02 Conical Pendulum

02 Conical Pendulum#

Aim#

To show that the period of motion of a conical pendulum changes only noticeably at large angles.

Subjects#

  • 1D50 (Central Forces)

Diagram#

../../../../../_images/figure_07.png

Fig. 19 .#

Equipment#

  • Ball suspended to a thread.

  • Clamping material.

  • Large paper circle, \(r = 35 \mathrm{~cm}\).

  • Small paper circle, \(r = 7.5 \mathrm{~cm}\).

  • Stopwatch with large display.

  • Small ball suspended to a thread

Presentation#

  1. Set up the conical pendulum as shown in the Diagram. Place the small paper circle under the pendulum and make the pendulum swing conically along the circumference of the paper circle. Measure the time needed for 10 periods. Repeat this procedure, but now with the large paper circle. In our set-up, the times measured are 18.2 and 17.5 sec respectively.

  2. Take the small simple pendulum by hand and make it swing conically. Gradually speed it up. At very large angles the increase in the angular speed is noticed easily.

Explanation#

Theory tells us that the period ( \(T\) ) of a conical pendulum is given by \(T=2 \pi \sqrt{\frac{l \cos \phi}{g}}\) (see Figure 20).

../../../../../_images/figure_15.png

Fig. 20 .#

So \(T \propto \sqrt{\cos \phi}\)

The table in Table 1 shows that from \(0^{\circ}\) to \(30^{\circ}, \sqrt{\cos \phi}\) only changes \(7 \%\), while from \(60^{\circ}\) to \(89^{\circ}\) this change is about \(82 \%\). So only at large angles \(\phi\), \(T\) changes noticeably.

Table 1 table#

\(\varphi(\%)\)

\(\sqrt{\cos \varphi}\)

0

1

15

0,98

30

0,93

45

0,84

60

0,71

75

0,51

80

0,42

85

0,30

89

0,13

Remarks#

  • When the pendulum is suspended vertically and not swinging, we have marked this central position on the table. The paper circles have a hole in their center so that it is easy to position the paper circles in the right place (hole and mark coincide).

  • Making the pendulum swing along the circumference of the paper circle needs some practice. Launch the suspended ball tangentially and give it a speed so that it just reaches a deflection equal to R of the paper circle (see Figure 21).

../../../../../_images/figure_31.png

Fig. 21 .#

Sources#

  • Mansfield, M and O’Sullivan, C., Understanding physics, pag. 70

  • Roest, R., Inleiding Mechanica, pag. 55-56

  • Young, H.D. and Freedman, R.A., University Physics, pag. 141-142

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