Spagna 2008

Cuelgue sucesivamente del hilo las arandelas necesarias para que los valores de la masa $m$ sean los indicados en la primera columna de la tabla 1 de la hoja de respuestas. Para cada valor de $m$, mida y anote en la columna adecuada de la tabla I de la figura 5, al final del enunciado: • El tiempo que tarda $m$ en descender una altura $h=1,00\text{m}$. Es conveniente repetir tres veces la medida ($t_1$, $t_2$ y $t_3$) y calcular el valor medio, $\bar{t}$. • La caída de potencial $\Delta V$ en la resistencia $R$. Como su valor puede variar ligeramente durante el descenso de $m$, es conveniente anotar los valores máximo y mínimo observados y tomar el promedio, $\overline{\Delta V}$, para cálculos posteriores.

Topic: Electromagnetic Induction, Newtonian Mechanics, Circuits Metodi: Kinematic Equations, Experimental Data Analysis, Physical Modeling Competenze: Measurement & Instrumentation, Experimental Data Analysis, Error Propagation Objects: Resistor, String Fonte: Testo (PDF) — p.3

Sulla fila si appendono successivamente le arande necessarie per rendere i valori di massa $m$ Le informazioni relative alle domande di risposta sono riportate nella prima colonna della tabella 1. Per ogni valore $m$, misura e annotate nella colonna appropriata del tabella I di figura 5, alla fine della frase: • El tiempo que tarda $m$ en descender una altura $h=1,00\text{m}$. È opportuno ripetere la misura tre volte ($t_1$, $t_2$ e $t_3$) e calcolare il valore medio, $\bar{t}$. • Il calo di potenziale $\Delta V$ nella resistenza $R$. Come il loro valore può variare leggermente durante il a $m$, è opportuno annotare i valori massimi e minimi osservati e prendere la media, $\overline{\Delta V}$, per ulteriori calcoli.

Topic: Electromagnetic Induction, Newtonian Mechanics, Circuits Metodi: Kinematic Equations, Experimental Data Analysis, Physical Modeling Competenze: Measurement & Instrumentation, Experimental Data Analysis, Error Propagation Objects: Resistor, String Fonte: Testo (PDF) — p.3

Hang the needle shafts from the thread in succession so that the mass values $m$ are the The following information is provided in the first column of Table 1 of the reply sheet: For each $m$ value, measure and annotate in the appropriate column of Table I of Figure 5, at the end of the sentence: • The time it takes $m$ to descend to a height $h=1,00\text{m}$. It is advisable to repeat the measure three times ($t_1$, $t_2$ and $t_3$) and calculate the mean value, $\bar{t}$. • The potential drop $\Delta V$ in the resistance $R$. As its value may vary slightly during the decrease of $m$, it is appropriate to note the maximum and minimum values observed and take the average, $\overline{\Delta V}$, for further calculations.

Topic: Electromagnetic Induction, Newtonian Mechanics, Circuits Metodi: Kinematic Equations, Experimental Data Analysis, Physical Modeling Competenze: Measurement & Instrumentation, Experimental Data Analysis, Error Propagation Objects: Resistor, String Fonte: Testo (PDF) — p.3

A partir de las medidas anteriores, calcule para cada $m$, y anote en las columnas correspondientes de la tabla II de la figura 6, la velocidad $v$ de la masa, la intensidad $I$ que circula por el circuito y la fem $ϵ$ del generador.

Topic: Circuits, Electromagnetic Induction Metodi: Kirchhoff’s Laws, Physical Modeling, Kinematic Equations Competenze: Experimental Data Analysis, Mathematical Modeling, Physical Reasoning Objects: — Fonte: Testo (PDF) — p.3

A partire dalle misure precedenti, calcola per ogni $m$ e annotate le colonne corrispondenti della tabella di riferimento: tabella II di figura 6, velocità $v$ della massa, intensità $I$ che circola nel circuito e fem $ϵ$ del

• Generatore.

Topic: Circuits, Electromagnetic Induction Metodi: Kirchhoff’s Laws, Physical Modeling, Kinematic Equations Competenze: Experimental Data Analysis, Mathematical Modeling, Physical Reasoning Objects: — Fonte: Testo (PDF) — p.3

From the above measurements, calculate for each $m$ and note in the corresponding columns of the Table II of Figure 6, mass speed $v$, power $I$ circulating through the circuit and fem $ϵ$ of the The generator.

Topic: Circuits, Electromagnetic Induction Metodi: Kirchhoff’s Laws, Physical Modeling, Kinematic Equations Competenze: Experimental Data Analysis, Mathematical Modeling, Physical Reasoning Objects: — Fonte: Testo (PDF) — p.3

Aplicando (1), calcule en cada caso el valor de la constante $k$ del generador y anótelo en la última columna de la tabla 2. Calcule el valor medio de $k$ y haga una estimación de su incertidumbre (margen de error), $\Delta k$.

Topic: Electromagnetic Induction, Circuits Metodi: Experimental Data Analysis, Physical Modeling, Statistical Averaging Competenze: Experimental Data Analysis, Error Propagation, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.3

Applicando (1), calcola in ogni caso il valore della costante $k$ del generatore e annotate l’ultima colonna della tabella 2. Calcolare il valore medio di $k$ e calcolare l’incertezza (margine di errore), $\Delta k$.

Topic: Electromagnetic Induction, Circuits Metodi: Experimental Data Analysis, Physical Modeling, Statistical Averaging Competenze: Experimental Data Analysis, Error Propagation, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.3

Applying (1), calculate the value of the generator constant $k$ in each case and enter it in the last column of table 2. Calculate the mean value of $k$ and estimate its uncertainty (margin of error), $\Delta k$.

Topic: Electromagnetic Induction, Circuits Metodi: Experimental Data Analysis, Physical Modeling, Statistical Averaging Competenze: Experimental Data Analysis, Error Propagation, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.3

Represente gráficamente en el papel milimetrado los puntos experimentales $(x,y)=(m,v)$.

Topic: Electromagnetic Induction, Circuits Metodi: Experimental Data Analysis, Physical Modeling Competenze: Graph Linearization, Experimental Data Analysis, Diagrammatic Reasoning Objects: — Fonte: Testo (PDF) — p.3

Rappresenta graficamente sui pezzi di carta millimetrica i punti sperimentali $(x,y)=(m,v)$.

Topic: Electromagnetic Induction, Circuits Metodi: Experimental Data Analysis, Physical Modeling Competenze: Graph Linearization, Experimental Data Analysis, Diagrammatic Reasoning Objects: — Fonte: Testo (PDF) — p.3

Graphically represent the experimental points $(x,y)=(m,v)$ on the millimeter paper.

