[SOLVED] Physics 396 homework set 2

1. Consider a free particle (a particle with no net force acting on it) examined in an inertial
reference frame with coordinates (x, y, z). A free particle obeys the equations of motion
d2x
dt2 = d2y
dt2 = d2z
dt2 = 0, (1)
as free particles dont accelerate relative to inertial frames.This particular free particle
follows a trajectory described by
x(t) = x0 + vxt
y(t) = y0
z(t) = z0 + vzt, (2)
where x0, y0, z0, vx, vz are constants in time.a) Explicitly show that this free particle, with a trajectory described by Eq. (2), obeys
the three equations of motion given in Eq. (1).
Consider the non-inertial reference frame (x0
, y0
, z0
), which rotates with respect to the
inertial frame with a constant angular velocity ! about the common z = z0 axis.The
coordinate transformation from the inertial to non-inertial reference frame is of the form
x0
(t) = cos(!t)x + sin(!t)y
y0
(t) = sin(!t)x + cos(!t)y
z0
(t) = z, (3)
b) What are the equations of motion describing x0
(t), y0
(t), and z0
(t) in the rotating
frame?2. Consider an inertial reference frame with coordinates (x, y, z) and a non-inertial
reference frame with coordinates (x0
, y0
, z0
). At t = t
0 = 0, the two coordinate systems
align and are momentarily at rest relative to one another.The non-inertial reference
frame is uniformly accelerating with respect to the inertial frame at a rate ~a = g|.a) Construct the coordinate transformations linking these two frames of reference.
Now consider a free particle that follows a trajectory described by
x = x0 + vxt
y = y0
z = z0 + vzt, (4)
where x0, y0, z0, vx, vz are constants in time.b) What are the equations of motion describing x0
(t), y0
(t), and z0
(t) in the uniformly
accelerating frame?3. Consider an object of inertial mass, mI , and gravitational mass, mG, in the earths uniform
gravitational field, where the force of gravity acting on the object is given by
F~ = mGg| (5)a) For a freely falling object, under the influence of gravity alone, construct the equations
of motion analogous to Eq. (1) for a non-zero net force. For the moment, assume
that mI 6= mG.
b) Now setting mI = mG, show that these equations reduce to the form given above in
Prob. 2b). In one sentence, what does this imply?4. Consider a spherically-symmetric massive object of radius R, mass M with mass density
(r) = rn for r < R, (6)
where n is an arbitrary power.a) Find an expression for in terms of M, R, and n.
b) What values can n have if one demands that the total mass of the object is finite?c) Using Gauss law in integral form for Newtonian gravitation, calculate the gravitational
field vector both inside and outside the massive object. Write this expression in
terms of G, M, n and the radial distance r.d) Calculate the gravitational potential of the massive object both inside and outside
the massive object.