Spagna 2008

Como aplicación del teorema del impulso y de la cantidad de movimiento, deduzca la expresión de la fuerza F $\Delta$ que el agua ejerce sobre el álabe en función de las velocidades u y v, del área A y de la densidad del agua $\rho$ .

Topic: Conservation of Momentum, Fluid Mechanics, Newtonian Mechanics Metodi: Impulse-Momentum Theorem, Conservation Laws, Free-Body Diagram Competenze: Physical Reasoning, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.2

Come applicazione del teorema dell’impulso e della quantità di movimento, dedurre l’espressione della forza F $\Delta$ che l’acqua esercita sull’alabe a seconda delle velocità u e v, dell’area A e della densità dell’acqua $\rho$ .

Topic: Conservation of Momentum, Fluid Mechanics, Newtonian Mechanics Metodi: Impulse-Momentum Theorem, Conservation Laws, Free-Body Diagram Competenze: Physical Reasoning, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.2

As an application of the theorem of momentum and quantity of motion, deduce the expression of the force F $\Delta$ The water is applied to the ground at speeds u and v, area A and area The water density $\rho$.

Topic: Conservation of Momentum, Fluid Mechanics, Newtonian Mechanics Metodi: Impulse-Momentum Theorem, Conservation Laws, Free-Body Diagram Competenze: Physical Reasoning, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.2

Determina el trabajo W $\Delta$ que realiza la fuerza F $\Delta$ sobre el álabe en el intervalo de tiempo t $\Delta$ .

Topic: Conservation of Energy, Newtonian Mechanics, Fluid Mechanics Metodi: Energy Conservation Method, Kinematic Equations, Physical Modeling Competenze: Physical Reasoning, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.2

Determina il lavoro W $\Delta$ che realizza la forza F $\Delta$ sull’alabe nell’intervallo di tempo t $\Delta$ .

Topic: Conservation of Energy, Newtonian Mechanics, Fluid Mechanics Metodi: Energy Conservation Method, Kinematic Equations, Physical Modeling Competenze: Physical Reasoning, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.2

Determine the work W $\Delta$ which performs the force F $\Delta$ on the alabe in the time interval t $\Delta$ .

Topic: Conservation of Energy, Newtonian Mechanics, Fluid Mechanics Metodi: Energy Conservation Method, Kinematic Equations, Physical Modeling Competenze: Physical Reasoning, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.2

Determina el rendimiento del proceso, definido como c E W $\Delta$ $\Delta$

/ $\eta$ , en donde c E $\Delta$ es la energía cinética de la masa m $\Delta$ contenida en el elemento de volumen x $A\Delta$ , que es la máxima energía que se puede transferir del agua al álabe.

Topic: Conservation of Energy, Fluid Mechanics, Newtonian Mechanics Metodi: Energy Conservation Method, Physical Modeling, Kinematic Equations Competenze: Physical Reasoning, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.2

Determina il rendimento del processo, definito come c E W $\Delta$ $\Delta$

/ $\eta$ , in cui c E $\Delta$ È l’energia. Cinetica della massa m $\Delta$ contenuto nell’elemento volume x $A\Delta$ , che è la massima energia che si può trasferire l’acqua dall’alabe all’alabe.

Topic: Conservation of Energy, Fluid Mechanics, Newtonian Mechanics Metodi: Energy Conservation Method, Physical Modeling, Kinematic Equations Competenze: Physical Reasoning, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.2

It determines the performance of the process, defined as c E W $\Delta$ $\Delta$

/ $\eta$ , where c E $\Delta$ It’s energy. Kinetics of the mass m $\Delta$ contained in the volume element x $A\Delta$ , which is the maximum energy that is It can transfer water from the water to the alabaster.

Topic: Conservation of Energy, Fluid Mechanics, Newtonian Mechanics Metodi: Energy Conservation Method, Physical Modeling, Kinematic Equations Competenze: Physical Reasoning, Mathematical Modeling Objects: — Fonte: Testo (PDF) — p.2

El rendimiento anterior depende de la velocidad u del álabe. Determina el valor 0 u de la velocidad del álabe para la cual dicho rendimiento es máximo. ¿Cuál es el valor del máximo rendimiento?

Topic: Conservation of Energy, Fluid Mechanics, Newtonian Mechanics Metodi: Calculus-Integration, Energy Conservation Method, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning Objects: — Fonte: Testo (PDF) — p.2

La resa precedente dipende dalla velocità o dall’alabe. Determina il valore 0 o della velocità del il quale è il massimo. Qual è il valore del massimo rendimento?

