1. Physics I LECTURE 22 11/29/10

2. Administrative Notes Exam III In Class Wednesday, 9am Chapters 9-11
9 problems posted on Website in Practice Exam Section. At least 1 of these problems will be on EXAM III.
If you have questions, start a discussion thread on Facebook, that way my response is seen by everyone in the class.
Review Session Tuesday Night, 6:00pm, OH150

3. Outline Oscillations Simple Harmonic Motion What do we know? Units
Work by Constant Force
Scalar Product of Vectors
Work done by varying Force
Work-Energy Theorem
Conservative, non-conservative Forces
Potential Energy
Mechanical Energy
Conservation of Energy
Dissipative Forces
Gravitational Potential Revisited
Power
Momentum and Force
Conservation of Momentum
Collisions
Impulse
Conservation of Momentum and Energy
Elastic and Inelastic Collisions2D, 3D Collisions
Center of Mass and translational motion
Angular quantities
Vector nature of angular quantities
Constant angular acceleration
Torque
Rotational Inertia
Moments of Inertia
Angular Momentum
Vector Cross Products
Conservation of Angular Momentum
Oscillations
Simple Harmonic Motion
What do we know?
Units
Kinematic equations
Freely falling objects
Vectors
Kinematics + Vectors = Vector Kinematics
Relative motion
Projectile motion
Uniform circular motion
Newton’s Laws
Force of Gravity/Normal Force
Free Body Diagrams
Problem solving
Uniform Circular Motion
Newton’s Law of Universal Gravitation
Weightlessness
Kepler’s Laws

4. Review of Lecture 21
Discussed cross product definition of angular momentum and torque
Why would we ever use cross products instead of simpler scalar expressions?
3D vectors
Point masses not moving in uniform circle
Conservation of Angular Momentum
Newton’s 2nd Law for rotational motion
No external torque, angular momentum conserved.

5. Oscillations (Chapter 14)
Imagine we have a spring/mass system, where the mass is attached to the spring, and the spring is massless. A
vmax A

6. Oscillations
So we can say that the mass will move back and forth (it will oscillate) with an Amplitude of oscillation A.
Can we describe what is going on mathematically?
Would like to determine equation of motion of the mass. In order to do this, we need to know the force acting on the block.
Force depends on position: Hooke’s Law

7. Oscillations

8. Possible Solutions
What if x=At?
What if x=Aebt?

9. Possible Solutions
What if x=Acos(bt)?
What if x=Acos(bt+Φ)?

10. Possible Solutions What if
If we start with the mass displaced from equilibrium by a distance A at t=0, then we can determine x(t).

11. What does motion vs. time look like?
Plot x at t=0, π/2ω, π/ω, 3π/2ω, 2π/ω, 5π/2ω x A t -A

12. Harmonic Motion: terminology
Displacement: Distance from equilibrium
Amplitude of oscillation: max displacement of object from equilibrium
Cycle: one complete to-and-fro motion, from some initial point back to original point.
Period: Time it takes to complete one full cycle
Frequency: number of cycles in one second

13. Simple Harmonic Motion
A form of motion where the only force on the object is the net restoring force, which is proportional to the negative of the displacement.
Such a system is often referred to as a simple harmonic oscillator
The simple harmonic oscillator’s motion is described by:

14. What is Φ?

15. More terminology
So the Φ term is known as the phase of the oscillation. It basically shifts the x(t) plot in time.
The term ω, which for a spring mass system, is equal to , is known as the angular frequency.

16. Velocity and Acceleration for SHO
If we know x(t), we can calculate v(t) and a(t)

17. Velocity and Acceleration for SHOIf

18. Example A SHO oscillates with the following properties:
Amplitude=3m
Period = 2s
Give the equation of motion for the SHO

19. Example A SHO oscillates with the following properties:
Amplitude=3m
Period = 2s
At t=0s, x=3m.
Give the equation of motion for the SHO

20. Example A SHO oscillates with the following properties:
Amplitude=3m
Period = 2s
At t=0s, x=1.5m
Give the equation of motion for the SHO

21. Example A SHO oscillates with the following properties:
Amplitude=3m
Period = 2s
At t=0s, v=2m/s.
Give the equation of motion for the SHO

22. Energy of SHO
The total energy of a simple harmonic oscillator comes from the potential energy in the spring, and the kinetic energy of the mass.

23. Example
A spring mass system with m=4kg and k=400N/m is displaced +0.2m from equilibrium and released.
A) What is the equation of motion for the mass?

24. Example
A spring mass system with m=4kg and k=400N/m is displaced +0.2m from equilibrium and released.
B) What is the total energy of the system?

25. Example
A spring mass system with m=4kg and k=400N/m is displaced +0.2m from equilibrium and released.
C) What is the Kinetic Energy and Potential Energy of the system at t=2s?

26. SHO and Circular Motiony A
You will notice that we use the same variable for both angular velocity and angular frequency of a simple harmonic oscillator.
If we imagine an object moving with uniform circular motion (angular velocity=ω) on a flat surface. Starting, at t=0s, at θ=0.
We know that θ(t)=ωt
We can write the x-position of the object as:
And the y-position as: θ x

27. The pendulum
A simple pendulum consists of a mass (M) attached to a massless string of length L.
We know the motion of the mass, if dropped from some height, resembles simple harmonic motion: oscillates back and forth.
Is this really SHO? Definition of SHO is motion resulting from a restoring force proportional to displacement.

28. Simple Pendulum L θ Δx We can describe displacement as:
The restoring Force comes from gravity, need to find component of force of gravity along x
Need to make an approximation here for small θ… θ Δx

29. Simple Pendulum L
Now we have an expression for the restoring force
From this, we can determine the effective “spring” constant k
And we can determine the natural frequency of the pendulum θ Δx

30. Simple Pendulum L θ Δx If we know We can determine period T
And we can the equation of motion for displacement in x
…or θ θ Δx