Download Sol.

Associate Professor: C. H.LIAO
 2.1
 2.2
 2.3
 2.4
 2.5
 2.6
 2.7
Introduction 48
Newton's Laws 49
Frames of Reference 53
The Equation of Motion for a Particle 55
Conservation Theorems 76
Energy 82
Limitations of Newtonian Mechanics 88

The science of mechanics need certain fundamental
concepts such as distance and time. The combination of the
concepts of distance and time allows us to define the
velocity and acceleration of a particle.

The third fundamental concept, mass, requires some
elaboration, which we give when we discuss Newton's laws.

Physical laws must be based on experimental fact.

We cannot expect a priori that the gravitational attraction

between two bodies must vary exactly as the inverse square
of the distance between them. But experiment indicates
that this is so.
The postulate then assumes the status of a physical law. If
some experiments disagree with the predictions of the law,
the theory must be modified to be consistent with the facts.

We begin by simply stating in conventional
form Newton's laws of mechanics :
I. A body remains at rest or in uniform
motion unless acted upon by a force.
 II. A body acted upon by a force moves in
such a manner that the time rate of change of
momentum equals the force.
 III. If two bodies exert forces on each other,
these forces are equal in magnitude and
opposite in direction.



The First Law, for example, is meaningless
without the concept of "force," a word Newton
used in all three laws.
In fact, standing alone, the First Law conveys
a precise meaning only for zero force; that is,
a body remaining at rest or in uniform. A
body moving in this manner is termed a free
body (or free particle).

The Second Law provides an explicit statement:
Force is related to the time rate of change of
momentum.
To demonstrate the significance of Newton‘s Third Law, let us
paraphrase it in the following way, which incorporates the
appropriate definition of mass:

III‘. If two bodies constitute an ideal, isolated system, then the
accelerations of these bodies are always in opposite directions,
and the ratio of the magnitudes of the accelerations is constant.
This constant ratio is the inverse ratio of the masses of the
bodies.
the more general conservation of linear
momentum.
A reference frame is called an inertial frame if
Newton's laws are indeed valid in that frame.
 That is, if a body subject to no external force
moves in a straight line with constant velocity
(or remains at rest), then the coordinate
system establishing this fact is an inertial
reference frame.
 If Newton's laws are valid in one reference
frame, then they are also valid in any
reference frame in uniform motion (i.e., not
accelerated) with respect to the first system.






1. Make a sketch of the problem, indicating forces,
velocities, and so forth.
2. Write down the given quantities.
3. Write down useful equations and what is to be
determined.
4. Strategy and the principles of physics must be
used to manipulate the equations to find the quantity
sought. Algebraic manipulations as well as
differentiation or integration is usually required.
Sometimes numerical calculations using a computer
are the easiest, if not the only, method of solution.
5. Finally, put in the actual values for the assumed
variable names to determine the quantity sought.
If a block slides without friction down a fixed, inclined plane
with θ = 30°, what is the block's acceleration?
Sol: The total force Fnet is constant:
If the coefficient of static friction between the block and plane in the previous
example is μs = 0.4, at what angle θ will the block start sliding if it is initially at
rest?
Sol:
Mter the block in the previous example begins to slide, the
coefficient of kinetic (sliding) friction becomes μk = 0.3. Find the
acceleration for the angle θ = 30°.
Sol:
where Cw is the dimensionless drag
coefficient, ρ is the air density, v is the
velocity, and A is the cross-sectional
area of the object (projectile) measured
perpendicularly to the velocity.
As the simplest example of the resisted motion of a particle,
find the displacement and velocity of horizontal motion in a
medium in which the retarding force is proportional to the
velocity.
Find the displacement and velocity of a
particle undergoing vertical motion in a
medium having a retarding force
proportional to the velocity.
Sol.:
Sol.:
When y = O
For t = T
if θ= 55 ° we have Vo = 1450 m/s and θ= 5
the range become
Big Bertha's actual range was 120 km.
The difference is a result of the real
effect of air resistance.
The maximum predicted height is:
The range R is
Next, we add the effect of air resistance to the motion of the projectile
in the
previous example. Calculate the decrease in range under the
assumption that
the force caused by air resistance is directly proportional to the
projectile's
velocity.
Sol.:
keep only terms in the expansion through k3,
p.s. : Read the force every 50 m/ s for Figure 2-3c and every 100 m/ s
Sol.:
Atwood's machine consists of a smooth pulley with two masses
suspended from
a light string at each end (Figure 2-11). Find the acceleration of the
masses and
the tension of the string (a) when the pulley center is at rest and (b)
when the
pulley is descending in an elevator with constant acceleration α
Sol.:
The equations of motion
(a) when the pulley center is
at rest.
(b) when the pulley is
descending in an elevator with
constant acceleration α.
Sol.:
∵
If the particle is free, that is, if the particle encounters no force
I. The total linear momentum p of a particle is conserved when the
total force on it is
zero.
II. The angular momentum of a particle subject to no torque is
conserved.
III. The total energy E of a particle in a conservative force field is a
constant in time.
the component of linear
momentum in a
direction in which the
force vanishes is
constant in time.
The total energy of a particle to be the sum of the kinetic and
potential energies
Sol.:
The angular momentum
The velocity of the outside edge
equilibrium point
P.S.: At x=x0 is an equilibrium point, Equilibrium may be stable,
unstable, or neutral.
∵ U0 at x = 0 can define to be
zero
Sol.:






We have implied that these are all measurable quantities and that
they can be specified with any desired accuracy, depending only
on the degree of sophistication of our measuring instruments.
When we attempt to make precise measurements on microscopic
objects, however, we find a fundamental limitation in the
accuracy of the results.
This momentum is uncertain by an amount ap. The product
△ x △ p is a measure of the precision with which we can
simultaneously determine the electron‘s position and momentum;
△x → 0, △ p → 0 implies a measurement with all imaginable
precision.
for if △x → 0, then we must have △ p → 0 for Heisenberg's
uncertainty principle to be satisfied.
Newtonian mechanics is therefore subject to fundamental
limitations when small distances or high velocities are
encountered.
Thanks for your
attention.
Problem:
 2-3, 2-7, 2-12, 2-17, 2-19, 2-23, 2-28, 229, 2-36, 2-41, 2-42, 2-49, 2-54