Simple Harmonic Motion


 Erika Patterson
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1 Simple Harmonic Motion Theory Simple harmonic motion refers to the periodic sinusoidal oscillation of an object or quantity. Simple harmonic motion is eecuted by any quantity obeying the Differential Equation & + ω, () = where, & & denotes the second Derivative of with respect to t, and ω is the angular frequency of oscillation. This Ordinary Differential Equation has an irregular Singularity at,. The general solution is = Asin( ω + B cos( ω () = C cos( ω t + φ) (3) where the two constants A and B (or C andφ ) are determined from the initial conditions. Many physical systems undergoing small displacements, including any objects obeying Hooke s Law, ehibit simple harmonic motion. This equation arises, for eample, in the analysis of the flow of current in an electronic CL circuit (which contains a capacitor and an inductor ). If a damping force such as Friction is present, an additional term β & must be added to the Differential Equation and motion dies out over time. Adding a damping force proportional to &, the first derivative of with respect to time, the equation of motion for damped simple harmonic motion is & + β& + ω, (4) =
2 where β is the damping constant. This equation arises, for eample, in the analysis of the flow of current in an electronic CLR circuit, (which contains a capacitor, an inductor, and a resistor ). This Ordinary Differential Equation can be solved by rt looking for trial solutions of the form = e. Plugging this into (4) gives ( r rt + β r + ω ) e = (5) r + β r + ω = (6) This is a Quadratic Equation with solutions r = ( β ± β 4ω ). (7) There are therefore three solution regimes depending on the Sign of the quantity inside the Square Root, α = β 4ω. (8) The three regimes are. α > is Positive: overdamped,. α = is Zero: critically damped, 3. α < is Negative: underdamped.
3 Underdamped simple harmonic motion occurs when β 4ω < (9) so α β 4ω < () Define ω α = 4ω β, () then the solutions satisfy r ± = β ± iγ, () where r ± ± β β 4ω, (3) and are of the form ( / iγ ) t = e β ± (4) 3
4 Using the Euler Formula e i = cos + isin (5) this can be rewritten. [ cos( ω. ± i sin( ω. ] ( β / ) t = e (6) We are interested in the real solutions. Since we are dealing here with a linear homogeneous ODE, linear sums of linearly independent solutions are also solutions. Since we have a sum of such solutions in (64), it follows that the Imaginary and real part separately satisfy the ODE and are therefore the solutions we seek. The constant in front of the sine term is arbitrary, so we can identify the solutions as ( β / ) t e cos( ω. (7) = [ ] ( β / ) t = e [ sin( ω. ] (8) so the general solution is ( β / ) t = e Acos( ω. + Bsin( ω. (9) [ ]  Eperiment  Object: To study Hooke s law, and simple harmonic motion of a mass oscillating on a spring.  Apparatus: Rotary motion sensor, thin string, uniform spring, balance, weight hanger and weight, computer Pasco Model 7 Interface, printer. 3 Procedure:  Adjust the apparatus. Hang the pan on the spring and add enough weight (about 4g) so that the spring and the scale are vertical. Take that position of the pan as the equilibrium position.  Start the data collection add etra mass m such as g, g, 3g and 4g to further stretch the spring. For each etra mass m, leave the pan get down very slowly to the maimum position and stop the data collection. Enlarge the 4
5 position s table you can record the etra stretch. Repeat this with the different weights. Plot mg vs., draw a straight line through the origin and all the points. (g is the gravity acceleration constant ). The slope of the line is the spring constant k 3 Remove the etra masses and come back to the equilibrium position (with 4g). Displace the whole weight by a small vertical distance (an inch or two). Release the system and start the collection data at the same time. When the oscillations (vibrations) vanish, stop the data collection. On the graph of the position vs. time, use the sine fit to fit all sinusoidal plots. You can read the period of oscillations and calculate the value of the corresponding frequency ω. Now, using the formula k ω = calculate the real m mass m.what is your conclusion? 4 Use the computer pencil to draw the line that connects the top peaks of the position s waves. 5 Use the Natural Eponent fit to fit that line. The eponent C gives you the value of β /. Infer the value of the damping constant β. 6 Repeat the same eperiment for the etra mass of 4 more grams. 7 Print all graphs and tables. 5
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