Class Notes (1,000,000)

CA (610,000)

UTSC (30,000)

Physics and Astrophysics (100)

PHYA10H3 (30)

Brian Wilson (10)

Lecture 8

# PHYA10H3 Lecture Notes - Lecture 8: Propagation Constant, Angular Frequency, Opata Language

by OC770797

Department

Physics and AstrophysicsCourse Code

PHYA10H3Professor

Brian WilsonLecture

8This

**preview**shows half of the first page. to view the full**1 pages of the document.**Note-Taker: Facilitor:

Collaborator:

PHYA10 Practical Week 8

This is a lab experiment week. You will have to submit a formal lab report next week during

your practical. Guidelines for the lab report can be found on Blackboard.

You have a cart on an air track with a spring on either side (two springs in total). Your

task is to evaluate how close this system is to an ideal harmonic oscillator (Chapter 14 of

your text). We have not covered harmonic oscillators in class yet, so what follows is a very

brief introduction of the theory needed to perform this experiment.

The cart (mass m) has a restoring force of F=−2k∆xwhere kis the spring constant

(for one of the springs). The factor of two arises because there are two springs. In the ideal

case, there are no other forces. Speciﬁcally, we are ignoring friction, and we are assuming

the spring force scales linearly with distance.

With these assumptions, we have md2x

dt2=−2k(x−xeq) as the equation of motion. This

has the following solutions:

x(t) = xeq +Asin(ωt +α)

v(t) = Aω cos(ωt +α) = Aω sin(ωt +α+π

2)

a(t) = −Aω2sin(ωt +α) = Aω2sin(ωt +α+π)

where xeq is the equilibrium position of the system, Ais the maximum displacement of the

cart from the equilibrium position, and ω=2π

T=q2k

mis the so-called angular frequency of

the harmonic motion, and Tis how long it takes for the system to do one full oscillation.

Finally, αis the phase constant, which is another thing you will be measuring.

Questions which you should answer about your system

•What value for kdo you get (by measuring the period)?

•Is your system harmonic? Is x(t) as theory predicts (sinusoidal, within uncertainties)?

Is Aa constant or is it decaying due to friction? Quantify your answers.

•Is v(t) out of phase from x(t) by exactly π/2? Is a(t) out of phase from x(t) by exactly

π? What is αfor your data? What does αrepresent physically for your system and

data?

•Is energy conserved? If not, how fast is energy being lost (power)? The energy should

be E=1

2mv2+kx2since there are two springs. (You can get LoggerPro to plot

functions such as kx2.) What are the maximum and minimum values of the kinetic

energy, and the potential energy?

•Is energy going back and forth between kinetic and potential? What do each of the

graphs look like? Are they sinusoidal? If so, how does the frequency for the energies

diﬀer from the frequency of the position? Explain the diﬀerence mathematically and

physically.

###### You're Reading a Preview

Unlock to view full version