Physics 1901 Advanced

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Physics 1901 (Advanced)

• Prof Geraint F. Lewis
• Rm 560, A29
• gfl_at_physics.usyd.edu.au
• www.physics.usyd.edu.au/gfl/Lecture

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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 Fermats principle).
• It is also central to general relativity and
  quantum mechanics!

not in exam
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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?

• Newtons 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

• Newtons 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. Havent
we covered this?
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Collisions
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Collisions
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Collisions

• By Newtons 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!

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Collisions

• In a collision, internal forces cancel (due to
  Newtons 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
• Newtons cradle
• Can we explain Newtons 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