Transcript ClassicalMechanics_2..

Physics 1901 (Advanced)
A/Prof Geraint F. Lewis
Rm 557, A29
[email protected]
www.physics.usyd.edu.au/~gfl/Lecture
http://www.physics.usyd.edu.au/~gfl/Lecture
Energy
 Using Newton’s laws can be tricky as
we have to keep track of the
components of vector quantities
 This can lead to coupled differential
equations which can be hard to solve
 Many problems can be simplified by
examining them in terms of Energy.
http://www.physics.usyd.edu.au/~gfl/Lecture
What is Energy?
 Force is a well defined concept
 In physics, energy is a specific, yet
abstract thing
 Energy comes in a large number of
forms; thermal, kinetic, electrical etc
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What is Energy?
In these lectures, we will consider two
forms of energy;
 Potential Energy: Energy that a body
has by virtue of its position
 Kinetic Energy: Energy that a body
has by virtue of its motion
Coupled with the conservation of energy
we have a powerful toolbox for problems.
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Work & Kinetic Energy
The concept of work can be understood when a
force is applied to a body to chance its motion
Work is done on an object when a force changes
its point of application & is defined to be;
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What’s the dot? (Sec 1.10)
The dot product allows us to multiply two vectors;
Given the components of a vector, the dot product is
simple to calculate;
Notice that the result is a scalar!
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Why the dot?
For a constant force;
Only the force in the direction of motion contributes to the
work done on an object. This is selected by the dot product.
Work has units of N m which equals Joules (i.e. it is energy)
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Why work?
From the kinematic equations;
A force acting on a body results in a change of kinetic energy.
This is known as the Work-Kinetic Energy Theorem.
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Negative Work
Friction opposes the direction of motion (=180)
Negative work done on an object reduces the amount of
kinetic energy it has.
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Using Energy
A mass of 10kg is acted on by a force of 10N
at an angle of 30o. The force acts over a
distance of 5m. What is the change in velocity
due to the action of the force?
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Using Energy
Why would we prefer to consider energy rather
than examine a system using Newton’s laws?
 In many problems, the force acting on a body is not
constant and varies with position
 Using F=ma becomes problematic as this results in
accelerations being a function of position
 The overall equations can become quite messy
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Using Energy
Calculating the work
done by a variable force
is equivalent to area
under the forcedistance curve along
the path of the object.
This can be much
simpler than dealing
with vectors.
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Example: A spring
A mass is pushed up against a spring,
compressing it by a distance X. The mass is
then released. What is its velocity as it passes
through x=0?
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What is Kinetic Energy?
 Energy only makes sense when we talk about
changes or transfers of energy
 Kinetic energy is a measure of the amount of
work that one object can do on another
 We will examine this more closely when we
look at collisions, but a more massive object or
a faster moving object does more work (i.e.
when it hits something).
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Potential Energy
The work done by the force of
gravity as an object is changes
position is;
The kinetic energy is reduced
Where did the energy go?
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Potential Energy
So, the energy extracted by gravity is somehow
stored in the gravitational field (although this
energy is not localized).
Using conservation of energy, we can define the
change in gravitational potential energy to be
As well as putting energy into the gravitational
field, we can extract it; the force is conservative
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Potential Energy
Given this, we can further define the gravitational potential
energy to be;
where h is the height above some point.
Consider a mass at rest at height h2. It is release and falls
to h1. Work done by gravity on the mass is
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Potential Energy
Note that only the difference of gravitational
potential energy between points appears in these
equations, and the absolute values of potential
energy do not matter.
You are free to choose the zero-point, so do so
to ease the problem you are looking at.
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Potential Energy: An example
A cart is released from a height h and slides down
a friction less track. It encounters a loop of radius
R. What is its velocity at the top of the loop?
(Assume the cart is fixed to the track). What
happens if we consider friction?
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Springs & Gravity
We can use a similar argument to gravity to define
the elastic potential energy stored in a spring.
Unlike gravitational potential energy, the zeropoint is not arbitrary as UE(x=0) = 0.
The total (mechanical) energy is conserved so
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Springs & Gravity
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Conservative Forces
The change in gravitational potential is the same for each.
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Conservative Forces
 Energy only on the difference between the
initial and final states
 Independent of the path
 Reversible
 If start point and end point are the same, then
the work done is zero
 Can define a potential energy function
Conservative forces allow energy storage!
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Conservative Forces: Springs
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Force & Potential Energy
Work done is related to potential energy via
Remembering the definition of work, this is
The force is the gradient of the potential!
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Force & Potential Energy
In 3-D;
This can be quite useful when you have complicated
potential functions. Consider, however, gravity & springs
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Force & Potential Energy
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Force & Potential Energy
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Energy Diagrams
For an object with a total
energy E, the potential
curve can be used to
calculate the kinetic
energy.
In this case, the mass
oscillates be § A. The mass
is stuck in a potential
well and cannot get to any
other values of x.
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Non-conservative forces
 When moving a mass in a gravitational field,
the amount of work done by gravity is
independent of the path taken.
 The same is not true of friction as it always
opposes the direction of motion.
 Whereas gravity can do positive and negative
work on an object, friction only does negative.
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