Transcript ClassicalMechanics_1..

Physics 1901 (Advanced)
Prof Geraint F. Lewis
Rm 560, A29
[email protected]
www.physics.usyd.edu.au/~gfl/Lecture
Semester 1 2009
http://www.physics.usyd.edu.au/~gfl/Lecture
Physics@Sydney
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Optics & Photonics
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Take advantage of this expertise & think about
research projects (TSP, Special Projects and
Honours).
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Physics 1901 (Advanced)
Three module course consisting of
 Mechanics (15 lectures)
 Thermal Physics (10 lectures)
 Waves & Chaos (13 lectures)
It is assumed you have prior physics knowledge.
Stream changes made by the HECS deadline.
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Learning
What you learn from this course depends
upon the effort you put in
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Lectures are a guide to course material
Read your module/unit outlines
University Physics by Young & Freeman
Online resources: WebCT & Junior Physics
6hrs/week independent study
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Tutorials
 Interactive Workshop Tutorials
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Work in small groups (up to 4)
Worksheets & Hands-on demonstrations
A chance to ask questions
A place to clarify ideas
 Not assessed; up to you.
 No worksheets if you don’t attend.
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Labs
 Labs are 3 hours
 Work in groups of 4
 Read in advance
 Get it done faster
 Better chance of learning something
 Level 4, Carslaw Building
 Lab manuals from the CO-OP
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Assessment
Lab
Mastering Physics
Progressive Test
Lab Skills Test
Exam
20%
10%
5%
5%
60%
It is important to know concepts & ideas, not just
manipulate formulae; look at previous exam papers.
It is important to know the meaning of Academic Honesty
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If you need help
 Talk to me;
 Email me your question or to make an
appointment (no walk-ins)
 See a duty tutor
 Consult the web resources
 Serious personal problems or illness it
is important to complete a Special
Consideration Form ASAP!
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Physics 1901: Mechanics
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Physics
 is the study of the changeable
properties of natural objects
 Position, mass, temperature, charge
Physics is predictive
Know the properties of something now,
calculate the properties of something later
Semester 1 2009
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Classical Mechanics
(why classical?)
 Modern physics
 General Relativity
 Quantum Mechanics
 Classical mechanics
 Physics of “human experience”
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Classical Mechanics
(what & why?)
Simply put, classical mechanics is “how do
things respond to forces?”
 The concepts of classical mechanics
underpin the rest of physics
 Have implications in all sciences!
 Applied classical mechanics = Engineering?
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Course Layout
Lecture
Content
1-3
4-5
6-7
8-9
10
11-12
13-14
15
Kinematics, dynamics & Newton’s Laws
Semester 1 2009
Applications of Newton’s Laws
Work & Kinetic Energy
Potential Energy & Energy Conservation
Momentum, Impulse & Collisions
Rotation of Rigid Bodies
Dynamics of Rotational Motion
Gravitation
http://www.physics.usyd.edu.au/~gfl/Lecture
Kinematics (Review Ch 1-3)
 Kinematics is the description of motion
Let’s start with motion in one dimension
xo is the initial position of an object
vo is the initial velocity of an object
a is the (constant) acceleration of an object
What are its properties after a time t ?
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Velocity & Acceleration
Velocity is the change of distance over time
Acceleration is the change of velocity over time
(Differential equations!)
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Kinematic Equations
You do not need to memorize such equations as they will be
given in an exam. You should be able to derive them from the
definitions of velocity and acceleration!
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Non-Constant Acceleration
Generally, we will consider only constant
acceleration (cos this makes life easier).
Remember this is not generally true.
is called the jerk
Can use these to derive more general kinematic equations.
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More than one dimension: Vectors
Once we consider motion in more than one
dimension, vectors make life simpler.
The kinematic equations can be applied in each direction separately.
You decide the coordinate system!
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Decomposing Vectors
Vectors have a length &
direction. To use them we
need to decompose the
vector into its components.
(this is important!)
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Adding Vectors
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Monkey & Hunter
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Galileo & Inertia
The Principle of Inertia
If a body is left alone, it remains
where it is or continues along with
uniform motion.
Why the universe behaves like this is
a mystery, but without it science
would be quite tricky.
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Isaac Newton
Developed concept of Dynamics
Considered the motion of a body as
it is being influenced by something.
Developed three fundamental laws
of motion.
Amongst the most powerful
scientific laws!
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What is the ‘something’?
“In order to use Newton’s laws, we have
to find some formula for the force;
these laws say pay attention to the
forces. If an object is accelerating,
some agency is at work; find it”
Richard Feynman
Lectures on Physics
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Universal Forces
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Gravity
Electro-magnetic Forces
Strong Force
Weak Force
All forces are some form of the above!
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Newton’s First Law
“A body acted on by no net force moves
with constant velocity (which may be
zero) and zero acceleration”
This just reiterates Galileo’s ideas of inertia.
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Newton’s Second Law
“If a net external force acts on a body, the
body accelerates. The direction of the
acceleration is the same as the direction
of the net force. The net force vector is
equal to the mass of the body times its
acceleration”
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What is Mass?
The amount of substance in a body
The source of gravity
The ‘coefficient’ of inertia
Why these quantities are the same is
another mystery of the Universe.
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Newton’s Third Law
“If body A exerts a force on body B (an
‘action’), then body B exerts a force on
body A (a `reaction’). These two forces
have the same magnitude but are
opposite in direction. These two forces
act on different bodies”
(Be careful with the minus sign! This is a vector equation!)
