1. Physics 218 Chapter 15
Prof. Rupak Mahapatra
Physics 218, Lecture XXII

2. Checklist for Today Midterm 3 average 61
Collect your exams from your TAs.
Last lecture next Monday
Will cover up to Chpater 18
Ch 14 and 15 Home work due Wed this week
Ch 18 Home work due Mon next week
Physics 218, Lecture XXII

3. Angular Quantities Momentum  Angular Momentum L Energy
Position  Angle q
Velocity  Angular Velocity w
Acceleration  Angular Acceleration a
Force  Torque t
Mass  Moment of Inertia I
Today we’ll finish:
Momentum  Angular Momentum L
Energy
Physics 218, Lecture XXII

4. Rotational Kinetic Energy
KEtrans = ½mv2
 KErotate = ½Iw2
Conservation of Energy must take rotational kinetic energy into account
Physics 218, Lecture XXII

5. Rotation and Translation
Objects can both Rotate and Translate
Need to add the two
KEtotal = ½ mv2 + ½Iw2
Rolling without slipping is a special case where you can relate the two
V = wr
Physics 218, Lecture XXII

6. Rolling Down an Incline
You take a solid ball of mass m and radius R and hold it at rest on a plane with height Z. You then let go and the ball rolls without slipping.
What will be the speed of the ball at the bottom?
What would be the speed if the ball didn’t roll and there were no friction?
Note:
Isphere = 2/5MR2 Z
Physics 218, Lecture XXII

7. A bullet strikes a cylinder
A bullet of speed V and mass m strikes a solid cylinder of mass M and inertia I=½MR2, at radius R and sticks. The cylinder is anchored at point 0 and is initially at rest.
What is w of the system after the collision?
Is energy Conserved?
Physics 218, Lecture XXII

8. Rotating Rod
A rod of mass uniform density, mass m and length l pivots at a hinge. It has moment of inertia I=ml/3 and starts at rest at a right angle. You let it go:
What is w when it reaches the bottom?
What is the velocity of the tip at the bottom?
Physics 218, Lecture XXII

9. Person on a Disk
A person with mass m stands on the edge of a disk with radius R and moment ½MR2. Neither is moving.
The person then starts moving on the disk with speed V.
Find the angular velocity of the disk
Physics 218, Lecture XXII

10. Same Problem: Forces Same problem but with Forces
Physics 218, Lecture XXII

11. Chapter 18: Periodic Motion
This time:
Oscillations and vibrations
Why do we care?
Equations of motion
Simplest example: Springs
Simple Harmonic Motion
Next time:
Energy }
Concepts }
The math
Physics 218, Lecture XXIV

12. Physics 218, Lecture XXIV

13. What is an Oscillation?
The good news is that this is just a fancy term for stuff you already know. It’s an extension of rotational motion
Stuff that just goes back and forth over and over again
“Stuff that goes around and around”
Anything which is Periodic
Same as vibration
No new physics…
Physics 218, Lecture XXIV

14. Examples Lots of stuff Vibrates or Oscillates: Radio Waves
Guitar Strings
Atoms
Clocks, etc…
In some sense, the Moon oscillates around the Earth
Physics 218, Lecture XXIV

15. Why do we care? Lots of engineering problems are oscillation problems
Buildings vibrating in the wind
Motors vibrating when running
Solids vibrating when struck
Earthquakes
Physics 218, Lecture XXIV

16. What’s Next You’ll see why we do this later
First we’ll “model” oscillations with a mass on a spring
You’ll see why we do this later
Then we’ll talk about what happens as a function of time
Then we’ll calculate the equation of motion using the math
Physics 218, Lecture XXIV

17. Simplest Example: Springs
What happens if we attach a mass to a spring sitting on a table at it’s equilibrium point
(I.e., x = 0) and let go?
What happens if we attach a mass, then stretch the spring, and then let go? k
Physics 218, Lecture XXIV

18. Questions What are the forces? Hooke’s Law: F= -kx
Does this equation describe our motion?
x = x0 + v0t + ½at2
Physics 218, Lecture XXIV

19. The forces No force Force in –x direction Force in +x direction
Physics 218, Lecture XXIV

20. More Detail
Time
Physics 218, Lecture XXIV

21. Some Terms Amplitude: Max distance Period: Time it takes to get
back to here
Physics 218, Lecture XXIV

22. Overview of the Motion
It will move back and forth on the table as the spring stretches and contracts
At the end points its velocity is zero
At the center its speed is a maximum
Physics 218, Lecture XXIV

23. Simple Harmonic Motion
Call this type of motion
Simple Harmonic Motion
(Kinda looks like a sine wave)
Next: The equations of motion:
Use SF = ma = -kx
(Here comes the math. It’s important that you know how to reproduce what I’m going to do next)
Physics 218, Lecture XXIV

24. Equation of Motion
A block of mass m is attached to a spring of constant k on a flat, frictionless surface
What is the equation of motion? k
Physics 218, Lecture XXIV

25. Summary: Equation of Motion
Mass m on a spring with spring constant k:
x = A sin(wt + f)
Where
w2 = k/m
A is the Amplitude
is the “phase”
(phase just allows us to set t=0 when we want)
Physics 218, Lecture XXIV

26. Simple Harmonic Motion
At some level sinusoidal motion is the definition of Simple Harmonic Motion
A system that undergoes simple harmonic motion
is called a
simple harmonic oscillator
Physics 218, Lecture XXIV

27. Understanding Phase: Initial Conditions
A block with mass m is attached to the end of a spring, with spring constant k. The spring is stretched a distance D and let go at t=0
What is the position of the mass at all times?
Where does the maximum speed occur?
What is the maximum speed?
Physics 218, Lecture XXIV

28. Paper which tells us what happens as a function of time
Check:
This looks like a cosine. Makes sense…
Spring and Mass
Paper which tells us what happens as a function of time
Physics 218, Lecture XXIV

29. Example: Spring with a Push
We have a spring system
Spring constant: K
Mass: M
Initial position: X0
Initial Velocity: V0
Find the position at all times
Physics 218, Lecture XXIV

30. Simple Harmonic Motion
What is MOST IMPORTANT?
Simple Harmonic Motion
X= A sin(wt + f)
What is the amplitude?
What is the phase?
What is the angular frequency?
What is the velocity at the end points?
What is the velocity at the middle?
Physics 218, Lecture XXIV