Transcript ClassicalMechanics_3..

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
Variational Principle
Suppose you have to rescue
a swimmer in trouble. You
can run fast on the sand, but
swim slowly in the water.
Which path should you take
to reach the swimmer in the
shortest time?
Look at the action for all
possible paths and choose
the minimum time path.
[not in exam]
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Least Action
The principle of least action is
very important in physics. In
optics, light is seen to take
the minimum time path
between two points (this is
known as Fermat’s principle).
It is also central to general
relativity and quantum
mechanics!
[not in exam]
Semester 1 2009
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Lagrangian
Euler and Lagrange reformulated classical
mechanics in terms of least action. The most
important quantity is the Lagrangian which is
simply the kinetic energy minus the potential
energy. If we consider a object moving vertically
in a gravitational field, then;
where
[not in exam]
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Euler-Lagrange Equation
Euler and Lagrange showed that the least action
path obeys the Euler-Lagrange equation;
For our object in a gravitational field, this is
[not in exam]
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Collisions
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Collisions: How to analyze?
 Newton’s laws?
 Work & Energy?
Each is applicable in a large number of
complex problems.
When things collide, application of
either can be problematic.
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Momentum
Newton’s second law;
However, Newton actually said;
(this is important in relativity!)
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Impulse
We can define an impulse
Hence, force acting over time changes
the momentum of an object.
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Impulse
A cricket ball with a mass
of 0.25kg heads towards a
bat at 27m/s. It is hit by
the bat and leaves with a
speed of 43m/s. What is
the average force on the
ball if the bat and ball are
in contact got 0.01s? What
if the contact time is
0.1sec?
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What next?
Remember, if there is no net force
acting, the momentum is constant;
No net force means momentum is
conserved.
Haven’t we covered this?
Semester 1 2009
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Collisions
Semester 1 2009
http://www.physics.usyd.edu.au/~gfl/Lecture
Collisions
Semester 1 2009
http://www.physics.usyd.edu.au/~gfl/Lecture
Collisions
By Newton’s third law, the car & truck
exert equal and opposite forces on one
another.
If we consider the car and truck
together, the net force is zero.
Again, taken together, momentum must
be conserved in a collision!
Semester 1 2009
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Collisions
 In a collision, internal forces cancel
(due to Newton’s third law)
 As long as no external forces are
acting, the total momentum is
conserved.
 YOU define the object(s) of interest.
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Collisions: Example
A truck of mass 3000kg
collides head-on with a
stationary car of mass 800kg.
The truck is initially traveling
at 20m/s. What is the velocity
after the collision if both the
truck and car move together?
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Types of Collision
Momentum is conserved in all collisions.
But we can define two kinds of collision;
 Elastic: Both energy and momentum
momentum are conserved
 Inelastic: Only momentum is
conserved in collisions.
Where does the energy go?
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Elastic Collisions
In elastic collisions, both kinetic
energy and momentum are
conserved.
 Billiards & snooker
 Newton’s cradle
Can we explain Newton’s Cradle?
What about that basketball and
tennis ball trick?
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Systems
In this free body example, we only considered
the action-reaction force between blocks 1 & 2
when we examined this situation as two
separate systems.
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External Forces
Considering this as a single system, then all
internal forces occur in equal & opposite pairs.
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External Forces
But;
So the individual external forces on each part of
the system change the individual momenta &
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External Forces
Only external forces change the total momentum
of a system.
Parts of a system can change momentum, move
relative to each other etc due to internal forces,
but changes in total momentum arise only from
the application of external forces.
Remember: What comprises a system is a matter
of choice (and convenience).
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External Forces
The parts of the system do not have to be connected!
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Centre of Mass
For the collection of objects
(pool balls, cars, planets
etc) we can define the
centre of mass.
This is weighted average
position of all the individual
masses.
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Centre of Mass
The centre of mass is a vector and its component are
With similar expressions of ycm and zcm.
Note in the continuous limit where we consider a
distribution of density rather than point masses;
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Centre of Mass
The centre of mass is not a physical thing!
If we differentiate the centre of mass with respect to time
then we find;
If the total mass is M = m1 + m2 + … then
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Centre of Mass
So, the momentum of the centre of mass is equal
to the momentum of the entire system. But;
Only external forces can change the momentum of
the centre of mass!
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Centre of Mass
Semester 1 2009
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