Topic: Conservation of Energy, Fluid Mechanics, Newtonian Mechanics Metodi: Calculus-Integration, Energy Conservation Method, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning Objects: — Fonte: Testo (PDF) — p.2

The previous performance depends on the speed or the algae. Determine the value 0 or the speed of the the maximum yield for which such performance is obtained. What’s the value of the maximum yield?

Topic: Conservation of Energy, Fluid Mechanics, Newtonian Mechanics Metodi: Calculus-Integration, Energy Conservation Method, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning Objects: — Fonte: Testo (PDF) — p.2

Si el radio de la noria es R, ¿con qué velocidad angular 0 $\omega$ debe girar para que el rendimiento sea máximo? La noria que se va a construir para la Expo2008, tiene un diámetro m R 5, 16 2

, la superficie efectiva de sus álabes planos es 2 5,0 m A = y se pretende elevar un caudal de agua /s m 10 5,1 3 2 $-$ $\times$

G .

Topic: Rotational Dynamics, Conservation of Energy, Fluid Mechanics Metodi: Kinematic Equations, Physical Modeling, Torque & Angular Momentum Analysis Competenze: Mathematical Modeling, Physical Reasoning Objects: Wheel Fonte: Testo (PDF) — p.2

Se il raggio della noria è R, a che velocità angolare 0 $\omega$ deve girare per rendere

• Il massimo? La noria che verrà costruita per l’Expo2008 ha un diametro m R 5, 16 2 = , la superficie effettiva di Le sue albi piane sono 2 5,0 m A = e si vuole aumentare un flusso d’acqua /s m 10 5,1 3 2 $-$ $\times$ = G .

Topic: Rotational Dynamics, Conservation of Energy, Fluid Mechanics Metodi: Kinematic Equations, Physical Modeling, Torque & Angular Momentum Analysis Competenze: Mathematical Modeling, Physical Reasoning Objects: Wheel Fonte: Testo (PDF) — p.2

If the radius of the northeast is R, what angular velocity is it? 0 $\omega$ must be rotated so that the output is What’s the maximum? The corridor that will be built for Expo2008 has a diameter m R 5, 16 2

, the effective surface area of Its flattened fins are 2 5,0 m A = And it’s meant to raise a water flow. /s m 10 5,1 3 2 $-$ $\times$

G .

Topic: Rotational Dynamics, Conservation of Energy, Fluid Mechanics Metodi: Kinematic Equations, Physical Modeling, Torque & Angular Momentum Analysis Competenze: Mathematical Modeling, Physical Reasoning Objects: Wheel Fonte: Testo (PDF) — p.2

Determina la expresión analítica de la velocidad media de la corriente de agua, v, que provoca el giro de la noria, en condiciones de rendimiento máximo, y calcula su valor. Toma como valor de la aceleración de la gravedad: 2 m/s 8,9

g .

x $\Delta$

A A u x $\Delta$

A A u v “antes del choque” “después del choque” Fig. 5 a Fig. 5 b XIX OLIMPIADA ESPAÑOLA DE FÍSICA. Solución 1o La variación de la componente horizontal del momento lineal de la masa m $\Delta$ contenida en el elemento de volumen x $A\Delta$ , durante el choque con el álabe es igual al impulso que éste ejerce sobre dicho elemento de volumen:

( ) v u m t F $-$ $\Delta$

$\Delta$ $\Delta$ ‘

Como x A m $\Delta$

$\Delta$ $\rho$ y, en virtud del principio de acción y reacción, la fuerza F $\Delta$ que el agua ejerce sobre el álabe es igual a ’ F $\Delta$ $-$ , se tiene

( ) u v x A t F $-$ $\Delta$

$\Delta$ $\Delta$ $\rho$

Además, como t v x $\Delta$

$\Delta$ se obtiene

( ) u v v A F $-$

$\Delta$ $\rho$

2o El trabajo W $\Delta$ que F $\Delta$ realiza sobre el álabe en el tiempo t $\Delta$ es

’x F W $\Delta$ $\Delta$

$\Delta$

Como t u x $\Delta$

$\Delta$ ’ es la distancia recorrida por el álabe en el tiempo que dura el choque, el trabajo viene dado por

( ) t u v v u A W $\Delta$ $-$

$\Delta$ $\rho$

3o Para el cálculo del rendimiento es necesario determinar la energía cinética de la masa m $\Delta$ . Esta energía es

