Assignment:
Question 1. Consider an isolated system comprising two particles of masses m1 and m2. Whose position vectors, in an inertial frame, are x1 and x2 and whose velocity vectors are v1 and v2. The interaction of the particles may be described by an energy function.
i) Which aspect of Newtonian relativity requires U to depend only on the separation vector x1 – x2?
ii) Which further aspect of Newtonian relativity requires U to depend only on the magnitude r = | x1 – x2| of the separation vector?
iii) Suppose that U = -k/r2, where k is a positive constant. Use energy conservation to determine the forces acting between the particles?
iv) Answer the following giving a brief explanation in each case.
a) Is the force law in ii) attractive or repulsive?
b) Does it satisfy Newton’s 3rd law?
c) Is it an inverse square law?
d) Does it conserve linear momentum?
e) Does it conserve angular momentum?
v) Suppose that we choose an inertial frame S in which the particles’ positions and velocities are related by,
at time t = 0. Show that these relationships persist at all subsequent times?
vi) Suppose that observer O’ uses a frame S’ with axes parallel to those of S and with origin moving at constant velocity with respect to that of S.
a) May O’ use the same energy function to predict the motion of S’?
b) Will O’ assign the same value to the kinetic energy as does an observer O in S?
c) Explain, briefly, why your answer to (a) and (b) are consistent.
vii) Suppose that an observer O’’ uses a frame S’’ with one axis coinciding with an axis of S and the other two axes rotating at constant angular speed with respect to those in S.
a) May O’’ use the same energy function to predict the motion in S’’
b) Will O’’ assign the same value to the potential energy as does an observer O in S?
c) Explain, briefly, why your answers to (a) and (b) are consistent?