- Derive Equation (2.12).
- Consider a system consisting of
point particles. Let
be the position vector of the
th particle, and
let
be the external force acting on this particle. Any internal forces are
assumed to be central in nature. The resultant force and torque (about the origin) acting on the system are
and

respectively. A*point of action*of the resultant force is defined as a point whose position vector satisfies*line of action*of the resultant force. - Consider an isolated system consisting of two extended bodies (which can, of course, be modeled as collections of point
particles),
and
. Let
be
the resultant force acting on
due to
, and let
be the resultant force acting on
due to
. Demonstrate that
, and that both forces have the same
line of action.
- An extended body is acted upon by two resultant forces,
and
.
Show that these forces can be only replaced by a single equivalent force,
,
provided:
- and are parallel (or antiparallel). In this case, the line of action of is parallel to those of and .
- and are not parallel (or antiparallel), but their lines of action cross at a point. In this case, the line of action of passes through the crossing point.

- Deduce that if an isolated system consists of three extended bodies,
,
, and
, where
is
the resultant force acting on
(due to
and
),
is the resultant force acting on
, and
is the resultant force acting on
, then
and the
forces either all have parallel lines of action or have lines of action that cross at a common point.
- A particle of mass
moves in one dimension and has an instantaneous displacement
. The particle is
released at rest from
,
subject to the force
, where
,
. Demonstrate that the time needed
for the particle to reach
is
- A particle of mass
moves in one dimension and has an instantaneous displacement
.
The particle is released at rest from
, subject to the force
, where
. Show that
the particle will reach the origin with a speed
after a time
has elapsed.
(Modified from Smart 1951.)
- A particle moves in one dimension and has an instantaneous displacement
.
The particle is released at rest from
and accelerates such that
, where
, and
. Show that
the particle will reach the origin after a time
has elapsed, and that its speed is then infinite. (Modified from Smart 1951.)
- A particle of mass
, moving in one dimension with an initial (i.e., at
) velocity
, is subject to a force
- A particle of mass
moving in one dimension is subject to a force
- A particle moving in one dimension with simple harmonic motion has speeds
and
at displacements
and
, respectively, from its mean position. Show that the period of the motion is
- The potential energy for the force between two atoms in a diatomic molecule has the approximate
form
- Assuming that one of the atoms is relatively heavy and remains at rest, while the other, whose mass is , moves in a straight line, find the equilibrium distance and the period of small oscillations about the equilibrium position.
- Assuming that both atoms have the same mass , and move in a straight line, find the equilibrium distance and the period of small oscillations about the equilibrium position.

- Two light springs have spring constants
and
, respectively, and are used in a vertical
orientation to support an object of mass
. Show that the angular frequency of oscillation
is
if the springs are connected in parallel, and
if the springs are connected in series.
- A body of uniform cross-sectional area
and mass density
floats in a liquid
of density
(where
), and at equilibrium displaces a volume
. Show
that the period of small oscillations about the equilibrium position is
- A particle of mass
executes one-dimensional simple harmonic oscillation under the action of a
conservative force such that its instantaneous displacement is
- Using the notation of Section 2.9, show that
the total momentum and angular momentum of a two-body system take the
form
and

respectively, where , and .- If the force acting between the bodies is conservative, such that
, demonstrate that the
total energy of the system is written
- If the force acting between the particles is central, so that , demonstrate, from the equation of motion, , that is constant in time.

- If the force acting between the bodies is conservative, such that
, demonstrate that the
total energy of the system is written