3 2 2 2 1 2 1 2 1 v t A v t v A v m Ec $\Delta$

$\Delta$

$\Delta$

$\Delta$ $\rho$ $\rho$

Por lo que el rendimiento es

2 ) ( 2 v u v u $-$

$\eta$

4o El rendimiento será máximo para el valor u = u0 que haga

0

du $d\eta$ $\Rightarrow$ 0 2

$-$ o u v

Con lo que

v u 2 1 0 = y 5,0

$\eta$

5o La velocidad angular de la noria es R u /

$\omega$ . Si el rendimiento ha de ser el máximo, la velocidad angular deberá ser

R v 2 0 = $\omega$

6o La energía disponible es el trabajo realizado sobre el álabe en condiciones de rendimiento máximo, 2 / 0 v u

t v A W u u $\Delta$

$\Delta$

3 4 1 0 $\rho$

Para elevar una masa de agua M $\Delta$ a una altura R h 2

es necesario consumir una energía

g R M E 2 $\Delta$

$\Delta$

que podemos poner en función del caudal G recordando que t G M $\Delta$

$\Delta$ $\rho$ , con lo que XIX OLIMPIADA ESPAÑOLA DE FÍSICA.

t g R G g R M E $\Delta$

$\Delta$

$\Delta$ 2 2 $\rho$

Igualando la energía disponible con la consumida (balance energético) se tiene

t g R G t v A $\Delta$

$\Delta$ 2 4 1 3 $\rho$ $\rho$

Que nos permite obtener la velocidad de la corriente de agua que provoca el giro de la noria en rendimiento máximo en función del caudal de agua elevada

3 / 1 8 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛

A G g R v

Sustituyendo los valores dados en el enunciado, se obtiene el siguiente valor

km/h 9,6 m/s 7,2

= v

Topic: Fluid Mechanics, Conservation of Energy, Rotational Dynamics Metodi: Energy Conservation Method, Continuity Equation, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning Objects: Wheel Fonte: Testo (PDF) — p.2

Determina l’espressione analitica della velocità media del flusso d’acqua, v, che provoca il giro di il noria, in condizioni di massimo rendimento, e calcola il suo valore. Prendi come valore l’accelerazione di gravità: 2 m/s 8,9

g .

x $\Delta$

A A u x $\Delta$

A A u v Prima dello shock Dopo lo scontro Fig. 5 a Fig. 5 b XIX Olimpiada di Fisica spagnola. Soluzione 1o La variazione del componente orizzontale del momento lineare della massa m $\Delta$ contenuto nell’elemento di volume x $A\Delta$ , durante l’impatto con l’alabe è pari al impulso che esso esercita su tale elemento di volume:

( ) v u m t F $-$ $\Delta$

$\Delta$ $\Delta$ ‘

Come x A m $\Delta$

$\Delta$ $\rho$ e, in virtù del principio di azione e reazione, la forza F $\Delta$ che l’acqua esercita sull’alabe è uguale a ’ F $\Delta$ $-$ , si ha

( ) u v x A t F $-$ $\Delta$

$\Delta$ $\Delta$ $\rho$

Inoltre, come t v x $\Delta$

$\Delta$ si ottiene

( ) u v v A F $-$

$\Delta$ $\rho$

2o Il lavoro W $\Delta$ che F $\Delta$ realizza sull’alabe nel tempo t $\Delta$ es

’x F W $\Delta$ $\Delta$

$\Delta$

Come t u x $\Delta$

$\Delta$ ’ è la distanza percorsa dall’alabe nel tempo che dura lo scontro, il lavoro viene dato per

( ) t u v v u A W $\Delta$ $-$

$\Delta$ $\rho$

3o Per il calcolo della prestazione è necessario determinare l’energia cinetica della massa m $\Delta$ . Questa energia es

3 2 2 2 1 2 1 2 1 v t A v t v A v m Ec $\Delta$

$\Delta$

$\Delta$

$\Delta$ $\rho$ $\rho$

Per quanto riguarda il rendimento

2 ) ( 2 v u v u $-$

$\eta$

4o Il rendimento sarà massimo per il valore u = u0 che fa

0

du $d\eta$ $\Rightarrow$ 0 2

$-$ o u v

Con ciò che

v u 2 1 0 = y 5,0

$\eta$

5o La velocità angolare della noria è R u /

$\omega$ . Se il rendimento deve essere massimo, la velocità angolare deve essere

