Abstract
The pure rolling of a bicycle wheel is the focus of this chapter. The wheel is launched at a very low velocity and the motion of a point of its edge is investigated by video analysis using both laboratory and center of mass reference frames. The condition of pure rolling is experimentally observed. This experiment is an example that several movements can be understood in a simple way by using the center of mass as a reference frame.
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Notes
- 1.
The uncertainty was estimated using the following equation:
\( \delta {v}_n=\frac{1}{\Delta {t}_{n+1; n-1}}\sqrt{{\left(\delta {x}_{n+1}\right)}^2+{\left(\delta {x}_{n-1}\right)}^2} \)
In this case, \( \Delta {t}_{n+1; n-1}=\frac{2}{30}\; s \) and δx n + 1 = δx n + 1 = δx = 0.2 cm. So, the uncertainty is
\( \delta {v}_n=\frac{0.2}{2/30}\sqrt{2}=3\sqrt{2}\approx 4\kern0.24em \mathrm{cm}/\mathrm{s}. \)
- 2.
The representation \( \dot{\theta}(t) \) corresponds to the first time derivative \( \frac{d\theta (t)}{d t} \).
- 3.
The uncertainty of \( \theta = \arctan \left(\frac{Y}{X}\right) \) was estimated using the following equation \( \delta \theta =\sqrt{{\left(\frac{\partial \theta}{\partial X}\delta X\right)}^2+{\left(\frac{\partial \theta}{\partial Y}\delta Y\right)}^2} \). The partial derivatives are \( \frac{\partial \theta }{\partial X}\delta X=\left(\frac{Y}{X^2}\right)\frac{\delta X}{1+{\left(Y/X\right)}^2}\kern1.32em ;\frac{\partial \theta }{\partial Y}\delta Y=\left(\frac{1}{X}\right)\frac{\delta Y}{1+{\left(Y/X\right)}^2} \). As δX = δY the final expression is \( \delta \theta =\left(\frac{\delta X}{X}\right)\frac{1}{1+{\left( Y/ X\right)}^2} \).The uncertainty δX can be estimated as \( \delta X=\delta x\sqrt{2}=0.2\sqrt{2}\approx 0.3\kern0.24em \mathrm{cm} \). Calculating the uncertainty for each value presented in Table I you can verify that \( \delta \theta =\left(\frac{0.3}{X}\right)\frac{1}{1+{\left( Y/ X\right)}^2}\le 0.01\kern0.24em \mathrm{rad} \)
- 4.
The uncertainty was estimated using the following equation:
\( \delta r=\sqrt{{\left(\frac{\partial r}{\partial v}\delta v\right)}^2+{\left(\frac{\partial r}{\partial \omega}\delta \omega \right)}^2} \)
Calculating the partial derivatives one finds
\( \delta r=\sqrt{{\left(\frac{\delta v}{\omega}\right)}^2+{\left(\frac{v}{\omega^2}\delta \omega \right)}^2}=\sqrt{{\left(\frac{0.2}{3.014}\right)}^2+{\left(\frac{87.3}{(3.014)^2}\times 0.007\right)}^2}=0.09\kern0.24em \mathrm{cm} \)
The same result can be obtained using the relative uncertainty equation
\( {\left(\frac{\delta r}{r}\right)}^2={\left(\frac{\delta v}{v}\right)}^2+{\left(\frac{\delta \omega}{\omega}\right)}^2. \)
Reference
D. Micha, M. Ferreira, Física no esporte—Parte 1: saltos em esportes coletivos. Uma motivação para o estudo da mecânica através da análise dos movimentos do corpo humano a partir do conceito de centro de massa. Rev. Bras. Ens. Fis. 35(3), 3301–3301 (2013)
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de Jesus, V.L.B. (2017). Pure Rolling by Video Analysis. In: Experiments and Video Analysis in Classical Mechanics . Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-52407-8_9
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DOI: https://doi.org/10.1007/978-3-319-52407-8_9
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