01 Going Round in Circles

01 Going Round in Circles#

Aim#

To see/feel centripetal force.

Subjects#

  • 1D50 Central Forces

Diagram#

../../../../../_images/figure_06.png

Fig. 16 .#

Equipment#

  • Tube, with rounded edges, \(l=15 \mathrm{~cm}\).

  • Piece of rope, \(l=1.5 \mathrm{~m}\).

  • Two rubber stoppers \(\left(m_{1}\right)\).

  • A number of weights \(\left(m_{2}\right)\). We use thick washers.

  • Paperclip.

  • (Stopwatch).

Presentation#

Diagram shows the components and how to use them. Swinging the tube a little makes the mass \(m_{1}\) go round in circles above your head. The demonstrator needs to make \(m_{1}\) go round at a certain frequency to balance the system.

../../../../../_images/figure_14.png

Fig. 17 .#

  • When he slows down to a lower frequency \(m_{2}\) will move down, making the circle of \(m_{1}\) smaller and smaller. Speeding up makes \(m_{2}\) go upward and \(m_{1}\) goes round in larger and larger circles.

  • The demonstrator makes \(m_{1}\) go round in a stable circle. Then he grabs \(m_{2}\) and pulls it downwards. \(m_{1}\) speeds up dramatically, going round in smaller and smaller circles.

If time permits the relationship between the variables in this demonstration can be verified more exactly. Just below the tubing a paperclip is fixed to the rope used as a marker to make \(m_{1}\) go round in a circle with fixed \(R\). A stopwatch can be used to time the frequency.

  1. When \(m_{1}\) is doubled by adding another rubber stopper to it, a lower frequency is needed to balance the system.

  2. When \(m_{2}\) is increased, a higher frequency is needed to balance the system.

  3. When half the rope length is used (shifting the paperclip) a higher frequency is needed to balance the system.

Explanation#

Analysis shows that movement at a constant speed ( \(v\) ) of a mass ( \(m_{1}\) ) in a circle with radius \(R\) can be described by \(a_{c}=\frac{v^{2}}{r} \omega^{2} R\). In our demonstration the tension ( \(T\) ) in the string provides the force needed for \(a_{c}: T=m_{1} a_{c}\), and \(m_{2} g=m_{1} a_{c} \Rightarrow a_{c}=\frac{m_{2}}{m_{1}} g\), (see Figure 18).

../../../../../_images/figure_22.png

Fig. 18 .#

  • First, the demonstrator showed a balanced situation. Then he slowed down \(\omega\). The centripetal acceleration, describing such a slower movement, will be lower \(\left(a_{c}=\omega^{2} R\right)\). But \(T\) is a fixed value, so \(T\) will pull \(m_{1}\) inwardly. As \(a_{c}=\omega^{2} R\) shows, this proces is cumulative and \(m_{1}\) ends at the centre of the circle.

  • When \(m_{2}\) is pulled downwards, the string tension increases substantially and so \(a_{c}\) will. According to \(a_{c}=\omega^{2} R\) and seeing that \(R\) decreases, \(\omega\) increases much.

  1. Doubling \(m_{1}\) means that the provided \(a_{c}\) by the tension of the string will be halved \(\left(a_{c}=\frac{m_{2}}{m_{1}} g\right)\). To make \(m_{1}\) still go round in the same circle \(\omega\) has to be a factor \(\sqrt{2}\) lower \(\left(a_{c}=\omega^{2} R\right)\).

  2. Increasing \(m_{2}\) will make the string tension higher and so the provided \(a_{c}\) is higher \(\left(T=m_{1} a_{c}\right)\). To make \(m_{1}\) still go in the same circle, whas to increase.

  3. When \(R\) is halved. The tension in the string has remained the same, so the provided \(a_{c}\) has remained the same. To make \(m_{1}\) still go in a (smaller) circle, \(\omega\) has to increase a factor \(\sqrt{2}\left(a_{c}=\omega^{2} R\right)\).

Remarks#

  • Practice the demonstration before you show it. A practiced hand is needed to make \(m_{1}\) go round properly.

  • We use rubber stoppers as masses moving around in circles for safety reasons.

  • The spinning mass should be light compared to the hanging weight (about a factor 3), because otherwise the angle between the string and the vertical does not approach \(90^{\circ}\) : there will be more friction and due to the slanting rope (making a cone of our circle) the analysis becomes a different one.

  • In the last part of the presentation (grabbing and pulling \(m_{2}\) downwards) the demonstrator will feel that quite a lot of force is needed. It is of course most instructive for the students if they feel this force themselves (during coffee break?).

Sources#

  • Ehrlich, Robert, Turning the World Inside Out and 174 Other Simple Physics Demonstrations, pag. 72-73

  • Mansfield, M and O’Sullivan, C., Understanding physics, pag. 68-71 and74-75

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