Elements of classical and relativistic mechanics, I
1) A particle moves x1, x2 and its position vector satifies
−
→r = b cos ω−→e1+ b sin ωt−→e2, ω = const b = const.
Show that
a) the velocity −→v is perpendicular to −→r ,
b) the accelaration −→a is directed toward the origin; find its magnitude, c) −→
L = −→r × m−→v is constant vector; find its magnitude.
2) Show that the angular momentum −→
L = −→r × −→p in the central force motion is constant.
3) Consider a particle moving in the plane with velocity −→v and accelera- tion −→a . Describe −→v i −→a via radial coordinates, i.e. using the orthogonal vectors:
−
→er = sin θ−→e1+ cos θ−→e2,
−
→eθ = cos θ−→e1− sin θ−→e2.
4) Using the result of the previous problem we arrive at the following formula for the acceleration
−
→a = −→er d2r
dt2 − ω2r
+ −→eθ
αr + 2ωdr dt
.
Explain the meaning of all terms in the above equation.
5) Decompose velocity and acceleration onto normal and tangent compo- nent.
1
