SIMPLE HARMONIC MOTION Ken Cheney

Transcription

1 SIMPLE HARMONIC MOTION Ken Cheney INTRODUCTION GENERAL Probably no tools that you will learn in Physics are more widely used than those that deal with simple harmonic motion. Here we will be investigating mechanical examples but the same equations, and hence the same solutions, appear everywhere from acoustics to electronics to Quantum Mechanics. APPLICATIONS The mechanical applications of simple harmonic motion theory are, strangely, not usually used to produce vibrations but are used to avoid destructive vibrations. If a system vibrates naturally at the same frequency as a driving force the amplitude of motion can easily become vary large. This match of frequencies is called a resonance. You can consider the practical consequences of this large amplitude the next time you are flying. Notice that the wings of the plane flex up and down and consider whether this vibration might match some frequency produced by the engines. Resonance is commonly used when you want to detect a vibration. Such detectors include radio tuners, NMR devices, sonar detectors, and even gravity wave detectors (no luck here yet). PHYSICS INVOLVED The critical point is that if there is a linear restoring force or torque the motion that results will be of the form y = A sin(ωt) where y is the displacement or rotation, A is the amplitude of the motion, ω is the angular frequency, and t is the time. This very simple result keeps reoccurring for any situation where there is a linear restoring force. LB1BSH.DOC 1.060, May 3, 2002 Ken Cheney 1

2 LINEAR CASE Let F be the force that results from a displacement x and k be the constant of proportionality. F = kx Equation 1 The force acts on a mass m which moves according to Newton s Second Law: F = ma = kx or m d 2 x dt = kx 2 or d 2 x dt = k 2 m x Equation 2 Equation 2 tells us that if we take the second derivative of x with time we get the same x back along with the constants (-k/m). If we thinkover the functionswe knowwe find thatsin and cos return after two derivatives and produce a minus sign too. e t also returns after two derivatives but does not produce a minus sign. To get the constants let s try x(t)=sin(ωt) Equation 3 in Equation 2. The first derivative is ω cos(ωt) and the second derivative is ω 2 sin(ωt). Substituting in Equation 2 gives: ω 2 sin(ωt)= k m sin(ωt) Canceling the common terms yields: ω 2 = k/m Equation 4 So our guess for x satisfies Equation 2 identically if the constants are related in this manner. Finally, we notice that if we multiply the right side of Equation 3 by a constant A the constant will not be affected by the derivatives and so will cancel when x is substituted in Equation 2. x = A sin(ωt) Equation 5 Notice that the constant A and the frequency ω are quite independent. 2 SIMPLE HARMONIC MOTION Ken Cheney

3 To summarize If F = kx Equation 1 Then x(t)=a sin(ωt) Equation 5 and ω= k/m Equation 4 ANGULAR CASE Suppose an object with moment of inertia I is suspended with a linear restoring torque: Τ= γθ Equation 1a Where γis the constant of proportionality and θis the angle of rotation. We then have: Τ=Iα= γθ and I d 2 θ dt = γθ 2 and the equivalent of Equation 2 is: d 2 θ dt = γ 2 I θ Equation 2a Equations 2 and 2a are functionally the same, only the symbols have changed, therefor the functions of the solutions must also be the same. Let Θ 0 be a constant, then: θ=θ 0 sin(ωt) Equation 5a ω= γ/i exactly the same as Equations 4 and 5. Equation 4a EXPERIMENT GENERAL Check the results above (Equations 4 and 5) for at least three widely different cases. Possibilities are ships, pendulums, inverted pendulums, twisting wires,... The more unusual the restoring force the better. SIMPLE HARMONIC MOTION Ken Cheney 3

4 TIMERS We have a wide variety of timing devices ranging from computer managed photogates to stopwatches. Most people do more faster by using a stopwatch to time a number of cycles of their experiment. By using a number of cycles the accuracy is quite good and the setup time for the timers is almost zero. AMPLITUDE Try a wide range of amplitudes from very small to as large as your equipment permits. In the ideal case discussed above all amplitudes will yield the same frequency but as discussed below this will not necessarily be the case. MASS AND MOMENT OF INERTIA If the mass of the "spring" providing the restoring force is appreciable a common rule of thumb is to add a third of the mass of the spring to the mass of the oscillating object. Clearly the spring is being accelerated but different parts of the spring have different amplitudes so it is quite non-trivial to include the spring properly in the system. ANALYSIS EXPECTED FREQUENCY Calculate the theoretical frequency and compare it with the experimental frequency. Be very sure to include all moving objects in the mass and moment of inertia calculations. Have the raw data and results in data tables, don t forget the percent difference between the frequencies. Analyze, with real numbers, where this experimental difference could have come from. AMPLITUDE Examine your results very carefully for any systematic changes in frequency due to amplitude. This is best done with graphs and, perhaps, curve fits. Ideally the plot of frequency verses amplitude would be a horizontal straight line but nature may not be so simple. If large amplitudes exceed the elastic limit or otherwise change the resistance force (by additional air friction perhaps) then the restoring force will not be linear and ω will change with amplitude. WRITE UP A formal report is required since you are to invent your own experiments. Have large, clear, well labeled sketches of the setup. Discuss in what ways your results did or did not agree with theory and why you think the discrepancies occurred. 4 SIMPLE HARMONIC MOTION Ken Cheney

5 GRADING CONSIDERATIONS The more ingenious the experiment the better; weights on springs are not ingenious. The wider the range of conditions the better, both different types of motion and a wide range of amplitudes. Good plots and curve fits, thoughtfully discussed, and neatly incorporated into the report are very helpful. Good numerical analysis of the source of any discrepancies is also quite helpful. SIMPLE HARMONIC MOTION Ken Cheney 5