Physics for the survivors 6. lecture ...after the test Goals of today: • Understand the oscillations and its relationship with the sinus function Group work!! • Groups of 2-3 people • Answer questions (Plickers) The pendulum – your favourite đ If we displace the mass out of equilibrium, it s wings Let us denote with đ the angle the current position of the mass and equilibrium Exercise 1: Show graphically the dependence of angle đ on time. The first point is given. Exercise 2: Plot the time-dependency of the rate of change of the angle đŊ. Exercise 3: Plot the dependence of the angle đ on time if the initial displacement angle is 2 times greater than in exercise 1. Exercise 4: Plot the dependence of the angle đ vs time for a pendulum moving 2x faster than in exercise 1. Exercise 1: Show graphically the dependence of angle đ on time. The first point is given. Exercise 2: Plot the time-dependency of the rate of change of the angle đŊ. Exercise 3: Plot the dependence of the angle đ on time if the initial displacement angle is 2 times greater than in exercise 1. Exercise 4: Plot the dependence of the angle đ vs time for a pendulum moving 2x faster than in exercise 1. Let's re-familiarize ourselves with 3 concepts: • Oscillation period • the time interval during which the body performs a complete oscillation and returns to its original position • symbol: T • SI unit: s (second) • Oscillation frequency • the number of oscillations per unit of time • symbol: ν (greek “nu”) vai f • SI units: Hz (hertz) or 1/s (1 over a second) • Oscillation amplitude • Maximum change in a quantity relative to equilibrium • symbol: A (sometimes) depends on the type of oscillator • Units: whatever are the units of the quantity undergoing oscillation TīŊ īŽīŊ 1 īŽ 1 T What is the quantity shown? A.Frequency B.Period C.Amplitude D.None of the above What is the quantity shown? A.Frequency B.Period C.Amplitude D.None of the above What is the quantity shown? A.Frequency B.Period C.Amplitude D.None of the above The formula for describing the oscillation đ = đ´ â sin(đđĄ) Sin(x) 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 0 0.258819 0.5 0.707107 0.866025 0.965926 1 0.965926 0.866025 0.707107 0.5 0.258819 0 -0.25882 -0.5 -0.70711 -0.86603 -0.96593 -1 -0.96593 -0.86603 -0.70711 -0.5 -0.25882 0 2*sin(x) sin(2x) How does the fluctuation graph change with A? How does the fluctuation graph change when changing đ? Fill in the table to the end and plot it! By increasing A, the amplitude of the oscillation _______________________ By increasing đ, the amplitude of the oscillation ______________________ Increasing A, the oscillation period _________________________ Increasing đ, the fluctuation period __________________________ The pendulum, a derivation. What are the forces? • Gravity (mg) • Tension The cosine part of mg is counteracted by the tension Arc length is equal to radius times angle (in radians) Length of the pendulum is constant so velocity is just the instantaneous change in the angle Acceleration is the change in velocity over time (or the second order change in position) Rearrange terms Small angle approximation A function with this property is the sine (or cosine or both). 2ī° x īŊ A sin( t īĢ īĻ ) īŊ A sin(īˇt īĢ īĻ ) īŊ T T īŊ 2ī° l g Suppose we have a pendulum of length l We displace it by an angle đ and measure that its period is T. Next we turn displace it by an angle 2đ. How will the oscillation period change? A.The period will be 2T B. The period will be 0.5 T C. The period will be 4 T D. The period will not change Very important! (for small deviations) Period and frequency are properties of a mathematical pendulum. There can be only one oscillation frequency for a pendulum of a given length. Amplitude is something that we can change almost arbitrarily (as long as the small deviation law holds). • The pendulum has an amplitude A (maximum displacement angle is A) and period T. What is the displacement after time period T? A. 0 B. A C. 2A D. 4A • The pendulum has an amplitude A and a period T. After how long has the pendulum traveled 6A? A. 3/4 T B. 1+1/4 T C. 1+1/2 T D. 2T • Are there non-conservative forces at play? • What energies are present here? • If you know the amplitude, can you tell me the energy? θ l In which position does the pendulum have the maximum potential energy? A. 1 and 3 B. 2 and 4 C. 1 and 4 D.Another answer 1 2 3 4 In which position does the pendulum have the maximum kinetic energy? A. 1 and 3 B. 2 and 4 C. 1 and 4 D. Another answer 1 2 3 4 In which position does the pendulum have the maximum total energy? A. 1 and 3 B. 2 and 4 C. 1 and 4 D. Another answer 1 2 3 4 • The pendulum has an amplitude A (maximum displacement angle is A) and period T. What is the distance travelled after time period T? A. B. 0 đđģđ¨ đđģđ C. đ¨ đđ đ đđ D. /đģđ¨ đ