Topic: Electromagnetic Induction, Circuits Metodi: Experimental Data Analysis, Physical Modeling Competenze: Graph Linearization, Experimental Data Analysis, Diagrammatic Reasoning Objects: — Fonte: Testo (PDF) — p.3

Ajuste estos puntos a una línea recta.

Topic: Electromagnetic Induction, Circuits Metodi: Graph Linearization, Experimental Data Analysis Competenze: Graph Linearization, Curve Fitting, Experimental Data Analysis Objects: — Fonte: Testo (PDF) — p.3

Aggiusta questi punti in una linea retta.

Topic: Electromagnetic Induction, Circuits Metodi: Graph Linearization, Experimental Data Analysis Competenze: Graph Linearization, Curve Fitting, Experimental Data Analysis Objects: — Fonte: Testo (PDF) — p.3

Adjust these points to a straight line.

Topic: Electromagnetic Induction, Circuits Metodi: Graph Linearization, Experimental Data Analysis Competenze: Graph Linearization, Curve Fitting, Experimental Data Analysis Objects: — Fonte: Testo (PDF) — p.3

Deduzca de este ajuste los valores de las dos constantes características del generador³, $k$ y $m_0$.

1 En nuestro montaje la velocidad límite se alcanza al poco de soltar $m$, como es fácil comprobar experimentalmente. 2 La fuerza de rozamiento dinámica (cuando hay deslizamiento relativo) es inferior a la fuerza de rozamiento estática máxima (límite de equilibrio, sin deslizamiento). Por ello, en nuestro sistema, la masa que hay que colgar para que el generador empiece a girar partiendo del reposo puede ser apreciablemente superior a $m_0$. 3 Puede haber discrepancias del orden del 5% entre los valores de $k$ obtenidos en los apartados 3 y 6, debidas principalmente a la dispersión de los valores reales de $R+r$ en los diversos montajes experimentales. Para cálculos posteriores, es preferible tomar el valor de $k$ obtenido en el apartado 6. XIX OLIMPIADA ESPAÑOLA DE FÍSICA.

Topic: Electromagnetic Induction, Circuits, Newtonian Mechanics Metodi: Graph Linearization, Physical Modeling, Experimental Data Analysis Competenze: Graph Linearization, Mathematical Modeling, Experimental Data Analysis Objects: — Fonte: Testo (PDF) — p.3

Della regolazione si deducono i valori delle due costanti caratteristiche del generatore3, $k$ e $m_0$.

1 Nel nostro montaggio la velocità limite è raggiunta poco dopo il rilascio $m$, come è facile verificare sperimentalmente. 2 La forza di rottura dinamica (quando c’è uno scivolo relativo) è inferiore alla forza di rottura statica massimo (limite di equilibrio, senza scorciatoie). In questo modo, nel nostro sistema, la massa che deve essere sospesa per il generatore inizia a girare a partire dal riposo può essere appreziatamente superiore a $m_0$. 3 Potranno esserci discrepanze dell’ordine del 5% tra i valori di $k$ ottenuti nei paragrafi 3 e 6, a causa di: principalmente a la dispersión de los valores reales de $R+r$ en los diversos montajes experimentales. Per calcoli Se il valore di $k$ è ottenuto in un secondo momento, è preferibile prendere il valore di $k$ ottenuto in un secondo momento. XIX Olimpiada di Fisica spagnola.

Topic: Electromagnetic Induction, Circuits, Newtonian Mechanics Metodi: Graph Linearization, Physical Modeling, Experimental Data Analysis Competenze: Graph Linearization, Mathematical Modeling, Experimental Data Analysis Objects: — Fonte: Testo (PDF) — p.3

Subtract from this setting the values of the two generator characteristic constants3, $k$ and $m_0$.

1 In our assembly the speed limit is reached at $m$, as is easily verified experimentally. 2 The dynamic friction force (when there is relative slip) is less than the static friction force maximum (balance limit, without slippage). The Commission has therefore decided to extend the scope of the The output of the generator starting from the resting position may be significantly higher than $m_0$. 3 There may be discrepancies of the order of 5% between the values of $k$ obtained in paragraphs 3 and 6, due to the fact that the value of MSK0/> is not principalmente a la dispersión de los valores reales de $R+r$ en los diversos montajes experimentales. For calculations The value of $k$ obtained in paragraph 6 is preferable. 19th Spanish Olympics in Physics.

Topic: Electromagnetic Induction, Circuits, Newtonian Mechanics Metodi: Graph Linearization, Physical Modeling, Experimental Data Analysis Competenze: Graph Linearization, Mathematical Modeling, Experimental Data Analysis Objects: — Fonte: Testo (PDF) — p.3

Haga una estimación de las incertidumbres de estas dos constantes.

Topic: Electromagnetic Induction, Circuits Metodi: Error Propagation, Experimental Data Analysis, Physical Modeling Competenze: Error Propagation, Estimation & Approximation, Experimental Data Analysis Objects: — Fonte: Testo (PDF) — p.4

Stima le incertezze di queste due costanti.

Topic: Electromagnetic Induction, Circuits Metodi: Error Propagation, Experimental Data Analysis, Physical Modeling Competenze: Error Propagation, Estimation & Approximation, Experimental Data Analysis Objects: — Fonte: Testo (PDF) — p.4

Make an estimate of the uncertainties of these two constants.

Topic: Electromagnetic Induction, Circuits Metodi: Error Propagation, Experimental Data Analysis, Physical Modeling Competenze: Error Propagation, Estimation & Approximation, Experimental Data Analysis Objects: — Fonte: Testo (PDF) — p.4

Deduzca la expresión analítica de la intensidad que circula por el circuito, $I$, en función de las constantes internas del generador ($k$ y $m_0$), y de los parámetros externos ($m$ y $R$). En particular, ¿cómo depende $I$ de $R$?

Topic: Circuits, Electromagnetic Induction, Newtonian Mechanics Metodi: Kirchhoff’s Laws, Physical Modeling, Conservation Laws Competenze: Mathematical Modeling, Physical Reasoning, Diagrammatic Reasoning Objects: Resistor Fonte: Testo (PDF) — p.4

Deduce l’espressione analitica dell’intensità che circola nel circuito, $I$, in funzione delle costanti le parti interne del generatore ($k$ e $m_0$), e dei parametri esterni ($m$ e $R$). In particolare, come $I$ dipende da $R$?

