1. Physics 1901 (Advanced) A/Prof Geraint F. Lewis Rm 557, A29

2. 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]

3. 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]

4. 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]

5. 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]

6. Collisions

7. 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.

8. Momentum Newton’s second law; However, Newton actually said;
(this is important in relativity!)

9. Impulse We can define an impulse
Hence, force acting over time changes the momentum of an object.

10. 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?

11. 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?

12. Collisions

13. Collisions

14. 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!

15. 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.

16. 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?

17. 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?

18. 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?

19. 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.

20. External Forces
Considering this as a single system, then all internal forces occur in equal & opposite pairs.

21. External Forces
But;
So the individual external forces on each part of the system change the individual momenta &

22. 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).

23. External Forces The parts of the system do not have to be connected!

24. 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.

25. Centre of Mass With similar expressions of ycm and zcm.
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;

26. 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

27. 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!

28. Centre of Mass