A.N., Inleiding in de Mechanica, pag. 129-131

  • Mansfield, M and O’Sullivan, C., Understanding physics, pag. 72-73

  • Roest, R., Inleiding Mechanica, pag. 91-93

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