RES.8-009 (Summer 2017), Problem Set 3 Solutions

Massachusetts Institute of Technology MITES 2017–Physics III Solutions 3: Damped and Forced Oscillators (Midterm Week) Preface: This problem set provides practice in understanding damped harmonic oscillator systems, solving forced oscillator equations, and exploring numerical solutions to di erential equations. 1. Equations of Underdamping Figure 1: Damped Oscillator (a) Given the information in the prompt and in the plot we want to determine the mass of the oscillating degree of freedom and the quality factor of the damped oscillation. The prompt tells us we apply a force Fapp = 4 N to pull a spring (of unknown spring constant k) a distance Δx = 0.2 m. By Hooke’s law, we find that the spring constant of the system k= Fapp = 20 N/m. Δx (1) The plot provides us with two pieces of information not included in the prompt. From the prompt, we can determine the period of oscillation and we can determine how much time it takes the amplitude to decay to a specific fraction of its initial value. From the figure we can estimate the period to be T = 3 s. For an underdamped oscillator, the period of motion is T = 2π 2π =p 2 , Ω ω0 − γ 2 (2) p where Ω = ω02 − γ 2 with ω02 = k/m. Noting that it takes t1/2 = 22 s, for the amplitude of oscillation to decay to half of its initial value we have Ae−γt1/2 = A(0.5), (3) which upon taking the logarithm of both sides and dividing by t1/2 yields γ= ln 2 ' 0.0315 s−1 . t1/2 (4) With a knowledge of k, T , and γ, as provided by Eq.(1), the plot, and Eq.(4), respectively, we can determine the mass through Eq.(2). Solving for ω02 given Ω2 , we find ω02 = k = Ω2 + γ 2 . m (5) Then using Eq.(2) to write Ω in terms of the period and Eq.(4) to write γ in terms of the ”half-life” of the oscillation, we find m= k ≈ 4.56 kg. 4π 2 /T 2 + (ln 2/t1/2 )2 (b) With Eq.(6) and Eq.(1), we find that the natural angular frequency of motion is r k ω0 = ≈ 2.09 rad/s. m (6) (7) Therefore, with Eq.(??) and Eq.(4), we find that the quality factor of the system is Q= ω0 ≈ 33. 2γ (8)  2. R factor For an underdamped oscillator, Amy defines the ”R factor” as R = π × (Number of oscillation cycles it takes to reach 1/e of the initial amplitude). (9) How does R compare to the quality factor Q for an underdamped oscillator? (We’re considering a very weakly damped oscillator) Solution: We want to compute R and thereby determine how R relates to Q for an underdamped oscillator. By the end of the problem, we will have found a new way to interpret Q in terms of kinematic variables rather than energy. For an underdamped oscillator, the position as a function of time is x(t) = Ae−γt cos(Ωt − φ), p (10) where Ω = ω02 − γ 2 . By definition, the Q factor for this oscillator is Q = ω0 /2γ. In order to find the number of oscillation cycles it takes to reach 1/e of the initial amplitude, we first need to determine how much time it takes to reach e−1 of the initial amplitude. By the fact that the amplitude has the time dependent A(t) = A0 e−γt , we can find the time it takes to satisfy the condition in the parentheses of Eq.(9) with Ae−γt1 = Ae−1 . (11) Solving for t1 , we then find that t1 = 1/γ. Now, given that a single oscillation cycle occurs over a time T = 2π/Ω ' 2π/ω0 (where we took the approximation given the assumption of very weakly damped motion), the number of period cycles we experience in time t1 is # of oscillation cycles = t1 ω0 ' . T 2πγ Therefore, Eq.(9) becomes R = π × # of oscillation cycles ' ω0 , 2γ (12) (13) for very weakly damped motion. Given Q = ω0 /2γ, we therefore find R ' Q. 3. Forced Oscillator (a) Our goal in this problem is to determine x(t) given the above initial conditions. First, we write the equation of motion for the system. Using the identity sin2 x = 1 − cos(2x), we find the external force can be written as F0 F (t) = F0 sin2 (ωt) = (1 − cos(2ωt)). (14) 2 Thus, Newton’s 2nd law for the system gives us F0 (1 − cos(2ωt)), 2 (15) F0 F0 − cos(2ωt), 2m 2m (16) mẍ = −kx + which can be written as ẍ + ω02 x = where ω02 = k/m. Defining X ≡x− F0 , 2mω02 (17) we can write Eq.(16) as ¨ + ω 2 X = − F0 cos(2ωt). (18) X 0 2m From here we can use the general solution to the driven undamped equation of motion to write the general solution for this case. Considering Eq. (4) and Eq. (11) in Lecture notes 04, we find that the solution to Eq.(18) is X(t) = B cos(ω0 t) + C sin(ω0 t) − F0 /2m cos(2ωt). ω02 − 4ω 2 (19) Given Eq.(17), we can write this solution in terms of x(t) as x(t) = F0 F0 /2m + B cos(ω0 t) + C sin(ω0 t) − 2 cos(2ωt). ω0 − 4ω 2 2mω02 Imposing the initial condition x(t = 0) = 0 and ẋ(t = 0) = 0, we find   F0 1 1 B= − , C = 0. 2m ω02 − 4ω 2 ω02 3 (20) (21) Therefore, the general solution to the given di erential equation is x(t) = F0 F0 /2m (cos(ω0 t) − cos(2ωt)) + (1 − cos(ω0 t)). 2 2 2mω02 ω0 − 4ω (22) F0 /2m F0 (cos(ω0 t) − cos(2ωt)) + sin2 (ω0 t) 2 2 mω02 ω0 − 4ω (23) or x(t) = (b) Resonance is defined as the state in which the amplitude of oscillation is maximized. Resonance is defined by the nonzero frequency which yields this maximum.The only term in Eq.(22) whose amplitude is maximized by a non-zero value of ω is the first term. To maximize the coeÿcient of the first term, we minimized the denominator, and we can do so by setting ω02 = 4ω 2 . Given k = mω02 , we find that the value of k which puts the system in resonance is k = 4ω02 (24)  4. Numerical Solution to Di erential Equations - Part II (a) The various relationships between ω0 and γ which determine the type of motion of our damped oscillatory are as follows: ω0 > γ ω0 < γ ω0 = γ leads to underdamped motion leads to damped motion leads to critically damped motion (25) (26) (27) (b) Mathematica plots on website  5. Ball in Bowl Figure 2: Ball in Bowl State (but do not answer) three precise physics questions we can ask about this system. Solution: There are many di erent questions we can ask about this system. Some of the questions below can be answered using methods, we have already covered in class. Other questions require methods outside the class (which is fine because I’m only asking you to ask questions and not answer them). – What is the equation of motion of the ball in the bowl in terms of θ(t) – What is the frequency of small oscillations about the point θ = 0? – As an integral, what is an expression for the total period of motion presuming the ball begins from rest at an angle θ0 – What is the first non-linear approximation to the simple harmonic oscillator equation of motion for this system? – At what radius of the smaller sphere does the angular frequency of motion disappear? – If we considered the bowl and sphere to be three-dimensional what are the resulting equations of motion? – If we were to shake the bowl vertically at frequency ω, such that the gravitational acceleration became g(t) = g + g0 cos(ωt)? for some quantity g0 , what would the resulting equation of motion be, and what would be the solution to that equation of motion?