Topic: Circuits, Electromagnetic Induction, Newtonian Mechanics Metodi: Kirchhoff’s Laws, Physical Modeling, Conservation Laws Competenze: Mathematical Modeling, Physical Reasoning, Diagrammatic Reasoning Objects: Resistor Fonte: Testo (PDF) — p.4

Subtract the analytical expression of the circuit-circulating intensity, $I$, based on the constants internas del generador ($k$ y $m_0$), y de los parámetros externos ($m$ y $R$). In particular, how does $I$ depend on $R$?

Topic: Circuits, Electromagnetic Induction, Newtonian Mechanics Metodi: Kirchhoff’s Laws, Physical Modeling, Conservation Laws Competenze: Mathematical Modeling, Physical Reasoning, Diagrammatic Reasoning Objects: Resistor Fonte: Testo (PDF) — p.4

Mediante las medidas que considere oportunas, compruebe experimentalmente la dependencia prevista de $I$ con $R$. Explique detalladamente las medidas que ha realizado y preséntelas en la tabla III de la figura 7. Discuta los resultados. Sugerencia: trabaje con una masa suspendida fija, por ejemplo $m=23,2\text{g}$.

Tabla I

$m$ (g)	$t_1$ (s)	$t_2$ (s)	$t_3$ (s)	$\bar{t}$ (s)	$\Delta V_{\text{max}}$ (V)	$\Delta V_{\text{min}}$ (V)	$\overline{\Delta V}$ (V)
17,5
23,2
29,1
34,8
40,7
46,4
52,3
58,0

Fig. 5

Tabla III

$R$ ($\Omega$)

Fig. 7

Tabla II

$m$ (g)	$v$ (m/s)	$I$ (A)	$ϵ$ (V)	$k$ (Vs/m)
17,5
23,2
29,1
34,8
40,7
46,4
52,3
58,0

Fig. 6

XIX OLIMPIADA ESPAÑOLA DE FÍSICA. Solución

Topic: Circuits, Electromagnetic Induction Metodi: Experimental Data Analysis, Kirchhoff’s Laws, Physical Modeling Competenze: Measurement & Instrumentation, Experimental Data Analysis, Physical Reasoning Objects: Resistor Fonte: Testo (PDF) — p.4

In base alle misure che ritiene opportune, verificare sperimentalmente la dipendenza prevista da $I$ con $R$. Esprimi in dettaglio le misure che hai adottato e presentale in tabella III di figura 7. Discutete dei risultati. Suggerimento: lavorare con una massa sospesa fissa, ad esempio $m=23,2\text{g}$.

Tabella I

$m$ (g)	$t_1$ (s)	$t_2$ (s)	$t_3$ (s)	$\bar{t}$ (s)	$\Delta V_{\text{max}}$ (V)	$\Delta V_{\text{min}}$ (V)	$\overline{\Delta V}$ (V)
17,5
23,2
29,1
34,8
40,7
46,4
52,3
58,0

Fig. 5

Tabella III

$R$ ($\Omega$)

Fig. 7

Tabella II

$m$ (g)	$v$ (m/s)	$I$ (A)	$ϵ$ (V)	$k$ (Vs/m)
17,5
23,2
29,1
34,8
40,7
46,4
52,3
58,0

Fig. 6

XIX Olimpiada di Fisica spagnola. Soluzione

Topic: Circuits, Electromagnetic Induction Metodi: Experimental Data Analysis, Kirchhoff’s Laws, Physical Modeling Competenze: Measurement & Instrumentation, Experimental Data Analysis, Physical Reasoning Objects: Resistor Fonte: Testo (PDF) — p.4

By means of measures which it considers appropriate, it shall experimentally verify the expected dependence on the $I$ con $R$. Please explain in detail the measures you have taken and present them in Table III in Figure 7. Discuss the results. Suggestion: work with a fixed suspended mass, e.g. $m=23,2\text{g}$.

The following table is inserted:

$m$ (g)	$t_1$ (s)	$t_2$ (s)	$t_3$ (s)	$\bar{t}$ (s)	$\Delta V_{\text{max}}$ (V)	$\Delta V_{\text{min}}$ (V)	$\overline{\Delta V}$ (V)
17,5
23,2
29,1
34,8
40,7
46,4
52,3
58,0

Fig. 5

The following table shows the following:

$R$ ($\Omega$)

Fig. 7

The following table shows the following:

$m$ (g)	$v$ (m/s)	$I$ (A)	$ϵ$ (V)	$k$ (Vs/m)
17,5
23,2
29,1
34,8
40,7
46,4
52,3
58,0

Fig. 6

19th Spanish Olympics in Physics. Solution

Topic: Circuits, Electromagnetic Induction Metodi: Experimental Data Analysis, Kirchhoff’s Laws, Physical Modeling Competenze: Measurement & Instrumentation, Experimental Data Analysis, Physical Reasoning Objects: Resistor Fonte: Testo (PDF) — p.4

Ejemplo de medidas reales:

Tabla I

Topic: Electromagnetic Induction, Newtonian Mechanics, Circuits Metodi: Kinematic Equations, Experimental Data Analysis, Physical Modeling Competenze: Measurement & Instrumentation, Experimental Data Analysis, Error Propagation Objects: — Fonte: Testo (PDF) — p.5

Esempio di misure reali:

Tabella I

Topic: Electromagnetic Induction, Newtonian Mechanics, Circuits Metodi: Kinematic Equations, Experimental Data Analysis, Physical Modeling Competenze: Measurement & Instrumentation, Experimental Data Analysis, Error Propagation Objects: — Fonte: Testo (PDF) — p.5

Example of actual measures:

The following table is inserted:

Topic: Electromagnetic Induction, Newtonian Mechanics, Circuits Metodi: Kinematic Equations, Experimental Data Analysis, Physical Modeling Competenze: Measurement & Instrumentation, Experimental Data Analysis, Error Propagation Objects: — Fonte: Testo (PDF) — p.5

Aplicando $v=h/t$, $I=\Delta V/R$ y $ϵ=(R+r)I$, se obtiene

Tabla II

Topic: Circuits, Electromagnetic Induction Metodi: Kirchhoff’s Laws, Physical Modeling, Kinematic Equations Competenze: Experimental Data Analysis, Mathematical Modeling, Physical Reasoning Objects: Resistor Fonte: Testo (PDF) — p.5

Applicando $v=h/t$, $I=\Delta V/R$ e $ϵ=(R+r)I$ si ottiene

Tabella II

Topic: Circuits, Electromagnetic Induction Metodi: Kirchhoff’s Laws, Physical Modeling, Kinematic Equations Competenze: Experimental Data Analysis, Mathematical Modeling, Physical Reasoning Objects: Resistor Fonte: Testo (PDF) — p.5