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Newton’s Third Law
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Using Newton’s Laws
 With no net force, a body remains at
rest or at constant velocity.
 With a net force, a body accelerates in
the direction of the net force, dependent
upon its mass.
 To every action, there is an equal and
opposite reaction.
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Complications: Weight
All masses are attracted to the centre
of the Earth.
Gravity produces an acceleration of
g=9.8m/s2 which means the force is
For example: a 51kg gymnast has a weight
of 500N (remember your units).
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Complications: Normal Forces
Weight acts through the centre of mass, but as I
am not accelerating when I stand on the ground,
the net force=0!
Hence, there is another force balancing weight,
supplied by the ground, called the normal force.
Do weight & the normal force represent an ActionReaction pair?
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Complications: Normal Forces
 Normal forces are
due to the repulsion
of atoms
 Normal forces are
normal to a surface
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Complications: Tension
Tension occurs in ropes and
strings and depends upon
the particular configuration
of the forces.
For a massless rope, the
tension is the same
throughout the rope.
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Complications: Tension
More correctly, the rope is
said to be the state of
tension.
This results in forces at
rope “edges”.
It’s important to remember
that the resultant forces
need not be in the same
direction (or of the same
magnitude).
Semester 1 2009
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Complications: Tension
When considering a rope with
mass, its weight must be
considered. In the static case
Remember, weight is a force so
its direction is important!!
Semester 1 2009
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Free-Body Diagrams
Split the problem
into smaller pieces.
Consider the forces
on particular parts.
Keeping track of
action-reaction pairs
is vital.
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Free-Body Diagram
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Free-Body Diagram: Example
A trolley of mass m1 is place
on a slope inclined at 15o. It
is attached via a light string
and pulley to a hanging sand
bucket. What mass of sand
m2 is needed such that the
trolley possesses uniform
motion?
(Assume no friction)
Semester 1 2009
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Free-Body Diagram: Example
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Solving Problems: A Guide
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Draw a ‘free-body’ diagram
Consider all of the forces acting
Choose axes to ease the solution
‘Decompose’ the forces
Equations of motion
Semester 1 2009
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Complications: Friction
Microscopically, surfaces
are not smooth but
consist of pits & peaks.
When you try and move
something these can
lock like a jigsaw puzzle
and resist movement.
What force is actually
causing the friction?
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Complications: Friction
Metals can have a more
complicated friction.
As surfaces come into
contact, atoms undergo cold
welding. Pull these apart
adds to the friction.
The number of atoms in
contact depends upon how
hard the surfaces are
pressed together.
Semester 1 2009
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Complications: Friction
Experimentally the amount of friction is found to
be proportional to the component of weight
perpendicular to the surface (equivalently the
normal force).
Static Friction: The frictional force resisting a
force attempting to move an object.
Kinetic Friction: The frictional force experience
by a moving object.
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Static Friction
As the object is not moving,
there must be no net force.
where s is the coefficient
of static friction.
The frictional force Ff balances the applied force
until a point where F=Ff.
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Kinetic Friction
Kinetic friction opposes a
moving object.
where K is the coefficient of
kinetic friction.
Unlike static friction, kinetic friction has a fixed
value independent of the applied force.
(Is this really true?)
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Friction
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Coefficients of Friction
Generally, s is larger than K (e.g. steel upon steel;
s=0.74 and K=0.57)
Semester 1 2009
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Worked Example (5-91)
Block A, with a weight of 3w,
slides down an inclined plane
S of slope angle 36.9o at a
constant speed, while plank
B with weight w rests on top
of A. The plank is attached
by a cord to the top of the
plane.
* Draw a diagram of the
forces acting on block A
* If the coefficient of kinetic
friction is the same between
A & B and A & S, determine
its value.
Semester 1 2009
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Complaining Horse
The horse claims that “due to Newton’s third law, no matter
how hard I pull on the cart, the cart pulls back on me with
the same force. How can I ever move the cart!”
Semester 1 2009
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Complications: Circular Motion
Consider a ball on a
string, moving in a
circle with uniform
speed.
What are the forces
acting on the ball?
Semester 1 2009
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Complications: Circular Motion
The forces are not in
equilibrium, and hence the
ball must be accelerating!
The acceleration points
towards the centre of the
circle.
DO NOT add fictitious forces!
(more on that in a moment)
Semester 1 2009
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Complications: Circular Motion
The length of the velocity vector
remains constant, and so the
acceleration is changing its direction.
For an object traveling with speed v to
move in a circle of radius r the
centripetal acceleration must be
(review chapter 3)
Semester 1 2009
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Fictitious Forces
Newton’s laws work perfectly in inertial frames
These are observers who are stationary or are in
uniform motion with respect to the situation being
examined, although quantities (such as velocity)
are relative.
When we consider accelerating (or rotating)
frames (non-inertial), Newton’s laws apparently
don’t hold anymore!
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Fictitious Forces
BUT we can make Newton’s laws hold in noninertial frames by inventing fictitious forces that
do not exist (by which we mean there is no
physical source for the force).
Hence in a rotating frame, we can add a
centrifugal force to balance the centripetal force!
(So, what is the force that you feel on a “stick to
the wall” fairground ride?)
Semester 1 2009
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Non-Constant Forces
In general, forces are not
constant. An example of
this is Hooke’s law for a
spring, where the force is
& k is the spring constant.
To calculate Newton’s laws with non-constant forces, we need
to integrate the various vector quantities (a very messy
process). What we will see next is that such problems are
more simply tackled using concepts of work & energy.
Semester 1 2009
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