Abstract
This chapter proposes the study of the movement of a glider on an air track driven by the pull of a wire that passes by a pulley connected to a suspended small mass. The experimental data on the kinematics of the physical system (glider + mass) are compared to the predictions provided by the principles of dynamics. An initial analysis is done based on the assumption that the pulley has no influence on the system. The comparison of the experimental data to the proposed model can suggest a more elaborate model that takes into account the movement of the pulley. In this case, it should be considered other physical concepts as moment of inertia and rotation of rigid body.
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Notes
- 1.
The apparatus used in this experiment is manufactured by the company CIDEPE (Centro Industrial de Equipamentos de Ensino e Pesquisa—www.cidepe.com.br). The experimental development as well as the data analysis is valid for any similar commercial or homemade equipment available in any didactic laboratory.
- 2.
The resolution concerning equipment with digital display corresponds to the digital increment. In the case of analogical display the resolution should be estimated by the experimentalist.
- 3.
The estimative was made via uncertainty propagation, considering the value of g as the standard value:
\( \delta a=\sqrt{{\left(\frac{\partial a}{\partial m}\delta m\right)}^2+{\left(\frac{\partial a}{\partial M}\delta M\right)}^2}=\frac{g}{m}{\left(\frac{1}{1+\frac{M}{m}}\right)}^2\sqrt{{\left(\frac{M\delta m}{m}\right)}^2+\delta {M}^2}=0.0004 m/{s}^2 \)
- 4.
The rotation kinetic energy E rotation of a disc of radius R around its central axis with angular velocity ω corresponds to the sum of the kinetic energy of all N particles that compose the disc:
$$ {E}_{r otation}=\sum_{i=1}^N\frac{1}{2}{m}_i{v}_i^2=\frac{1}{2}\left(\sum_{i=1}^N{m}_i{r}_i^2\right)\kern0.24em {\omega}^2=\frac{1}{2} I\;{\omega}^2 $$The relation v i = ω r i was used and it is related to each particle that makes uniform circular movement at a distance r i from the axis. The sum between parentheses is known as momentum of inertia. In the case of a disc:
\( I=\sum_{i=1}^N{m}_i{r}_i^2=\int {r}^2 dm=\sigma \iint {r}^2 dA=\sigma \underset{0}{\overset{R}{\int }}{r}^3 dr\underset{0}{\overset{2\pi}{\int }} d\theta =\sigma \frac{\pi\;{R}^4}{2}=\frac{M_p}{\pi\;{R}^2}\frac{\pi\;{R}^4}{2}=\frac{1}{2}{M}_p{R}^2 \)
Considering a homogeneous disc of density \( \sigma =\frac{M_p}{\pi\;{R}^2} \), and dA = r dr dθ the element of area in polar coordinates.
- 5.
The estimative was obtained by uncertainty propagation considering the standard value of g:
$$ \delta a=\sqrt{{\left(\frac{\partial a}{\partial m}\delta m\right)}^2+{\left(\frac{\partial a}{\partial M}\delta M\right)}^2+{\left(\frac{\partial a}{\partial {M}_p}\delta {M}_p\right)}^2} $$$$ \delta a=\frac{g}{m}{\left(\frac{1}{1+\frac{M}{m}+\frac{M_p}{2 m}}\right)}^2\sqrt{{\left( M+\frac{M_p}{2}\right)}^2{\left(\frac{\delta m}{m}\right)}^2+\delta {M}^2+{\left(\frac{\delta {M}_p}{2}\right)}^2}=0.0004\kern0.24em m/{s}^2 $$ - 6.
The uncertainty was estimated from the relation:
\( \delta {E}_{M0}=\sqrt{{\left(\frac{\partial {E}_{M0}}{\partial m}\delta m\right)}^2+{\left(\frac{\partial {E}_{M0}}{\partial h}\delta h\right)}^2}= g\sqrt{{\left( h\delta m\right)}^2+{\left( m\delta h\right)}^2}=3\times {10}^{-4} J \)
- 7.
The uncertainty was estimated from the relation:
\( \delta {E}_{M0}=\sqrt{{\left(\frac{\partial {E}_{M0}}{\partial m}\delta m\right)}^2+{\left(\frac{\partial {E}_{M0}}{\partial h}\delta h\right)}^2+{\left(\frac{\partial {E}_{M0}}{\partial {M}_p}\delta {M}_p\right)}^2+{\left(\frac{\partial {E}_{M0}}{\partial v}\delta v\right)}^2}\approx \frac{\partial {E}_{M0}}{\partial v}\delta v={10}^{-2} J \)
It is interesting to evaluate the contribution of each term. Proceeding in this way we find that the term related to the uncertainty of velocity is the most important, about 3 or 4 orders of magnitude higher than the others, as it can be seen below:
\( \frac{\partial {E}_{M0}}{\partial m}\delta m=\frac{v^2}{2}\delta m=4.9\times {10}^{-6} J \); \( \frac{\partial {E}_{M0}}{\partial M}\delta M=\frac{v^2}{2}\delta M=4.9\times {10}^{-6} J \)
\( \frac{\partial {E}_{M0}}{\partial {M}_p}\delta {M}_p=\frac{v^2}{2}\delta {M}_p=1.2\times {10}^{-5} J \); \( \frac{\partial {E}_{M0}}{\partial v}\delta v=\left( m+ M+\frac{M_p}{2}\right) v\delta v=9.8\times {10}^{-3} J \)
References
J.R. Pimentel, V.H. Zumpano, L.T. Yaginuma, Trilho de ar – uma proposta de baixo custo. Rev. Bras. Fis. 11, 15–23 (1989)
F. Laudares, M.C.S.M. Lopes, F.A.O. Cruz, Usando sensores magnéticos em um trilho de ar. Rev. Bras. Ens. Fis. 26(3), 236 (2004)
J.R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 2nd edn. (University Science Books, Sausalito, 1996)
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de Jesus, V.L.B. (2017). Dynamics. 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_5
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