Aplicando $v=h/t$, $I=\Delta V/R$ y $ϵ=(R+r)I$, se obtiene

The following table shows the following:

Topic: Circuits, Electromagnetic Induction Metodi: Kirchhoff’s Laws, Physical Modeling, Kinematic Equations Competenze: Experimental Data Analysis, Mathematical Modeling, Physical Reasoning Objects: Resistor Fonte: Testo (PDF) — p.5

En la última columna de la tabla anterior se presentan los valores de la constante $k=ϵ/v$ obtenidos en cada caso. El valor medio es

$\bar{k}=4,71\text{Vs/m}$

Una estimación razonable para la incertidumbre de esta constante, obtenida como promedio de $n=8$ determinaciones, es su error típico⁴

$\Delta k={\left[\frac{\sum (\bar{k}-k_i{)}^2}{n(n-1)}\right]}^{1/2}$

Se obtiene

$\Delta k=0,04\text{Vs/m}$

4 Si se toma como incertidumbre el error típico, el nivel de confianza es inferior al 65% pues se han realizado menos de 10 medidas. Para aumentar este nivel al 95% habría que duplicar con creces este margen de error. Sin necesidad de buscar en unas tablas de error, un resultado razonable con aproximadamente el 95% de confianza sería $k=(4,7\pm 0,1)\text{Vs/m}$.

$m$ (g)	$t_1$ (s)	$t_2$ (s)	$t_3$ (s)	$\bar{t}$ (s)	$\Delta V_{\text{max}}$ (V)	$\Delta V_{\text{min}}$ (V)	$\overline{\Delta V}$ (V)
17,5	14,40	14,37	14,44	14,40	0,15	0,15	0,15
23,2	8,72	8,75	8,72	8,73	0,25	0,23	0,24
29,1	6,90	6,40	6,50	6,60	0,34	0,32	0,33
34,8	5,37	5,34	5,28	5,33	0,42	0,39	0,405
40,7	4,56	4,29	4,53	4,46	0,52	0,49	0,505
46,4	3,84	3,62	3,69	3,72	0,59	0,57	0,58
52,3	3,22	3,28	3,28	3,26	0,68	0,66	0,67
58,0	2,82	2,88	2,78	2,83	0,75	0,73	0,74

$m$ (g)	$v$ (m/s)	$I$ (A)	$ϵ$ (V)	$k$ (Vs/m)
17,5	0,0694	0,020	0,33	4,76
23,2	0,115	0,032	0,52	4,54
29,1	0,152	0,044	0,72	4,75
34,8	0,188	0,054	0,89	4,74
40,7	0,224	0,067	1,09	4,86
46,4	0,269	0,077	1,26	4,68
52,3	0,307	0,089	1,46	4,76
58,0	0,354	0,099	1,61	4,55

XIX OLIMPIADA ESPAÑOLA DE FÍSICA.

Topic: Electromagnetic Induction, Circuits Metodi: Statistical Averaging, Experimental Data Analysis, Physical Modeling Competenze: Experimental Data Analysis, Error Propagation, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.5

L’ultima colonna della tabella precedente presenta i valori della costante $k=ϵ/v$ ottenuti in ogni caso. Il valore medio è

$\bar{k}=4,71\text{Vs/m}$

Una ragionevole stima dell’incertezza di questa costante, ottenuta come media di $n=8$ La determinazione è il suo errore tipico4

$\Delta k={\left[\frac{\sum (\bar{k}-k_i{)}^2}{n(n-1)}\right]}^{1/2}$

Si ottiene

$\Delta k=0,04\text{Vs/m}$

4 Se si prende come incertezza l’errore tipico, il livello di fiducia è inferiore al 65% perché meno di 10 misure. Per aumentare questo livello al 95%, si dovrebbe raddoppiare di gran lunga questo margine di errore. Senza bisogno di cercare in se si tratta di tabelle di errore, un risultato ragionevole con circa il 95% di fiducia sarebbe $k=(4,7\pm 0,1)\text{Vs/m}$.

$m$ (g)	$t_1$ (s)	$t_2$ (s)	$t_3$ (s)	$\bar{t}$ (s)	$\Delta V_{\text{max}}$ (V)	$\Delta V_{\text{min}}$ (V)	$\overline{\Delta V}$ (V)
17,5	14,40	14,37	14,44	14,40	0,15	0,15	0,15
23,2	8,72	8,75	8,72	8,73	0,25	0,23	0,24
29,1	6,90	6,40	6,50	6,60	0,34	0,32	0,33
34,8	5,37	5,34	5,28	5,33	0,42	0,39	0,405
40,7	4,56	4,29	4,53	4,46	0,52	0,49	0,505
46,4	3,84	3,62	3,69	3,72	0,59	0,57	0,58
52,3	3,22	3,28	3,28	3,26	0,68	0,66	0,67
58,0	2,82	2,88	2,78	2,83	0,75	0,73	0,74

$m$ (g)	$v$ (m/s)	$I$ (A)	$ϵ$ (V)	$k$ (Vs/m)
17,5	0,0694	0,020	0,33	4,76
23,2	0,115	0,032	0,52	4,54
29,1	0,152	0,044	0,72	4,75
34,8	0,188	0,054	0,89	4,74
40,7	0,224	0,067	1,09	4,86
46,4	0,269	0,077	1,26	4,68
52,3	0,307	0,089	1,46	4,76
58,0	0,354	0,099	1,61	4,55

XIX Olimpiada di Fisica spagnola.

Topic: Electromagnetic Induction, Circuits Metodi: Statistical Averaging, Experimental Data Analysis, Physical Modeling Competenze: Experimental Data Analysis, Error Propagation, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.5

The last column of the table above presents the values of the $k=ϵ/v$ constant obtained in every case. The mean value is

$\bar{k}=4,71\text{Vs/m}$

A reasonable estimate of the uncertainty of this constant, obtained as an average of $n=8$ The Commission’s decision to adopt the directive is not a matter for the Commission.

$\Delta k={\left[\frac{\sum (\bar{k}-k_i{)}^2}{n(n-1)}\right]}^{1/2}$

You get it

$\Delta k=0,04\text{Vs/m}$

4 If the typical error is taken as uncertainty, the confidence level is less than 65% because less than 10 measures. To increase this level to 95%, this margin of error would have to be increased by more than twice. No need to look in If the error table is a reasonable result with approximately 95% confidence, it would be $k=(4,7\pm 0,1)\text{Vs/m}$.

