1. The center of gravity of a system of particles is the point about which the external gravitational forces exert no net torque. For a uniform gravitational force, show that the center of gravity is identical to the center of mass for the system of particles.

2.
A projectile is fired at an angle of 45° with initial kinetic
energy *E*_{0}. At the top of its trajectory, the
projectile explodes into two fragments (an additional energy *E*_{0} is released during the explosion). One fragment of mass *m*_{1} travels straight down. What is the velocity (specify both
magnitude and direction) of the second fragment of mass *m*_{2}, and the velocity of the first?

3.
A fixed force center scatters a particle of mass *m* according to the force law *F*(*r*) = *k*/*r*^{3}. If the initial velocity of the particle
is *u*_{0},
show that the differential scattering cross section is

4.
The most energetic *a*-particles
available to Ernest Rutherford and his colleagues for the famous Rutherford
scattering experiment were 7.7 MeV. For the scattering of 7.7 MeV *a*-particles
from ^{238}*U* (initially at rest)
at a laboratory scattering angle of 90°, find the following:

a.
The recoil scattering angle of ^{238}*U* in the laboratory frame.

b.
The scattering angles of the *a*-particle
and ^{238}*U* in the
center-of-mass frame.

c.
The laboratory kinetic energy of the scattered *a*-particle and ^{238}*U*.

d.
The impact parameter *b*.

e.
The distance of closest approach *r*_{min}.

f. The differential cross section at a laboratory scattering angle of 90°.

g. The ratio of the probabilities of scattering at 90° to that of 5° in the laboratory frame.

5.
A rocket in outer space, in a negligible gravitational field,
starts from rest and accelerates uniformly with an acceleration *a* until it reaches its final speed *v*. The
initial mass of the rocket is *m*_{0}. How much work does the rocket's engine
do?

Practice the material and concepts discussed in Chapter 9.