Physics 114 Lecture 24: Oscillations (PDF)

11/18/24 Physics 114 General Physics I: Physics for the Life Sciences Lecture 24...

11/18/24 Physics 114 General Physics I: Physics for the Life Sciences Lecture 24 Prof. Yue Qiu Please check in with the Email: [email protected] UNC Check-In app. Office: 178 Phillips Hall 1 © 2024 All rights reserved.  2 Today’s Class  Module 23 Review  Oscillations II  Vertical Offsets  Phase Constant  Damped Oscillations © 2024 All rights reserved. 11/18/24  3 Poll Question 24.1 𝑥(cm) At right is a graph of the position of an oscillating mass versus time. What is the amplitude (A) of this oscillator? 𝑡 (s) A) 4 cm B) 2 cm C) 1 cm D) 0 cm Respond at pollev.com/yqiu © 2024 All rights reserved. 11/18/24  4 Poll Question 24.2 𝑥(cm) At right is a graph of the position of an oscillating mass versus time. What is the period (T), frequency (f), and angular frequency (𝝎) of this 𝑡 (s) oscillator? A) 8 s , 0.13 Hz , 0.8 rad/s B) 4 s , 0.25 Hz , 1.6 rad/s C) 8 s , 0.8 Hz , 0.13 rad/s D) 4 s , 1.6 Hz , 0.25 rad/s © 2024 All rights reserved. 11/18/24  5 Poll Question 24.3 𝑥(cm) At right is a graph of the position of an oscillating mass versus time. At which of the following times is this oscillator moving the fastest? 𝑡 (s) A) t = 2.5 s B) t = 3.0 s C) t = 3.5 s D) t = 4.0 s © 2024 All rights reserved. 11/18/24  6 Basic position equation for oscillations  The position of a mass in simple harmonic motion varies with time as 𝑥 𝑡 = 𝐴 cos(𝜔𝑡)  Here … Can also be written as: ω is the angular frequency 𝑥 𝑡 = 𝐴 cos(2𝜋𝑓𝑡) A is the amplitude 2𝜋 or 𝑥 𝑡 = 𝐴 cos( 𝑇 𝑡) © 2024 All rights reserved. 11/18/24  7 Vertical Offsets ▪ If the oscillation’s average value is not 0, then we must add a vertical offset. x 𝑥 𝑡 = 𝐴 cos 𝜔𝑡 + 𝐴 2A The oscillation gets a distance A from its average value, so the amplitude is A. A The average value is A above 0, so we must add a vertical offset of A. 0 1 3 time (t) 𝑇 𝑇 𝑇 2𝑇 2 © 2024 All rights reserved. 2 11/18/24  8 Phase Constant  When the oscillatory motion is NOT at its maximum at t = 0 s, we have to modify the equation of x(t) to include a phase constant (symbol φ or 𝜙): x(t) = A cos(ωt + φ)  Phase constant unit: radians  The phase constant can be positive, zero, or negative.  The phase constant shifts the x-t graph horizontally. © 2024 All rights reserved. 11/18/24  9 Determining the Phase Constant x A 𝑥 𝑡 = 𝐴 cos 𝜔𝑡 0 1 3 time (t) 𝑇 𝑇 𝑇 2𝑇 2 2 −A ▪ If the oscillation is at its maximum at t = 0, then the phase constant φ = 0. © 2024 All rights reserved. 11/18/24  10 Determining the Phase Constant x A positive value!! 𝑥 𝑡 = 𝐴 cos 𝜔𝑡 + 𝜑 A 𝑥 𝑡 = 𝐴 cos 𝜔𝑡 0 1 3 time (t) 𝑇 𝑇 𝑇 2𝑇 2 2 −A ▪ The phase constant φ is positive if the oscillation’s peak occurs before t = 0. (The graph is shifted to the left.) © 2024 All rights reserved. 11/18/24  11 Determining the Phase Constant x A negative value!! 𝑥 𝑡 = 𝐴 cos 𝜔𝑡 + 𝜑 A 𝑥 𝑡 = 𝐴 cos 𝜔𝑡 0 1 3 time (t) 𝑇 𝑇 𝑇 2𝑇 2 2 −A ▪ The phase constant φ is negative if the oscillation’s peak occurs after t = 0. (The graph is shifted to the right.) © 2024 All rights reserved. 