PHYA10H3 Lecture Notes - Lecture 8: Propagation Constant, Angular Frequency, Opata Language
DepartmentPhysics and Astrophysics
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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
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
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
•Is energy conserved? If not, how fast is energy being lost (power)? The energy should
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
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