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# Damped Oscillations

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Damped Oscillations
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# Damped Oscillations

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### Damped Oscillations

1. 1. Damped Oscillations Author: Hantao Mai Physics Learning Objective
2. 2. Key Questions Being Answered  1) What are damped Oscillations?  2) What variables are considered in calculating damped oscillations?  3) What is an underdamped oscillator?  4) What is a critically damped oscillator?  5) What is an over-damped oscillator  6) What is the energy and energy loss in a Damped Harmonic Oscillator?  7) What questions might be asked on damped oscillation?
3. 3. Question 1: What are damped Oscillations?  Definition: Oscillations In the presence of frictional forces are called damped oscillations.  The formula F=-bv is used to determine the drag force an object experiences when moving though a medium.  b = The damping constant or drag constant  V = velocity of the object moving through the medium  The negative sign in the formula indicates that the direction of the drag is opposite of the velocity of the object.
4. 4. Question 1: What are damped Oscillations? (cont.)  It has been shown that two facts are true about damped oscillations:  1) The amplitude of a damped oscillator decreases exponentially with time.  2) The oscillation frequency of a damped oscillator is lower than the oscillation frequency of an undamped oscillator.  The equation used to find the displacement of a damped oscillator is:  X(t) = Ae^(-bt/2m)cos(W(d)t + Φ)
5. 5. Question 2: What variables are considered in calculating damped oscillations?  As seen in the last slide, the equation to determine the displacement of a damped oscillator is:  X(t) = Ae^(-bt/2m)cos(W(d)t + Φ)  The variables considered include:  W(d) = (k/m) – (b^2/4m^2)  K is the spring constant  M is the mass  B is the damping constant  W(d) is known as the oscillation frequency in the presence of drag force.  W(0) (which is sqrt (k/m)) is called the natural frequency and is when b = 0.
6. 6. Question 3: What is an underdamped oscillator?  An Underdamped oscillator is when the natural frequency (W(0)) is larger than the fraction b/2m.  W(0) > (b/2m)  Underdamped oscillations are when the natural frequency (W(0)) is close to the oscillation frequency (W(D)).  The main effect of drag force in underdamped oscillations is to decrease the amplitude exponentially with time.
7. 7. Question 4: What is a critically damped oscillator?  A critically damped oscillator is when the natural frequency is equal to b/2m  W(0) = (b/2m)  In this case, the oscillation frequency (W(D)) is ) so there is no back- and-forth motion in the oscillator.  The question X(t) = (a(1) + a(2)t)e^(w(0)t) is used to find the position of a critically damped oscillator where:  A(1) and A(2) are determined by the initial conditions X(0) and X(1).  Critically damped oscillators reach its equilibrium position faster than any other value of the damping constant.  I.e. These oscillators quickly return to their natural position.
8. 8. Question 5: What is an over-damped oscillator?  An over-damped oscillator is when the value of (b/2m) surpasses that of the natural frequency.  W(0) < (b/2m)  As the value of b increases, it takes an increasing amount of time for the oscillator to reach back to its natural position.
9. 9. Question 6: What is the energy and energy loss in a Damped Harmonic Oscillator?  The Energy in a damped harmonic oscillator is given by the equation:  E(t) = (1/2)(kA^2e^(-bt/m))  E(0) = (1/2)kA^2  The fraction energy loss in one oscillation is given by the equation:  1 – e^(-bT(D)/m)  B is the damping constant  T(D) is the period 2pi/W(D)
10. 10. Question 7: What questions might be asked on damped oscillation?  Using the example given in the textbook, we are asked to find the damping constant (b) given the information:  M = 2kg  K = 10N/m  A= 25cm at t=0  The amplitude falls to 75% of its initial value after 4 oscillations.
11. 11. Question 7: What questions might be asked on damped oscillation? (cont.)  Step 1: Calculate the total energy  E(0) = (1/2)kA^2 = 0.312J  Step 2: Calculate the period  T = 2pi/(W(0)) = 2pi*Sqrt(m/k) = 2.81s  Step 3: Use the information that the amplitude falls off by 75% in for oscillation.  A(4T) = 3/4A  e^(-2bt/m)=3/4
12. 12. Question 7: What questions might be asked on damped oscillation? (cont.)  Step 4: Solve for b.  e^(-2bt/m)=3/4  -2bt/m = ln(3/4)  b = -m/2T*(ln3/4)  b = -(2/2*2.81)*ln(3/4)  b = 0.102kg/2
13. 13. Conclusion  The question that were asked in this power point only cover the main concepts of the ideas of damped oscillation.  If you have any questions that could be added to this power-point, feel free to message me and I will add them onto this slideshow. Thank You for Watching