10 Rolling Up-and-Down, Again and Again

10 Rolling Up-and-Down, Again and Again#

Aim#

Determining the coefficient of rolling friction and to give an impression how low the coefficient of rolling friction is.

Subjects#

  • 1K20 (Friction)

Diagram#

../../../../../_images/figure_051.png

Fig. 121 .#

Equipment#

  • U-shaped railtrack

  • Metal ball

Presentation#

Release the ball and it will roll down the track, climb the other track, and so on. But gradually the distance it rolls reduces (due to rolling friction).

After \(\mathrm{n}\) runs the coefficient of rolling friction can be determined by measuring the distance the ball travels upward in the \(\mathrm{n}\)-th run.

Explanation#

The potential energy of the ball equals (see Figure 122 and Figure 123)

../../../../../_images/figure_140.png

Fig. 122 .#

\(U_{p}(0)=m g s_{0} \sin (\alpha)=F s_{0}\)

Reacting the other side (1): \(U_{p}(0)-U_{p}(1)=F_{f}\left(s_{0}+s_{1}\right)\)

So: \(F\left(s_{0}-s_{1}\right)=F_{f}\left(s_{0}+s_{1}\right)\)

\(s_{1}=s_{0}\left[\frac{F-F_{f}}{F+F_{f}}\right]=s_{0}\left[\frac{1-\frac{F_{f}}{F}}{1+\frac{F_{f}}{F}}\right]=s_{0} b\)

Rolling back ( \(\mathrm{s}_{1}\) ) and up ( \(\mathrm{s}_{2}\) ) again:

\(s_{2}=s_{1} \cdot b=s_{0} \cdot b^{2}\)

The coefficient of friction \((\mu)\) is by definition \(F_{f} / F_{N}\).

In this case (see Figure 123): \(\mu=\frac{F_{f}}{F} \tan \alpha\).

../../../../../_images/figure_217.png

Fig. 123 .#

So the coefficient of friction can be determined by measuring \(s_{0}, s_{2}\) and \(\alpha\) and using the formulas above.

Sources#

  • Jordens, H.

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