$m$ (g)	$t_1$ (s)	$t_2$ (s)	$t_3$ (s)	$\bar{t}$ (s)	$\Delta V_{\text{max}}$ (V)	$\Delta V_{\text{min}}$ (V)	$\overline{\Delta V}$ (V)
17,5	14,40	14,37	14,44	14,40	0,15	0,15	0,15
23,2	8,72	8,75	8,72	8,73	0,25	0,23	0,24
29,1	6,90	6,40	6,50	6,60	0,34	0,32	0,33
34,8	5,37	5,34	5,28	5,33	0,42	0,39	0,405
40,7	4,56	4,29	4,53	4,46	0,52	0,49	0,505
46,4	3,84	3,62	3,69	3,72	0,59	0,57	0,58
52,3	3,22	3,28	3,28	3,26	0,68	0,66	0,67
58,0	2,82	2,88	2,78	2,83	0,75	0,73	0,74

$m$ (g)	$v$ (m/s)	$I$ (A)	$ϵ$ (V)	$k$ (Vs/m)
17,5	0,0694	0,020	0,33	4,76
23,2	0,115	0,032	0,52	4,54
29,1	0,152	0,044	0,72	4,75
34,8	0,188	0,054	0,89	4,74
40,7	0,224	0,067	1,09	4,86
46,4	0,269	0,077	1,26	4,68
52,3	0,307	0,089	1,46	4,76
58,0	0,354	0,099	1,61	4,55

19th Spanish Olympics in Physics.

Topic: Electromagnetic Induction, Circuits Metodi: Statistical Averaging, Experimental Data Analysis, Physical Modeling Competenze: Experimental Data Analysis, Error Propagation, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.5

Se presenta la gráfica pedida, con un aspecto similar al que tendría dibujada en papel milimetrado.

Topic: Electromagnetic Induction, Circuits Metodi: Experimental Data Analysis, Physical Modeling Competenze: Graph Linearization, Experimental Data Analysis, Diagrammatic Reasoning Objects: — Fonte: Testo (PDF) — p.6

Si presenta il grafico richiesto, con un aspetto simile a quello che avrebbe disegnato su carta millimetrica.

Topic: Electromagnetic Induction, Circuits Metodi: Experimental Data Analysis, Physical Modeling Competenze: Graph Linearization, Experimental Data Analysis, Diagrammatic Reasoning Objects: — Fonte: Testo (PDF) — p.6

The graphic ordered is presented, with a similar appearance to what it would have been drawn on millimeter paper.

Topic: Electromagnetic Induction, Circuits Metodi: Experimental Data Analysis, Physical Modeling Competenze: Graph Linearization, Experimental Data Analysis, Diagrammatic Reasoning Objects: — Fonte: Testo (PDF) — p.6

En la gráfica anterior también se ha trazado la recta que más aproximadamente pasa por los puntos experimentales. La pendiente y la ordenada en el origen pueden calcularse con buena precisión a partir de dos puntos alejados de esta recta. Pueden tomarse, por ejemplo, los puntos extremos de dicha recta, de coordenadas

$(x_1;y_1)=(0,0;-0,050),(x_2;y_2)=(60,0;0,362)$

Por tanto, la pendiente, $p$, y la ordenada en el origen, $c$, de la recta son⁵

$p=\frac{y_2-y_1}{x_2-x_1}=\frac{0,412\text{m/s}}{60,0\text{g}}\Rightarrow p=6,87{\text{m s}}^{-1}{\text{kg}}^{-1}$

$c=y_1\Rightarrow c=-0,050\text{m/s}$

Topic: Electromagnetic Induction, Circuits Metodi: Graph Linearization, Experimental Data Analysis Competenze: Graph Linearization, Curve Fitting, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.6

Il grafico precedente ha anche tracciato la retta che più approssimativamente passa attraverso i punti

• Esperienti. La pendice e la pendice di origine possono essere calcolate con buona precisione a partire da due punti distanti da questa retta. Si possono prendere, ad esempio, gli estremi di tale retta, coordinate

$(x_1;y_1)=(0,0;-0,050),(x_2;y_2)=(60,0;0,362)$

La pendice $p$ e la pendice ordinata all’origine $c$ della retta sono quindi5

$p=\frac{y_2-y_1}{x_2-x_1}=\frac{0,412\text{m/s}}{60,0\text{g}}\Rightarrow p=6,87{\text{m s}}^{-1}{\text{kg}}^{-1}$

$c=y_1\Rightarrow c=-0,050\text{m/s}$

Topic: Electromagnetic Induction, Circuits Metodi: Graph Linearization, Experimental Data Analysis Competenze: Graph Linearization, Curve Fitting, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.6

In the previous graph, the line that passes through the points is also drawn. The experimental ones. The slope and the order at the origin can be calculated with good precision from two points away from this straight line. The end points of this line can be taken, for example, from the Coordinates

$(x_1;y_1)=(0,0;-0,050),(x_2;y_2)=(60,0;0,362)$

Therefore, the slope $p$ and the slope at the origin, $c$, of the straight line are5

$p=\frac{y_2-y_1}{x_2-x_1}=\frac{0,412\text{m/s}}{60,0\text{g}}\Rightarrow p=6,87{\text{m s}}^{-1}{\text{kg}}^{-1}$

$c=y_1\Rightarrow c=-0,050\text{m/s}$

Topic: Electromagnetic Induction, Circuits Metodi: Graph Linearization, Experimental Data Analysis Competenze: Graph Linearization, Curve Fitting, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.6

Según la ecuación (2) del enunciado, la pendiente y la ordenada en el origen de la gráfica anterior son, respectivamente,

$p=\frac{g(R+r)}{k^2},c=-p{m}_0$

Por tanto, las constantes $k$ y $m_0$ del generador pueden deducirse en la forma

$k={\left(\frac{g(R+r)}{p}\right)}^{1/2},m_0=-\frac{c}{p}(3)$

Se obtiene

5 Aplicando el método de mínimos cuadrados se obtiene $p=6,866{\text{m⋅s}}^{-1}{\text{kg}}^{-1}$ y $c=-0,0496\text{m/s}$.

XIX OLIMPIADA ESPAÑOLA DE FÍSICA.

