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

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bluehedgehog628

15 Jan 2016

School

UTSC

Department

Physics and Astrophysics

Course

PHYA10H3

Professor

Brian Wilson

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Document Summary

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 x where k is 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. Speci cally, we are ignoring friction, and we are assuming the spring force scales linearly with distance.

Related Questions

Two particles of identical mass m and charge q aresuspended bymeans of uncharged threads of identical length L whosemasses arenegligible. Due to the forces present, each particle isin a stateof equilibrium. Due to the symmetry of theconfiguration, eachthread forms the angle (data) with the verticaldirection, asdepicted in the diagram. Standard gravity actsverticallydownward.

a) Let T denote the magnitude of the tension force inthethread and F denote the magnitude of the electrostaticforceexerted on a particle. By constructing a proper free-bodydiagramfor either particle, show that

sin(Î¸)*T= F, cos(Î¸)*T = mg

b)By combining these equations, show that

(sin(Î¸))^2*tan(Î¸)= (K_e*q^2)/(4*L^2*mg)

c) Assume that (Î¸) << 1 (radian), so thattan(Î¸) Ësin(Î¸). Determine an approximate value forÎ¸ when

m = 5.0 g, q = 25 nC, and L = 120 cm. Express this value ofÎ¸in degrees.

d) Furthermore, in the case for which tan(Î¸) Ësin(Î¸), showthat

xË ((2*K_e*q^2*L)/(mg))^(1/3)

Determine an approximate value for x based upon thedataprovided above.

Please provide step by step solutions please....

Learning Goal: Understand how to determine the constants in thegeneral equation for simple harmonic motion, in terms of giveninitial conditions.
A common problem in physics is to match the particular initialconditions - generally given as an initial position x_0 andvelocity v_) at t=0 - once you have obtained the general solution.You have dealt with this problem in kinematics, where theformula

1. x(t) = x_0 + v_0t + at^2

has two arbitrary constants (technically constants of integrationthat arise when finding the position given that the acceleration isa constant). The constants in this case are the initial positionand velocity, so "fitting" the general solution to the initialconditions is very simple.

For simple harmonic motion, it is more difficult to fit the initialconditions, which we take to be

x_0, the position of the oscillator at t = 0, and
v_0, the velocity of the oscillator at t = 0.

There are two common forms for the general solution for theposition of a harmonic oscillator as a function of time t:

2. x(t) = A\cos(omega t + phi) and
3. x(t) = C\cos(omega t) + S\sin(omega t),

where A, phi, C, and S are constants, omega is the oscillationfrequency, and t is time.

Although both expressions have two arbitrary constants--parametersthat can be adjusted to fit the solution to the initialconditions--Equation 3 is much easier to use to accommodate x_0 andv_0. (Equation 2 would be appropriate if the initial conditionswere specified as the total energy and the time of the first zerocrossing, for example.)

Find C and S in terms of the initial position and velocity of theoscillator.
Give your answers in terms of x_0, v_0, and omega. Separate youranswers with a comma.

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

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