Phase, Phase Constant and S.H.M.
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Physics 1051 – Workshop 2 Phase, Phase Constant and S.H.M. Physics 1051 - General Physics II Oscillations, Waves and Magnetism Workshop 2 - Contents I. Mathematics Used in S.H.M. a) Sine and Cosine b) The Unit Circle c) Phase II. Physics with Cosine and Sine a. Phase b. Phase Shift or Phase Constant III. Problem Solving in S.H.M IV. An Example 25/05/10 Physics 1051 – Bill Kavanagh Physics 1051 - General Physics II Oscillations, Waves and Magnetism 2 I - Mathematics Used in S.H.M. 25/05/10 Physics 1051 – Bill Kavanagh Physics 1051 - General Physics II Oscillations, Waves and Magnetism 3 Cosine and Sine ● ● ● Cosine and Sine are useful because they can describe the things in physics. − Simple Harmonic Motion − Mechanical Waves Here we will concentrate on Cosine since it describes position with time for S.H.M. What properties allow us to describe S.H.M.? − Values of Function repeat after a certain period − Has a Maximum and Minimum 25/05/10 Physics 1051 – Bill Kavanagh Physics 1051 - General Physics II Oscillations, Waves and Magnetism 4 Function Values and The Unit Circle ● Unit Circle tells Cosine and Sine Values x , y= cos , sin +y cos ,sin =0,1 R=1 = cos , sin =−1,0 =/ 2 =/ 4 cos , sin = 2 , 2 2 2 +x =0 cos , sin = x , y = 1, 0 =3/ 2 cos ,sin = 0,−1 25/05/10 Physics 1051 – Bill Kavanagh Physics 1051 - General Physics II Oscillations, Waves and Magnetism 5 Phase ● What is Phase, ? − the argument of the Sine, Cosine functions ◦ ◦ − like an “index” that tell us where the function is in it's oscillation (we should have an intuition for this) ◦ ◦ − 25/05/10 sin =sin phase cos=cos phase e.g. Where is Cos when phase is e.g. Where is Sine when phase is ? 0? Ans. Ans. trough zero this idea of phase is the same in physics except for a slight complication Physics 1051 – Bill Kavanagh Physics 1051 - General Physics II Oscillations, Waves and Magnetism 6 II - Physics with Cosine and Sine 25/05/10 Physics 1051 – Bill Kavanagh Physics 1051 - General Physics II Oscillations, Waves and Magnetism 7 Physics of Phase! ● The issue of phase comes into play since we use Cosine and Sine to describe S.H.M. x t = A cos − ● − v x t =−A sin − a x t =−A 2 cos How does in physics compare to that in math? SAME: meaning, plot of cos vs. − − DIFFERENCE in PHYSICS ◦ ◦ 25/05/10 phase is a function of time i.e. =t plot of cos vs. t isn't identical to cos vs. Physics 1051 – Bill Kavanagh Physics 1051 - General Physics II Oscillations, Waves and Magnetism 8 Cosine of Phase in Physics 25/05/10 Physics 1051 – Bill Kavanagh Physics 1051 - General Physics II Oscillations, Waves and Magnetism 9 What is Phase? ● ● This shift illustrates that is not always zero when t is zero. What is =t ? − it depends on t − we know it is proportional to − ● we now know the initial at t=0 has to give correct initial condition Correct form of : − 25/05/10 = t 0 such that initial position determines 0 Physics 1051 – Bill Kavanagh Physics 1051 - General Physics II Oscillations, Waves and Magnetism 10 III - Problem Solving with Phase in S.H.M. 25/05/10 Physics 1051 – Bill Kavanagh Physics 1051 - General Physics II Oscillations, Waves and Magnetism 11 Solving for Phi and other Paramters ● ● General Problem solving steps apply; − write given, find, formula − compare number of unknowns to equations Starting with position equation: x t = A cos t0 − − has 5 unknowns − 4 unknowns are required to solve for other ● Of course could find or be given A, , x t , t ● Several variations of this given information... 25/05/10 Physics 1051 – Bill Kavanagh Physics 1051 - General Physics II Oscillations, Waves and Magnetism 12 Initial Conditions as Givens ● Add in the position initial condition x 0 x 0= A cos0 − − 1 unknown required ◦ ◦ ● 4 subtract 2 (x and t) 2 subtract 1 (t =0 eliminates omega term) Add in initial velocity condition v 0 v 0=−Asin 0 − − 0 unknown required ◦ ◦ 25/05/10 1 subtract 1 (v in new equation doesn't have new unknown) can now find 0 Physics 1051 – Bill Kavanagh Physics 1051 - General Physics II Oscillations, Waves and Magnetism 13 IV – An S.H.M. Example 25/05/10 Physics 1051 – Bill Kavanagh Physics 1051 - General Physics II Oscillations, Waves and Magnetism 14 Example Find the earliest time (t>0) when the following oscillating mass with position function x t =7cm cos5rad / s t/6 has a velocity +20 m/s. 25/05/10 Physics 1051 – Bill Kavanagh Physics 1051 - General Physics II Oscillations, Waves and Magnetism 15
