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

2. 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.

3. 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

4. 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.

5. 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;

6. 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!

7. 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)

8. 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.

9. Negative Work Friction opposes the direction of motion (=180)
Negative work done on an object reduces the amount of kinetic energy it has.

10. 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?

11. 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

12. Using Energy
Calculating the work done by a variable force is equivalent to area under the force-distance curve along the path of the object.
This can be much simpler than dealing with vectors.

13. 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?

14. 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).

15. 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?

16. 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

17. 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

18. 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.

19. 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?

20. 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 zero-point is not arbitrary as UE(x=0) = 0.
The total (mechanical) energy is conserved so

21. Springs & Gravity

22. Conservative Forces
The change in gravitational potential is the same for each.

23. Conservative forces allow energy storage!
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!

24. Conservative Forces: Springs

25. 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!

26. Force & Potential Energy
In 3-D;
This can be quite useful when you have complicated potential functions. Consider, however, gravity & springs

27. Force & Potential Energy

28. Force & Potential Energy

29. 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.

30. 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.