R v 2 0 = $\omega$

6o L’energia disponibile è il lavoro effettuato sull’alabe in condizioni di massima efficienza, 2 / 0 v u

t v A W u u $\Delta$

$\Delta$

3 4 1 0 $\rho$

Per sollevare una massa d’acqua M $\Delta$ ad un’altezza R h 2

è necessario consumare un’energia

g R M E 2 $\Delta$

$\Delta$

che possiamo mettere in funzione del flusso G ricordando che t G M $\Delta$

$\Delta$ $\rho$ , con cui XIX Olimpiada di Fisica spagnola.

t g R G g R M E $\Delta$

$\Delta$

$\Delta$ 2 2 $\rho$

La produzione di energia disponibile è pari a quella consumata (equilibrio energetico).

t g R G t v A $\Delta$

$\Delta$ 2 4 1 3 $\rho$ $\rho$

Che ci permette di ottenere la velocità del flusso d’acqua che provoca il giro della noria in rendimento massimo in funzione dell’elevato flusso d’acqua

3 / 1 8 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛

A G g R v

Sostituendo i valori forniti nella frase, si ottiene il seguente valore:

km/h 9,6 m/s 7,2

= v

Topic: Fluid Mechanics, Conservation of Energy, Rotational Dynamics Metodi: Energy Conservation Method, Continuity Equation, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning Objects: Wheel Fonte: Testo (PDF) — p.2

Determines the analytical expression of the mean velocity of the water current v, which causes the rotation of the the noria, in maximum yield conditions, and calculate its value. Take the value of the acceleration of gravity: 2 m/s 8,9

g .

x $\Delta$

A A u x $\Delta$

A A u v Before the shock after the shock Fig. 5 a Fig. 5 b 19th Spanish Olympics in Physics. Solution 1o The variation of the horizontal component of the linear moment of mass m $\Delta$ contained in the element of The volume x $A\Delta$ , during the impact with the algae is equal to the impulse it exert on the element of a width of not more than 15 mm

( ) v u m t F $-$ $\Delta$

$\Delta$ $\Delta$ ‘

As x A m $\Delta$

$\Delta$ $\rho$ and, under the principle of action and reaction, force F $\Delta$ which the water exerts over the alabe is equal to ’ F $\Delta$ $-$ , you got it

( ) u v x A t F $-$ $\Delta$

$\Delta$ $\Delta$ $\rho$

In addition, as t v x $\Delta$

$\Delta$ It’s getting

( ) u v v A F $-$

$\Delta$ $\rho$

2o The work W $\Delta$ which F $\Delta$ He does over the alabe in time t $\Delta$ es

’x F W $\Delta$ $\Delta$

$\Delta$

As t u x $\Delta$

$\Delta$ ’ It’s the distance travelled by the alabe in the time the impact lasts, the work comes given by

( ) t u v v u A W $\Delta$ $-$

$\Delta$ $\rho$

3o For the calculation of the yield, the kinetic energy of the mass must be determined. m $\Delta$ . This energy es

3 2 2 2 1 2 1 2 1 v t A v t v A v m Ec $\Delta$

$\Delta$

$\Delta$

$\Delta$ $\rho$ $\rho$

So the performance is

2 ) ( 2 v u v u $-$

$\eta$

4o The yield will be maximum for the value u = u0 that makes

0

du $d\eta$ $\Rightarrow$ 0 2

$-$ o u v

With what

v u 2 1 0 = y 5,0

$\eta$

5o The angular velocity of the noria is R u /

$\omega$ . If the output is to be maximum, the speed angular shall be

R v 2 0 = $\omega$

6o The available energy is the work carried out on the algae under maximum performance conditions, 2 / 0 v u

t v A W u u $\Delta$

$\Delta$

3 4 1 0 $\rho$

To raise a body of water M $\Delta$ At a height R h 2

It is necessary to consume energy

g R M E 2 $\Delta$

$\Delta$

We can put that in terms of the flow rate G remembering that t G M $\Delta$

$\Delta$ $\rho$ , with which 19th Spanish Olympics in Physics.

t g R G g R M E $\Delta$

$\Delta$

$\Delta$ 2 2 $\rho$

The energy balance is equal to the energy available and consumed.

t g R G t v A $\Delta$

$\Delta$ 2 4 1 3 $\rho$ $\rho$

Which allows us to get the velocity of the water current that causes the spinning of the noria in Maximum yield by high water flow

3 / 1 8 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛

A G g R v

Substituting the values given in the statement, the following value is obtained

km/h 9,6 m/s 7,2

= v

Topic: Fluid Mechanics, Conservation of Energy, Rotational Dynamics Metodi: Energy Conservation Method, Continuity Equation, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning Objects: Wheel Fonte: Testo (PDF) — p.2