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Pokaza´c, ˙ze t =r m 2 Z x x0 dx0 pE − U(x0), gdzie m , E , U

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Fizyka Teoretyczna VI - Theoretical Physics VI

Zagadnienie Keplera; (literatura; np. Landau, Lifshitz: Mechanika, J. Tay- lor: Mechanika klasyczna)

1) Rowa˙zmy dwa cia la o masach m1i m2(−→r1, −→r2oznaczaj¸a wektory po lo˙zenia) oddzia lywuj¸ace na siebie wzajemnymi si lami−→

F12−→

F21 = −F12. Wprowad´zmy oznaczenia

→R = m1−→r1+ m2−→r2

m1+ m2 , −→r = −→r1− −→r2 oraz masy ca lkowitej i masy zredukowanej:

M = m1+ m2, µ = m1m2 m1 + m2.

Pokaza´c, ˙ze ca lkowita energia kinetyczna uk ladu mo˙ze by´c wyra˙zona:

Ekin= 1

2M ˙R2+1 2µ ˙r2.

Jak wygl¸ada wyra˙zenie ca lkowit¸a energi¸e w uk ladzie ´srodka masy, gdy energia potencjalna wyra˙za si¸e wzorem

U (−→r1, −→r2) = − α

|−→r1− −→r2|, α > 0 .

2) Niech t oznacza czas jaki cia lo porzebuje, aby przemie´sci´c si¸e w (ruchu jednowymiarowym) z punktu x0 do x. Pokaza´c, ˙ze

t =r m 2

Z x x0

dx0 pE − U(x0),

gdzie m , E , U (·) oznaczaj¸a odpowiednio masa cia la, energi¸e ca lkowit¸a, en- ergi¸e potencjaln¸a.

3) a) Rowa˙zmy cia lo poruszaj¸ace si¸e w polu si l centralnych. Pokaza´c, ˙ze moment p¸edu we wsp´o lrz¸ednych biegunowych (r, φ) ma posta´c

M = mr2φ = const .˙ 1

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b) Niech f oznacza pr¸edko´s´c polow¸a. Pokaza´c M = 2m ˙f . Z powy˙zych wynik´ow wywnioskuj drugie prawo Keplera.

4) Rozwa˙zmy ruch cia la masie m w polu si l centralnych; niech U (r) oznacza potencja l, M warto´s´c momentu p¸edu, E ca lkowit¸a energi¸e cia la. Pokaza´c ˙ze wsp´o lrz¸edna k¸atowa toru ruchu cia la okre´slona jest wzorem:

φ =

Z M

r2dr q

2m(E − U (r)) − Mr22

+ const (0.1)

5) Niech U (r) = −αr. Pokaza´c, przez sca lkowanie (0.3), ˙ze

φ = arccos

M rM q

2mE +mM2α22

+ const (0.2)

6) a) Pokaza´c, analizuj¸ac wz´or (0.4), ˙ze gdy E < 0, to tor ruchu jest elips¸a.

Opisa´c jej parametry: mimo´sr´od, wielk¸a i ma l¸a o´s, etc.

b)Pokaza´c, ˙ze gdy E > 0, to tor ruchu jest hiperbol¸a; opisa´c jej parametry.

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1) Consider two particles with masses m1 i m2 (−→r1, −→r2 denote position vectors) mutually interacting with the forces −→

F12

→F21 = −−→

F12. Introduce notations

→R = m1−→r1+ m2−→r2

m1+ m2 , −→r = −→r1− −→r2

moreover the total and reduced masses

M = m1+ m2, µ = m1m2 m1 + m2

. Show that the total kinetic energy can be expressed as:

Ekin= 1

2M ˙R2+1 2µ ˙r2.

Write the expression for the total energy in the center mass system. if the potential energy is given by

U (−→r1, −→r2) = − α

|−→r1− −→r2|, α > 0 .

2) a) Consider a particle in the central field. Show that the magnitude of the angular momentum in the polar coordinates (r, φ) equals

M = mr2φ = const .˙

b) Let f denote the sectorial velocity. Show that M = 2m ˙f . Combine the above results to obtain the second Kepler’s law.

3) Let us consider a particle of mass m moving in central field; let U (r) stand for the potential, M the magnitude of angular momentum, E the total energy. Show that the shape of path is given by:

φ =

Z M

r2dr q

2m(E − U (r)) − Mr22

+ const (0.3)

4) Let U (r) = −αr. Show by integration of (0.3) that φ = arccos

M rM q

2mE +mM2α22

+ const (0.4)

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5) a) Using formula (0.4), show that if E < 0 then the path is given by ellipse. Describe its parameters: eccentricity, major and minor semi axis.

b)Show that if E > 0 then then the path is given by hyperbola; describe its parameters.

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