11/18/24  12 Poll Question 24.4 x Which function below best describes the position of the mass on a spring as shown in the graph? Hint: you can determine the phase constant by looking at 𝑥(0) æ pö ( A. x(t) = Acos çw t - ÷ C. x(t) = Acos w t + 2p è 4ø ) æ pö æ pö B. x(t) = Acos çw t + ÷ D. x(t) = Acos çw t + ÷ è © 2024 All rights reserved. 2ø è 4ø 11/18/24  13 Method for finding phase constant  Find the value of the position x at t = 0.  Plug in t = 0 and the value of x at this time into your equation for the position as a function of time. x(t) = A cos(2πt/T + φ) x(0) = A cos(φ)  Solve for φ…and don’t forget that the value you get can be positive or negative!  Find the peak of x(t) nearest to t = 0. If that peak occurs before t = 0, then φ is positive. If that peak comes after t = 0, then φ is negative. © 2024 All rights reserved. 11/18/24  14 Poll Question 24.5 For the oscillations shown below, the period is 0.33 s. At t = 0 s, x = -1.29 cm. What is the phase constant? A. 1.9 rad B. ⎯ 0.63 rad C. ⎯ 1.9 rad D. 0.1 rad E. ⎯ 1.3 rad © 2024 All rights reserved. 11/18/24  15 Damped Oscillations  An oscillation that diminishes and eventually stops is called a damped oscillation.  In most physical situations, there is non-conservative force of some sort (e.g., friction, air resistance), which will tend to decrease the amplitude of the oscillation.  When there is damping, the amplitude of oscillation decreases exponentially with time: 𝐴 = 𝐴0 𝑒 −𝑡/𝜏 © 2024 All rights reserved. 11/18/24  16 Damped Oscillations 𝐴 = 𝐴0 𝑒 −𝑡/𝜏 𝑥 𝑡 = 𝐴0 𝑒 −𝑡/𝜏 cos(𝜔𝑡)  The constant τ is the time constant, A0 is the initial amplitude, and e is the base of the natural logarithm (e ≈ 2.718).  Although amplitude decreases exponentially with time, the period of oscillation stays roughly the same. © 2024 All rights reserved. 11/18/24  17 Time Constant & Damping Constant  For practical purposes, the time constant 𝜏 is the lifetime of the oscillation—the measure of roughly how long it takes for amplitude to decay. (A = A0/e = 0.37A0 after 𝜏.)  If 𝜏 ≫ 𝑇, the oscillation persists over many periods and the amplitude decrease is small.  If 𝜏 ≪ 𝑇, the oscillation will damp quickly.  The constant δ is the damping constant.  It is defined as δ = 1/τ 𝐴 = 𝐴0 𝑒 −𝑡/𝜏 𝐴 = 𝐴0 𝑒 −𝛿𝑡 © 2024 All rights reserved. 11/18/24  18 Method for finding τ or δ  Find the maximum value of the amplitude, A0.  Find the value of the amplitude at some later time t.  Plug in A0, t, and the later value of the amplitude into the equation for the amplitude as a function of time. 𝐴 = 𝐴0 𝑒 −𝑡/𝜏  Solve for τ (or δ, depending on the question) © 2024 All rights reserved. 11/18/24  19 Poll Question 24.6 What is the time constant for the oscillation shown in the position versus time graph below? 6 5 4 3 2 position (m) 1 0 -1 0 1 2 3 4 5 6 7 -2 -3 -4 -5 -6 time (s) A) -21.9 s B) -4.00 s C) 4.00 s D) 21.9 s E) none of the above © 2024 All rights reserved. 11/18/24  20 Poll Question 24.7 Given the spring constant k = 10 N/m, how much energy is lost after the first cycle of oscillation due to friction? 6 5 A)180 J 4 3 B) 125 J 2 C)55 J position (m) 1 0 -1 0 1 2 3 4 5 6 7 D)10 J -2 -3 -4 -5 -6 time (s) © 2024 All rights reserved. 11/18/24