1. Physics 218 Chapter 18
Prof. Rupak Mahapatra
Physics 218, Lecture XXIV

2. 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

3. Physics 218, Lecture XXIV

4. 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

5. 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

6. 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

7. 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

8. 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

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

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

11. More Detail
Time
Physics 218, Lecture XXIV

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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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