College Physics (II)

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College Physics (II)

• Qingxu Li
• Tel [phone], Email liqx_at_cqupt.edu.cn
• Room 306, College of Mathematics and Physics

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The most incomprehensible thing about the
universe is that it is comprehensible.


-Albert Einstein
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About the Course

• College Physics (II)
• Textbook General Physics, Bin Liang, et al.
• Contents Mechanics, Oscillation and Wave,
  Optics Electromagnetism, Relativity, Quantum
  Physics, etc.
• Course grade Final Exam (70) Performance
  (30)
• Exercises and Exam are to be finished in English

(Chapter 2-7, 10)
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Reference books
a. Principle of Physics, 3rd edition, Serway
and Jewett b. Feynman Lectures on Physics
(Volume I), by R. P. Feynman c.
???,???,???????,???
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A Brief Summary of Chapter 1
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Units, Dimension, Significant Figures, Order of
Magnitude, Vector
(??,??,????,???,??)
a. Units are indispensable for physical
quantities. b. Vectors are to be distinguished
from scalars.
c. Properties of Vectors
magnitude, direction, components, equality,
addition, dot product, cross product, etc.
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Position and Displacement Vectors
(?????????)
path ??,?? locus ?? distance ??
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Average Velocity and Instantaneous Velocity
(?????????)
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Fig 1.1 A particle moving in the xy plane
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Alternative Expressions
(????)
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Acceleration
(???)
The average acceleration of a particle over a
time interval is defined as
And the instantaneous acceleration is defined as
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Alternative Expressions
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Fig 1.2 The Velocity-Time diagram. The magnitude
of acceleration vector is the slope of the curve
vt.
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Problems Related to Kinematics
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• Mechanics
• Kinematics
• Dynamics

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Part II Dynamics
The Laws of Motion
(????)
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Nature and natures laws lay hid in night.
God said Let Newton be! and all was
light. --Alexander Pope
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The Concept of Force (????)
The force is a vector quantity.
The unit of force is newton, which is defined
as the force that, when acting on a 1-kg mass,
produces an acceleration of 1m/s2.
The dimension of force is
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Newtons First Law (??????)
Newtons first law of motion
In the absence of external forces, an object at
rest remains at rest and an object in motion
continues in motion with a constant velocity
(that is, with a constant speed in a
straight line)
(?????????,??????????,???????? ???????,????????
????????)
In simpler terms, when no force acts on a
body, its acceleration is zero.
(????,????????,????????)
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Comments on the First Law

• The first law tell us that an object has a
  tendency to maintain its
• original state of motion in the absence of the
  force. This tendency is
• called inertia, and the first law sometimes
  called the law of inertia.

(?????????????????????????????????? ??)-????,?
???????????????)
2. Newtons first law defines a special set of
reference frames called inertial frames. An
inertial frame of reference is one in which
the first law is valid.
(????????????????????-????????,???? ??)
3. Inertial mass is the measure of an objects
resistance to change in motion in response to an
external force. Inertial mass is different in
definition from gravitational mass, but they
have the same value, so we call them both simply
mass.
(?????????????????????????????????? ?????????
,??????????,?????)
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Mass and Weight (?????)
Mass and weight are two different quantities,
and should not be confused with each other.
The magnitude of an object is equal to the
magnitude of the gravitational force exerted by
the planet on which the objects resides. While
the mass of an object is the same everywhere.
A given object exhibits a fixed amount of
resistance to changes in motion regardless of its
location. E.g. A person of mass 60 kg on
Earth also has a mass of 60 kg on the moon. The
same person weighs 588 Newton on Earth,
but weighs 98 Newton on the moon.
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Newtons Second Law (??????)
Newtons second law of motion
The acceleration of an object is directly
proportional to the net force acting on it and
inversely proportional to its mass.
(?????????????????????????)
net force
(??)
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Net Force
The net force is also known as

1. the resultant force
2. the sum of the force
3. the total force
4. the unbalanced force

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The Mathematical Form of the Second Law
(?????????)
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Comments on the Second Law

• The Newtons second law is the central rule of
  classical mechanics,
• which bridges dynamics and kinematics and tells
  that force is the
• cause of the change of motion (not motion!).