$k=4,82\text{Vs/m},m_0=7,3\text{g}$

Topic: Electromagnetic Induction, Circuits, Newtonian Mechanics Metodi: Graph Linearization, Physical Modeling, Experimental Data Analysis Competenze: Graph Linearization, Mathematical Modeling, Experimental Data Analysis Objects: — Fonte: Testo (PDF) — p.6

Secondo l’equazione (2) della frase, la pendice e la pendice ordinate all’origine del grafico precedente sono: rispettivamente,

$p=\frac{g(R+r)}{k^2},c=-p{m}_0$

Le costanti $k$ e $m_0$ del generatore possono quindi essere dedotte in

$k={\left(\frac{g(R+r)}{p}\right)}^{1/2},m_0=-\frac{c}{p}(3)$

Si ottiene

5 Applicando il metodo dei minimi quadrati si ottiene $p=6,866{\text{m⋅s}}^{-1}{\text{kg}}^{-1}$ e $c=-0,0496\text{m/s}$.

XIX Olimpiada di Fisica spagnola.

$k=4,82\text{Vs/m},m_0=7,3\text{g}$

Topic: Electromagnetic Induction, Circuits, Newtonian Mechanics Metodi: Graph Linearization, Physical Modeling, Experimental Data Analysis Competenze: Graph Linearization, Mathematical Modeling, Experimental Data Analysis Objects: — Fonte: Testo (PDF) — p.6

According to the statement equation (2), the slope and the order at the origin of the above graph are: respectively,

$p=\frac{g(R+r)}{k^2},c=-p{m}_0$

Therefore, the $k$ and $m_0$ generating constants can be deduced as

$k={\left(\frac{g(R+r)}{p}\right)}^{1/2},m_0=-\frac{c}{p}(3)$

You get it

5 Using the least squares method, $p=6,866{\text{m⋅s}}^{-1}{\text{kg}}^{-1}$ and $c=-0,0496\text{m/s}$ are obtained.

19th Spanish Olympics in Physics.

$k=4,82\text{Vs/m},m_0=7,3\text{g}$

Topic: Electromagnetic Induction, Circuits, Newtonian Mechanics Metodi: Graph Linearization, Physical Modeling, Experimental Data Analysis Competenze: Graph Linearization, Mathematical Modeling, Experimental Data Analysis Objects: — Fonte: Testo (PDF) — p.6

Las incertidumbres de la pendiente y de la ordenada en el origen de la gráfica $v(m)$ pueden estimarse trazando las rectas que con pendientes máxima y mínima se ajustan razonablemente a los puntos experimentales, dentro de su margen de incertidumbre. En nuestro caso, la dispersión de los puntos experimentales respecto a la recta óptima es pequeña, del orden del radio de los puntos dibujados. Teniendo en cuenta, además, que ambas rectas deben pasar por el punto promedio de los experimentales, $(\bar{m},\bar{v})$, en la siguiente gráfica se presenta una estimación razonable de estas rectas.

Observando sobre la gráfica las coordenadas de los dos puntos extremos de cada una de estas dos rectas, como en el apartado 5, se obtienen sus pendientes y sus ordenadas en el origen

$p_{\text{max}}=\frac{(0,365+0,055)\text{m/s}}{60,0\text{g}}=7,00{\text{m⋅s}}^{-1}{\text{kg}}^{-1},c_{\text{min}}=-0,055\text{m/s}$

$p_{\text{min}}=\frac{(0,359+0,045)\text{m/s}}{60,0\text{g}}=6,73{\text{m⋅s}}^{-1}{\text{kg}}^{-1},c_{\text{max}}=-0,045\text{m/s}$

Por tanto, una estimación de las incertidumbres de $p$ y $c$ podría ser⁶

$\Delta p=0,13{\text{m⋅s}}^{-1}{\text{kg}}^{-1},\Delta c=0,005\text{m/s}$

Los correspondientes valores máximo y mínimo de las constantes $k$ y $m_0$ se obtienen aplicando (3) con los valores de $p$ y $c$ ajustados para cada una de estas dos rectas

6 Un cálculo estadístico conduce a que los errores típicos de $p$ y $c$ son, respectivamente, $0,11{\text{m⋅s}}^{-1}{\text{kg}}^{-1}$ y $0,004\text{m/s}$.

XIX OLIMPIADA ESPAÑOLA DE FÍSICA.

$k_{\text{max}}={\left(\frac{g(R+r)}{p_{\text{min}}}\right)}^{1/2}=4,87\text{Vs/m},m_{0,\text{max}}=-\frac{c_{\text{min}}}{p_{\text{max}}}=7,9\text{g}$

$k_{\text{min}}={\left(\frac{g(R+r)}{p_{\text{max}}}\right)}^{1/2}=4,78\text{Vs/m},m_{0,\text{min}}=-\frac{c_{\text{max}}}{p_{\text{min}}}=6,7\text{g}$

Los valores máximo y mínimo estimados para $m_0$ también pueden leerse directamente en la gráfica anterior, pues, teniendo en cuenta la ecuación (2) del enunciado, corresponden a las intersecciones de las dos rectas con la ordenada $v=0$.

En total, los resultados para $k$ y $m_0$, incluyendo una estimación de sus incertidumbres, serían

$k=(4,82\pm 0,05)\text{Vs/m}$

$m_0=(7,3\pm 0,6)\text{g}$

Nótese que las incertidumbres relativas de $k$ y $m_0$, son del orden del 1% y del 8%, respectivamente, por lo que la principal fuente de error para cálculos posteriores es la incertidumbre de $m_0$.

Topic: Electromagnetic Induction, Circuits Metodi: Error Propagation, Graph Linearization, Experimental Data Analysis Competenze: Error Propagation, Estimation & Approximation, Graph Linearization Objects: — Fonte: Testo (PDF) — p.7

Le incertezze di pendenza e di ordine nella fonte del grafico $v(m)$ possono essere stimate tracciando le linee che con pendici massimi e minimi si adattano ragionevolmente ai punti La Commissione ha adottato una decisione che non è stata adottata. Nel nostro caso, la dispersione dei punti La linea ottimale è piccola, nell’ordine del raggio dei punti disegnati. Considerando inoltre che entrambe le linee devono passare per il punto medio degli esperimenti, $(\bar{m},\bar{v})$, il seguente grafico presenta una ragionevole stima di queste rette.

