Difference between revisions of "Angular Momentum Conservation"

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[[File:Discos_velocidade_protocolo1.png|thumb|alt=|Figura1: velocidade de rotação]]
 
[[File:Discos_velocidade_protocolo1.png|thumb|alt=|Figura1: velocidade de rotação]]
  
5 discs with a total mass of 115g are acelerated by the hard-drive motor until they reach 1500rpm. In this moment the motor is disconnected, the discs rotate freely and their velocity is read. When a certain velocity that the user defines previously is reached, the servo let's 3 suspended discs with a total mass of 69g initially at rest fall on top of the o servo deixa cair sobre os discos em rotação 3 discos com 69g no total inicialmente em repouso.
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5 discs with a total mass of 115g are acelerated by the hard-drive motor until they reach 1500rpm. In this moment the motor is disconnected, the discs rotate freely and their velocity is read. When a certain velocity that the user defines previously is reached, the servo let's 3 suspended discs with a total mass of 69g initially at rest fall on top of the rotating discs.
  
No final da sessão obtém-se uma tabela com a velocidade dos discos em função do tempo, cujos valores servirão para criar um gráfico num programa ao critério do utilizador.
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In the end of the session a table is provided with the disc velocity function of time, whose values will be used in a plot created in some program the user chooses.
  
A '''Figura1''' é um gráfico criado no Microsoft Excel a partir da tabela de resultados de uma experiência em que o servo deixa os discos cair a 1000 rpm. Fazendo uma regressão entre o início da desaceleração e a queda dos discos, obtém-se uma reta cuja equação nos permite determinar a velocidade prevista dos discos em rotação no ponto em que os discos que caem deixam de deslizar sobre os que estavam em rotação.
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'''Figure1''' it's a plot created in Microsoft Excel using the table of results of an experience in which the servo let's the suspended discs fall when the discs below reach 1000 rpm.  
  
Usando as seguintes quantidades:
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Doing a linear regression between the deceleration and fall of the discs, it's possible to obtain the predicted velocity of the discs at the time that the falling discs stop sliding over the bottom discs.
  
L - momento angular
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Using the following quantities:
  
I - momento de inércia
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L - angular momentum
  
ω - velocidade angular
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I - moment of inertia
  
m - massa em rotação.
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ω - angular velocity
  
<div style="text-align: center;">Temos para a conservação do momento angular:</div>
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m - mass in rotation.
 +
 
 +
<div style="text-align: center;">We have for the angular momentum conservation:</div>
 
\[L_i=L_f\]
 
\[L_i=L_f\]
 
\[I_i \omega_i=I_f \omega_f\]
 
\[I_i \omega_i=I_f \omega_f\]

Revision as of 12:14, 12 October 2012

UNDER CONSTRUCTION

Description of the Experiment

This control room allows the confirmation of angular momentum conservation and the measuring of the moment of inertia of rotating discs.

Experimental Apparatus

The experimental apparatus is based in a PC hard-drive to which were added a servo and a group of discs that the servo keeps above the rotating discs. This setup allows the realization of the first protocol, centered in the study of angular momentum conservation.

The aparatus also has a group of braking resistors that can be connected in parallel with the motor windings. This setup allows the realization of the second protocol, centered in the study of the moment of inertia.

Protocol1 - Angular Momentum Conservation

Figura1: velocidade de rotação

5 discs with a total mass of 115g are acelerated by the hard-drive motor until they reach 1500rpm. In this moment the motor is disconnected, the discs rotate freely and their velocity is read. When a certain velocity that the user defines previously is reached, the servo let's 3 suspended discs with a total mass of 69g initially at rest fall on top of the rotating discs.

In the end of the session a table is provided with the disc velocity function of time, whose values will be used in a plot created in some program the user chooses.

Figure1 it's a plot created in Microsoft Excel using the table of results of an experience in which the servo let's the suspended discs fall when the discs below reach 1000 rpm.

Doing a linear regression between the deceleration and fall of the discs, it's possible to obtain the predicted velocity of the discs at the time that the falling discs stop sliding over the bottom discs.

Using the following quantities:

L - angular momentum

I - moment of inertia

ω - angular velocity

m - mass in rotation.

We have for the angular momentum conservation:

\[L_i=L_f\] \[I_i \omega_i=I_f \omega_f\] \[\frac{I_i}{I_f}=\frac{\omega_f}{\omega_i}\] \[\frac{\frac{m_i\left (r_1^2+r_2^2 \right )}{2}}{\frac{m_f\left (r_1^2+r_2^2 \right )}{2}}=\frac{\omega_f}{\omega_i}\] \[\frac{m_i}{m_f}=\frac{\omega_f}{\omega_i}\]

Obtém-se experimentalmente

\[\frac{\omega_f}{\omega_i}=\frac{623}{950}=0,656\]

enquanto que pela razão das massas

\[\frac{m_i}{m_f}=\frac{115}{115+69}=0,625\]

Fazendo um desvio à exatidão

\[\frac{\left|0,656-0,625\right|}{\left|0,625\right|}\times 100=4,9\%\]

Conclui-se que a razão das velocidades (experimental) difere 4,9% da razão das massas (teórica), que está de acordo com a conservação do momento angular.