2. The second law has an alternative expressions
In special relativity, the mass of an object will
vary with its velocity and thus vary with time.
The previous form is invalidated in this case but
the new form still holds. Of course, both form
are equivalent for non-relativistic cases.
3. The second law can also be expressed as
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Newtons Third Law (??????)
Newtons third law of motion
If two objects interact, the force exerted by
object 1 on object 2 is equal in magnitude but
opposite in direction to the force exerted by
object 2 on object 1.
(????????(??)????),??? 1 ???? ? 2 ?????? 2
?????1???????,????)
Forces always occurs in pairs, i.e., that a
single isolated force cannot exist.
(???????,????,????????????)
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Comments on the Third Law
The force that object 1 exerts on object 2 may
be called action force and the force of object 2
on object 1 the reaction force. The action force
is equal in magnitude to the reaction force
and opposite in direction. In all cases, the
action and the reaction forces act on different
objects and must be of the same type.
(?? 1 ????? 2 ??????????,????????
2 ?????1?????????????????????????? ????????????,??
?????????)
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Applications of Newtons Law
(???????)
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The Particle in Equilibrium
(?????????)
Objects that are either at rest or moving with
constant velocity are said to be in equilibrium.
From Newtons second law, this condition of
equilibrium can be expressed as
(???????????????????????????????,?? ??????????????
)
or
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The Accelerating Particle (????)
When a nonzero net force is acting on a
particle, the particle is accelerating, and the
second law tell us
(??????????????,???????,???????)
In practice, the above equation is broken into
components, so that two or three equations can be
handled independently.
(???????????????????,???????????? ????)
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The Atwood Machine (?????)
E.g. 1.1 When two objects with unequal masses are
hung vertically over a light, frictionless
pulley as in the figure, the arrangement
is called an Atwood machine. The device is
sometimes used in the lab to measure the
free-fall acceleration. Calculate the magnitude
of the acceleration of the two objects and the
tension in the string.
Fig 1.10 The Atwood machine.
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Forces of Friction
(???)
When an object moves either on a surface or
through a viscous medium such as air or water,
there is resistance to the motion. We call such
resistance a force of friction.
force of static friction force of kinetic friction
(????)
Force of friction
(????)
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Simplified model for force of friction

• The magnitude of the force of static friction
  between any two
• surfaces in contact can have the values

µs the coefficient of static friction n the
magnitude of normal force
2. The magnitude of the force of kinetic friction
acting between two surfaces is
µk the coefficient of kinetic friction
3. The values ofµk andµs depend on the nature of
the surfaces, but the former is generally
less than the latter.
4. The direction of the friction force on an
object is opposite to the actual motion or
the impending motion of the object relative to
the surface with which it is in contact.
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Fig 1.5 A graph of the magnitude of the friction
force versus that of the applied force.
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The Gravitational Force Newtons Law of
Universal Gravitation
??????????
Newtons Law of Universal Gravitation
Every particle in the Universe attracts
every other particle with a force that is
directly proportional to the product of the
masses of the particles and inversely
proportional to the square of the
distance between them.
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The electrostatic force Coulombs Law
The magnitude of the electrostatic force
between two charged particle separated by a
distance r is
The Coulomb constant
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The Fundamental Forces of Nature
(??????)