Guardando sulla grafica le coordinate dei due punti estremisti di ciascuno di questi due rettitudini, come previsto al paragrafo 5, le loro pendenti e le loro ordini sono ottenute all’origine

$p_{\text{max}}=\frac{(0,365+0,055)\text{m/s}}{60,0\text{g}}=7,00{\text{m⋅s}}^{-1}{\text{kg}}^{-1},c_{\text{min}}=-0,055\text{m/s}$

$p_{\text{min}}=\frac{(0,359+0,045)\text{m/s}}{60,0\text{g}}=6,73{\text{m⋅s}}^{-1}{\text{kg}}^{-1},c_{\text{max}}=-0,045\text{m/s}$

Un’estimazione delle incertezze di $p$ e $c$ potrebbe quindi essere6

$\Delta p=0,13{\text{m⋅s}}^{-1}{\text{kg}}^{-1},\Delta c=0,005\text{m/s}$

I valori massimi e minimi corrispondenti delle costanti $k$ e $m_0$ sono ottenuti applicando (3) con i valori $p$ e $c$ aggiustati per ciascuna di queste due linee

6 Un calcolo statistico porta a che gli errori tipici di $p$ e $c$ sono rispettivamente $0,11{\text{m⋅s}}^{-1}{\text{kg}}^{-1}$ e $0,004\text{m/s}$.

XIX Olimpiada di Fisica spagnola.

$k_{\text{max}}={\left(\frac{g(R+r)}{p_{\text{min}}}\right)}^{1/2}=4,87\text{Vs/m},m_{0,\text{max}}=-\frac{c_{\text{min}}}{p_{\text{max}}}=7,9\text{g}$

$k_{\text{min}}={\left(\frac{g(R+r)}{p_{\text{max}}}\right)}^{1/2}=4,78\text{Vs/m},m_{0,\text{min}}=-\frac{c_{\text{max}}}{p_{\text{min}}}=6,7\text{g}$

I valori massimi e minimi stimati per $m_0$ possono anche essere letti direttamente nel grafico Infatti, in base all’equazione (2) della frase precedente, le intersezioni delle due righe con la scritta $v=0$.

In totale, i risultati per $k$ e $m_0$, compresa una stima delle loro incertezze, sarebbero

$k=(4,82\pm 0,05)\text{Vs/m}$

$m_0=(7,3\pm 0,6)\text{g}$

Si noti che le incertezze relative di $k$ e $m_0$ sono rispettivamente dell’ordine dell’1% e dell’8%, per la principale fonte di errore per i successivi calcoli è l’incertezza di $m_0$.

Topic: Electromagnetic Induction, Circuits Metodi: Error Propagation, Graph Linearization, Experimental Data Analysis Competenze: Error Propagation, Estimation & Approximation, Graph Linearization Objects: — Fonte: Testo (PDF) — p.7

The slope and order uncertainties at the source of the graph $v(m)$ can be estimated Drawing the lines which with the highest and lowest slopes reasonably fit the points The Commission’s proposal for a directive on the protection of workers’ rights in the field of social protection is to be adopted by the Council. In our case, the scattering of the points The experimental reference to the optimum straight is small, in the order of the radii of the points drawn. Whereas, furthermore, both directions must pass through the average point of the experimental ones, $(\bar{m},\bar{v})$, a reasonable estimate of these straight lines is given in the following graph.

Looking at the coordinates of the two extreme points of each of these two rectangles on the graph, as in paragraph 5, their earrings and their orders are obtained at the origin

$p_{\text{max}}=\frac{(0,365+0,055)\text{m/s}}{60,0\text{g}}=7,00{\text{m⋅s}}^{-1}{\text{kg}}^{-1},c_{\text{min}}=-0,055\text{m/s}$

$p_{\text{min}}=\frac{(0,359+0,045)\text{m/s}}{60,0\text{g}}=6,73{\text{m⋅s}}^{-1}{\text{kg}}^{-1},c_{\text{max}}=-0,045\text{m/s}$

Therefore, an estimate of the uncertainties of $p$ and $c$ could be6

$\Delta p=0,13{\text{m⋅s}}^{-1}{\text{kg}}^{-1},\Delta c=0,005\text{m/s}$

The corresponding maximum and minimum values of the constants $k$ and $m_0$ are obtained by applying (3) to the values of $p$ and $c$ adjusted for each of these two straight lines

6 A statistical calculation leads to the typical errors of $p$ and $c$ being $0,11{\text{m⋅s}}^{-1}{\text{kg}}^{-1}$ and $0,004\text{m/s}$ respectively.

19th Spanish Olympics in Physics.

$k_{\text{max}}={\left(\frac{g(R+r)}{p_{\text{min}}}\right)}^{1/2}=4,87\text{Vs/m},m_{0,\text{max}}=-\frac{c_{\text{min}}}{p_{\text{max}}}=7,9\text{g}$

$k_{\text{min}}={\left(\frac{g(R+r)}{p_{\text{max}}}\right)}^{1/2}=4,78\text{Vs/m},m_{0,\text{min}}=-\frac{c_{\text{max}}}{p_{\text{min}}}=6,7\text{g}$

The estimated maximum and minimum values for $m_0$ can also be read directly on the graph The following are the intersections of the two points of the two straight lines with the order $v=0$.

In total, the results for $k$ and $m_0$, including an estimate of their uncertainties, would be:

$k=(4,82\pm 0,05)\text{Vs/m}$

$m_0=(7,3\pm 0,6)\text{g}$

Note that the relative uncertainties of $k$ and $m_0$ are of the order of 1% and 8%, respectively, for The main source of error for subsequent calculations is the uncertainty of $m_0$.

Topic: Electromagnetic Induction, Circuits Metodi: Error Propagation, Graph Linearization, Experimental Data Analysis Competenze: Error Propagation, Estimation & Approximation, Graph Linearization Objects: — Fonte: Testo (PDF) — p.7

La intensidad que circula en régimen estacionario por el circuito es, teniendo en cuenta la ley de Ohm y las ecuaciones (1) y (2) del enunciado

$I=\frac{ϵ}{R+r}=\frac{kv}{R+r}=\frac{k}{R+r}\frac{g(R+r)}{k^2}(m-m_0)$

Simplificando, queda

$I=\frac{(m-m_0)g}{k}(4)$

Esta corriente no depende de la resistencia del circuito. Por ello, tal y como indica el título de esta prueba experimental, el sistema opera como un generador de corriente. Nótese que la constante $k$ del generador describe la relación entre el peso motor efectivo y la intensidad que suministra el generador, por lo que en el SI las unidades de $k$ podrían expresarse como N/A.