1. The Gravitational Force
2. The Electromagnetic Force
3. The Strong Force (The Nuclear force)
4. The Weak Force

(??)
(???)
(??)
(??)
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Newtons Second Law Applied to a Particle in
Uniform Circular Motion
A particle moving in a circular path with
uniform speed experiences a centripetal
acceleration of magnitude
The acceleration vector is directed toward the
centre of the circle and is always perpendicular
to its velocity.
Apply Newtons second law to the particle along
the radial direction
centripetal force (???)
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Non-uniform Circular Motion
For non-uniform circular motion, there is, in
addition to the radial component of acceleration,
a tangential component, that is
The total force exerted on the particle is
The first term in the RHS is directed toward
the center of the circle and is responsible for
the centripetal acceleration and the second
term is tangent to the circle and responsible for
the tangential acceleration, which causes the
speed of the particle to change with time.
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Energy of a System (???????)
kinetic energy potential energy
Energy
(??)
(??)
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Work(?)
The work done by a force on a system is defined
as
(??????????????????)
For a finite displacement,
(????????)
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Work Done by a Constant Force (????)
For a constant force, the work reads
If the applied force is parallel to the direction
of the displacement,
And if the force is perpendicular to the
displacement, then
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From the definition of dot product, we get
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Work done by a Spring
Hookes Law
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The work done by the restoring force on a block
connected with a spring reads
restoring force ???
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Kinetic Energy (??)
The work done on a system in motion
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Define the kinetic energy of a particle is
From the above definition, we get
Work-kinetic energy theorem
(????)
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Work-kinetic energy theorem
When work is done on a system and the only
change in the system is in its speed, the work
done by the net force equals the change in
kinetic energy of the system.
(???????????????????????????,? ???????????????)
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E.g. 1.8 A 6.00 kg block initially at rest is
pulled to the right along a horizontal friction
less surface by a constant, horizontal force
of 12.0 N, as shown in the figure. Find the speed
of the block after it has moved 3.00m.
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Potential Energy
(??)
Kinetic Energy related to the motion of an
object Internal Energy related to the
temperature of a system Potential Energy related
to the configuration of a system
Example gravitational potential energy,
configuration ??,??
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Conservative Forces
(???)
The work done by a conservative force does not
depend on the path followed by the members of the
system, and depends on the initial and final
configurations of the system.
In other words, the work done by a
conservative on an object does not depend on the
path of the object, but depends on its initial
and final position.
From above definition, it immediately follows
that the work done by a conservative force when
an object is moved through a closed path is equal
to zero.
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Conservative Forces and Potential Energy
Conservative force
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Define potential energy function as
From above definition, we can get
and
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Gravitational Force
Consider a particle of mass m above the Earths
surface
The gravitational force on the particle due to
the Earth reads
The work done by the gravitational force
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Assuming
And we get
In summary, the gravitational potential
energy for any pair of particles varies as 1/r.
Furthermore, the potential energy is
negative because the force is attractive and we
have chosen the potential energy to be zero when
the particle separation is infinity.
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E.g. A particle of mass m is displaced through a
small vertical distance ?y near the Earths
surface. Show that expression for the change
in gravitational potential energy reduces to the
familiar relationship ?Ugmg ?y.
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Electrostatic Force (???)
The electrostatic force between two charged
particles reads
In a way similar to gravitational force case,
we can get electric potential energy
function(???)
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The Force of a Spring
According to Hooks law, a block connected to
a spring experiences a force
Therefore, the potential energy stored in a
block-spring system is
If the initial position of the block is xi0,
Ui is always chosen as zero, Then,
elastic potential energy
(????)
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????
???
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Mechanical Energy (???)
The sum of kinetic and potential energy is
defined as mechanical energy
(???????????????????)
If in an isolated system there are only
conservative forces which do work, the mechanical
energy will keep unchanged, as is called
conservation of mechanical energy.
(??????????????,???????????,??? ??????)
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The conservation of mechanical energy in a
system can be expressed as
The conservation of energy in an isolated
system can be expressed as
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Stability of Equilibrium
(??????)
Energy diagram An energy diagram shows the
potential energy of the system as a function of
the position of one of members of the system.
Stable equilibrium When the system locates
such a position that any movement away from this
position results in a force directed back toward
the position. (this type of force is called
restoring force.)
In general, positions of stable equilibrium
correspond to those positions for which the
potential energy function has a relative minimum
value on an energy diagram. And positions of
unstable equilibrium correspond to those
positions for which the potential energy has a
relative maximum value.
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Energy Transfer
(????)
system and its environment
(?????)
isolated systems Vs non-isolated systems
(???? Vs ?????)
Work is one means of energy transfer between
the system and its environment. If positive work
is done on the system, energy is transferred from
the system to the environment, whereas negative
work indicates that energy is transferred from
the system to the environment.
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Heat and Thermal Conduction
(?????)
Except for work, energy can also be
transferred through thermal conduction.
The work done on a system may also increase
its internal energy, in addition to change its
kinetic energy. The internal energy of an object
is associated with its temperature. And its well
known that heat can be transferred from a warmer
object to another object. The energy transfer
caused by a temperature difference between
two regions in space is called thermal conduction.
internal energy ?? heat ?,?? thermal conduction
???
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Mechanical Wave
(???)
Mechanical wave are a means of transferring
energy by allowing a disturbance to propagate
through into air or another medium. This is the
method by which energy leaves a radio through the
loudspeaker-sound-and by which energy enters your
ears to stimulate the hearing process.
disturbance ?? propagate ?? medium ??,??
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Principle of Conservation of Energy
(??????)
We can neither create nor destroy energy energy
is conserved.
(???????,???????????)
If the amount of energy in a system changes,
it can only be due to the fact that energy has
crossed the boundary by a transfer mechanism.
(????????????,?????????????????? ????????????)
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isolated and non-isolated systems
For isolated systems, energy is always conserved,
so
While for non-isolated systems, we have
Work
Heat
matter transfer
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Power(??)
Power the time rate of energy transfer.
(?????????)
Average power If the work done by a force is
W in the time interval ?t , then the average
power during this time interval is defined as
The instantaneous power
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For work done by a varying force
In general, power is defined for any type of
energy transfer and the most general expression
for power is, therefore
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Unit of Power (?????)
Unit of Power Watt
Other mostly used units of power
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Momentum and Impulse (?????)
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kinematics mechanics (force) energy momentum
motion
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Momentum (??)
The (linear) momentum of an object of mass m
moving with a velocity v is defined to be the
product of the mass and velocity
Momentum is a vector quantity and its
direction is the same as that for velocity And
it has dimension ML/T. In SI system,
the momentum has the units kgm/s.
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Momentum and Force
As pointed out before, the Newtons second
law can be rewritten as
From above equation, we see that if the net
force on an object is zero, the time derivative
of the momentum is zero, and therefore the
momentum of the object must be constant. Of
course, if the particle is isolated, then no
forces act on it and the momentum remains
unchangedthis is Newtons first law.
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Momentum and Isolated Systems
The total momentum of an isolated system remains
constant.
(????????????)
The total momentum
for an isolated system
Thus, we have
The law of conservation of linear momentum!
(??????)
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Impulse and Momentum (?????)
Assuming a net varying force acts on a particle,
then we get
thus the change in the momentum of the particle
during the time interval ?t tf - ti reads
The Impulse of a force is defined as
impulse-momentum theorem (??-????)
Also valid for a system of particles(????????)
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Impulse is an interaction between the system
and its environment. As a result of this
interaction, the momentum of the system changes.
(?????????????????,??????????)
The impulse approximation We assume that one of
the forces exerted on a particle acts for a short
time but is much greater than other force
present. this simplification model allows us to
ignore the effects of other forces, because these
effects are be small during the short time during
which the large force acts.
(????????????????????,???? ?????????????????????,
????? ??????????)
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Collisions
(??)
When two objects collide, it is a good
approximate in many cases to assume that the
forces due to the collision are much larger than
any external forces present, so we can use the
simplification model the impulse approximation.
Elastic collision (????) Inelastic
collision(?????) Perfectly inelastic collision
Collisions
Momentum is conserved in all cases, but
kinetic energy is conserved only in elastic
collisions.
(?????????????,????????????)
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For the collision is elastic, we get the
third equation for conservation of kinetic energy
Solve the set of equations composed of above
three equations, we obtain
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The Centre of Mass (??)
The center of mass of a system of particles is
defined as
For an extended object reads
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The centre of mass of a homogeneous, symmetric
body must lie on an axis of symmetry.
The centre of mass of a system is different
from its centre of gravity. Each portion of a
system is acted on by the gravitational force.
The net effect of all of these forces is
equivalent to the effect of a single force Mg
acting at a special point called the center of
gravity. The centre of gravity is the average
position of the gravitational force on all parts
of the object. If g is uniform over the system,
the centre of gravity coincides with the
centre of mass. In most cases, for objects or
systems of reasonable size, the two points can be
considered to be coincident.
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E.g. 0.1 A system consists of three particles
located at the corners of a right triangle as in
the figure. Find the centre of mass of the system.
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E.g. 0.2 A rod of length 30.0 cm has a linear
density