Topic: Circuits, Electromagnetic Induction, Newtonian Mechanics Metodi: Kirchhoff’s Laws, Physical Modeling, Conservation Laws Competenze: Mathematical Modeling, Physical Reasoning, Diagrammatic Reasoning Objects: Resistor Fonte: Testo (PDF) — p.8

L’intensità che circola in modo stazionario sul circuito è, tenendo conto della legge di Ohm e Le due parti sono state presentate in un’unica versione:

$I=\frac{ϵ}{R+r}=\frac{kv}{R+r}=\frac{k}{R+r}\frac{g(R+r)}{k^2}(m-m_0)$

In semplicità, resta

$I=\frac{(m-m_0)g}{k}(4)$

Questo corrente non dipende dalla resistenza del circuito. Pertanto, come indica il titolo di questa prova sperimentare, il sistema funziona come un generatore di corrente. Nota che la costante $k$ del generatore descrive il rapporto tra il peso motore effettivo e l’intensità fornita dal generatore, il SI le unità di $k$ potrebbero essere espresse come N/A.

Topic: Circuits, Electromagnetic Induction, Newtonian Mechanics Metodi: Kirchhoff’s Laws, Physical Modeling, Conservation Laws Competenze: Mathematical Modeling, Physical Reasoning, Diagrammatic Reasoning Objects: Resistor Fonte: Testo (PDF) — p.8

The intensity circulating in stationary mode over the circuit is, taking into account Ohm’s law and the equations (1) and (2) of the statement

$I=\frac{ϵ}{R+r}=\frac{kv}{R+r}=\frac{k}{R+r}\frac{g(R+r)}{k^2}(m-m_0)$

Simply put, it remains

$I=\frac{(m-m_0)g}{k}(4)$

This current does not depend on the resistance of the circuit. Therefore, as the title of this test indicates, Experimental, the system operates like a power generator. Note that the generator constant $k$ It describes the relationship between the actual engine weight and the power supplied by the generator, so that in the SI units of $k$ could be expressed as N/A.

Topic: Circuits, Electromagnetic Induction, Newtonian Mechanics Metodi: Kirchhoff’s Laws, Physical Modeling, Conservation Laws Competenze: Mathematical Modeling, Physical Reasoning, Diagrammatic Reasoning Objects: Resistor Fonte: Testo (PDF) — p.8

En particular, para $m=23,2\text{g}$ la ecuación (4) prevé una corriente

$I=32,4\text{mA}$

Propagando las incertidumbres de $k$ y $m_0$ obtenidas en el apartado 7, se obtiene una incertidumbre para esta corriente

$\Delta I=1,2\text{mA}.$

Como se dispone de dos resistencias de 15 $\Omega$, puede trabajarse experimentalmente con tres valores de la resistencia de carga del generador: $R_1=7,5\Omega$ si se conectan en paralelo, $R_2=15\Omega$ si se conecta una sola resistencia y $R_3=30\Omega$ si se conectan en serie. En la tabla III se presentan medidas de la caída de potencial $\Delta V$ en cada caso y de la intensidad deducida como $I=\Delta V/R$. También podría medirse directamente la intensidad usando el multímetro como amperímetro, conectado en serie con la resistencia y el generador.

El acuerdo entre las previsiones del modelo y los resultados experimentales es correcto, dentro del margen de incertidumbre estimado.

$R$ ($\Omega$)	$\Delta V_{\text{max}}$ (V)	$\Delta V_{\text{min}}$ (V)	$\overline{\Delta V}$ (V)	$I$ (mA)
7,5	0,25	0,23	0,24	32
15	0,52	0,48	0,50	33
30	1,01	0,97	0,99	33

Tabla III

Topic: Circuits, Electromagnetic Induction Metodi: Experimental Data Analysis, Kirchhoff’s Laws, Error Propagation Competenze: Measurement & Instrumentation, Experimental Data Analysis, Error Propagation Objects: Resistor Fonte: Testo (PDF) — p.8

In particolare, per $m=23,2\text{g}$ l’equazione (4) prevede un corrente

$I=32,4\text{mA}$

Propagando le incertezze di $k$ e $m_0$ di cui al paragrafo 7, si ottiene un’incertezza per: Questo corrente

$\Delta I=1,2\text{mA}.$

Poiché si dispone di due resistenze di 15 $\Omega$, si possono lavorare sperimentalmente con tre valori della resistenza di carico del generatore: $R_1=7,5\Omega$ se collegati in parallelo, $R_2=15\Omega$ se collegati in singolo resistenza e $R_3=30\Omega$ se collegati in serie. Il tabella III presenta le misure per la diminuzione del Potenzialmente $\Delta V$ in ogni caso e dell’intensità dedotta come $I=\Delta V/R$. Potrebbe anche essere misurato direttamente l’intensità utilizzando il un multimetro come un amperimetro, collegato in serie con resistenza e Il generatore.

L’accordo tra le previsioni del modello e risultati La Commissione ha adottato una decisione che prevede che il sistema di controllo dei dati margine di incertezza stimata.

$R$ ($\Omega$)	$\Delta V_{\text{max}}$ (V)	$\Delta V_{\text{min}}$ (V)	$\overline{\Delta V}$ (V)	$I$ (mA)
7,5	0,25	0,23	0,24	32
15	0,52	0,48	0,50	33
30	1,01	0,97	0,99	33

Tabella III

Topic: Circuits, Electromagnetic Induction Metodi: Experimental Data Analysis, Kirchhoff’s Laws, Error Propagation Competenze: Measurement & Instrumentation, Experimental Data Analysis, Error Propagation Objects: Resistor Fonte: Testo (PDF) — p.8

In particular, for $m=23,2\text{g}$ the equation (4) provides a current

$I=32,4\text{mA}$

The uncertainty of $k$ and $m_0$ obtained in paragraph 7 is spread to This current

$\Delta I=1,2\text{mA}.$

Since two resistors of 15 $\Omega$ are available, three values of the Generator load resistance: $R_1=7,5\Omega$ if connected in parallel, $R_2=15\Omega$ if connected in single resistance and $R_3=30\Omega$ if connected in series. Table III shows the measures of the fall in the potencial $\Delta V$ en cada caso y de la intensidad deducida como $I=\Delta V/R$. It could also be measured directly the intensity using the multi-meter as ampere, connected in series with resistance and the generator.

The agreement between the forecasts of the model and results The experimental is correct, within the the estimated margin of uncertainty.

$R$ ($\Omega$)	$\Delta V_{\text{max}}$ (V)	$\Delta V_{\text{min}}$ (V)	$\overline{\Delta V}$ (V)	$I$ (mA)
7,5	0,25	0,23	0,24	32
15	0,52	0,48	0,50	33
30	1,01	0,97	0,99	33

The following table shows the following:

Topic: Circuits, Electromagnetic Induction Metodi: Experimental Data Analysis, Kirchhoff’s Laws, Error Propagation Competenze: Measurement & Instrumentation, Experimental Data Analysis, Error Propagation Objects: Resistor Fonte: Testo (PDF) — p.8