• where x is the distance from one end, measured in
  meters.
• What is the mass of the rod?
• How far from the x 0 end is its center of mass?

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Motion of a System of Particles
??????
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E.g. 0.2 A rod of length 30.0 cm has a linear
density

• where x is the distance from one end, measured in
  meters.
• What is the mass of the rod?
• How far from the x 0 end is its center of mass?

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Motion of a System of Particles
??????
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Outline of Rotational Motion
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Rigid body model A rigid body is any system of
particles in which the particles remain fixed in
position with respect to one another.
Rotation about a fixed axis Every particles on a
rigid body has the same angular speed and the
same angular acceleration.
rigid body ?? rotation about a fixed axis ????
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Rotational kinematics
(???)
The rigid body under constant angular
acceleration (??????????)
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Relations Between Rotational and Translational
Quantities (????????????)
translational motion ??
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Rotational Kinetic Energy
(????)
the moment of inertial
rotational kinetic energy
For an extended system
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The Rigid Body under a Net Torque (????????)
The net torque acting on the rigid body is
proportional to its angular acceleration.
(????)
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The Rigid Body in Equilibrium (?????????)
The torque vector
(??)
Two conditions for complete equilibrium of an
object
translational equilibrium
rotational equilibrium
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Work and Energy in Rotational Motion
(?????????)
Work done by a torque
The power of a torque
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Angular Momentum
(???)
The angular momentum of the particle relative to
the origin is defined as
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Conservation of Angular Momentum (?????)
The total angular momentum of a system remains
constant if the net external torque acting on the
system is zero.
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???????,???????????,??? ??,??????????????,?
???????? ?,??????????????,?????????? ?,???????????
??
???????,???
?????????????????????????
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A Brief Summary of Part I

1. Kinematics of a Particle
2. Newtons Laws of Motion
3. Work and Energy
4. Momentum and Impulse
5. Motion of a System of Particles
6. Rotations of a Rigid Body about a Fixed Axis

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The End of Part I