# Simple Harmonic Motion

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1 Simple Harmonic Motion Restating Hooke s law The equation of motion Phase, frequency, amplitude Simple Pendulum Damped and Forced oscillations Resonance

2 Harmonic Motion A lot of motion in the real world does not fit some of our earlier models (linear or circular motion, uniform acceleration). Many phenomena are repetitive or oscillatory. The most basic oscillation is the block/spring system:

3 x distance from equilibrium. M Hooke s Law: F k F m a d x dt k m k m x k m x x x x(t) x equilibrium What is the equation of motion, x(t)?

4 does 1 x ( t) x0 + v0t + g t work? does x( t) A cos ( ω t + φ ) work?

5 Is the motion of a bouncing ball described by SHM? Is this motion SHM?

6 Quick Quiz Which of the following functions also describe simple harmonic motion? A) x(t) e -t B) x(t) tan (ωt) C) x(t) sin (ωt) D) All of the above E) None of the above

7 Generic Simple Harmonic Oscillator d x dt ω x(t) x( t) A cos( ω t + ϕ ) but what do A, ω, φ mean?

8 x A cos( ωt + ϕ) A amplitude of the motion maximum value of x (x ranges from +A to A) φ phase constant -- gives the initial position at t0 ω x( t 0) xo A cos ϕ angular frequency and is related to the period of the motion [radians/sec]

9 x A cos ( ω t + ϕ) This quantity in brackets is an angle, called the phase angle (or just the phase) of the motion xa at phase angles of multiples of π radians:..., 4π, π, 0, π, 4π, x(t) t

10 x A cos ( ω t + ϕ) φ phase constant -- gives the initial position at t 0 x t

11 The period T is the time needed to repeat the cycle: At t 0, position is x 0 A cos φ One period later, at t T, position is x A cos (ωt + φ) In time T, the phase angle (ωt+φ) has increased by (π) radians ωt π radians ω π / T

12 Two types of frequency! Angular frequency: ω π / T in [rad/s] Frequency: f 1/ T in [Hz]or [s -1 ] cycles per second

13 Quick Quiz A mass hung from a spring is made to oscillate vertically. The mass goes from its highest position to its lowest position in seconds. What is the angular frequency of this oscillation? A) 1.5 rad/s B) 3 rad/s C) 1.5 Hz D) 3 Hz

14 Quick Quiz What is the difference between these oscillators? A) Phase constant B) Amplitude C) Angular Frequency D) More than one of the above E) None of the above

15 Quick Quiz What is the difference between these oscillators? A) Phase constant B) Amplitude C) Angular Frequency D) More than one of the above E) None of the above

16 Example What is the ratio of the periods for these two oscillators?

17 Example The block is at its equilibrium position and is set in motion by hitting it (and giving it an initial velocity) at time t 0. Its motion is SHM with amplitude 5 cm and period seconds. Write the function x(t). M v 0 x

18 Quick Quiz The figure shows four oscillators at t 0. Which one has the phase constant

19 Quick Quiz M v 0 x 0 The block is at x 0 +5 cm, with positive velocity v 0, at time t 0. Its motion is SHM with amplitude 10 cm and period seconds. If x(t) A cos (ωt + φ), the phase constant f should be: A) 0 o B) 30 o C) 60 o D) -30 o E) -60 o

20 Velocity and Acceleration ) cos( ) ( ) sin( ) ( ) cos( ) ( x t A dt dv t a t A dt dx t v t A t x ω ϕ ω ω ϕ ω ω ϕ ω a(t) ω x(t) max max : Note ω ω A a A v

21 Position, Velocity and Acceleration x(t) t v(t) t a(t) t

22 Question: Where in the motion is the velocity largest? Where in the motion is acceleration largest?

23 Quick Quiz An object moves with simple harmonic motion. If the amplitude and the period are both doubled, the object s maximum speed is A. quartered. B. halved. C. quadrupled. D. doubled. E. Unchanged.

24 Example An object oscillates with SHM along the x-axis. Its displacement from the origin varies with time according to the equation: x(t)(4.0m)cos(πt+π/4) where t is in seconds and the angles in radians. a) determine the amplitude b) determine the frequency c) determine the period d) its position at t0 sec e) calculate the velocity at any time, and the v max f) calculate the acceleration at any time, and a max

25

26 Back to Hooke s Law: F k x d dt x ω x where ω k m Generic SHO Hooke s SHO ω π / T T π m k

27 Example A 7.0 kg mass is hung from the bottom end of a vertical spring fastened to the ceiling. The mass is set into vertical oscillations with a period of.6 s. Find the spring constant (aka force constant of the spring).

28 Example A block with a mass of 00g is connected to a light spring with a spring constant k5.0 N/m and is free to oscillate on a horizontal frictionless surface. The block is displaced 5.0cm from equilibrium and released from rest. a) find the period of its motion b) determine the maximum speed of the block c) determine the maximum acceleration of the block

29

30 Simple Pendulum Gravity is the restoring force taking the place of the spring in our block/spring system. L Instead of x, measure the displacement as the arc length s along the circular path. Write down the tangential component of Fma: θ T Restoring force But s Lθ mg mg sin θ ma sin θ t m d s dt s mg mg sin θ d θ dt g L sin θ

31 SHM: Simple pendulum: d x dt d θ dt ω x g L x sin θ The pendulum is not a simple harmonic oscillator! However, take small oscillations: sinθ θ (radians) if θ is small. Then d θ g sin θ dt L g θ L

32 For small θ : This looks like d θ dt g θ L d x ω x, with angle θ instead of x. dt The pendulum oscillates in SHM with an angular frequency ω g L and the period is π/ω, etc.

33 Question: How high is the ceiling?

34 Quick Quiz What happens to the period of a simple pendulum if the mass is doubled? a) Increases by a factor of two b) Increases by a factor of c) Stays the same d) Decreases by a factor of two e) Decreases by a factor of

35 Quick Quiz Pendulum clocks ( grandfather clocks ) often have a swinging arm with an adjustable weight. Suppose the arm oscillates with T1.05sec and you want to adjust it to 1.00sec. Which way do you move the weight??

36 Quick Quiz A simple pendulum hangs from the ceiling of an elevator. If the elevator accelerates upwards, the period of the pendulum: a) Gets shorter b) Gets larger c) Stays the same

37 Energy of a SHO Recall for a spring: E Tot K + U 1/mv + 1/kx But we know that: x A cos(ωt+φ) v -A ω sin(ωt+φ)

38 Energy in SHM M Look again at the block & spring ( ) ( ) energy) mechanical (total constant a ) ( cos ) ( sin ) ( cos ) ( sin! ) ( cos ) ( sin ka t t ka t k t m A U K k t ka kx U t A m mv K φ ω φ ω φ ω φ ω ω φ ω φ ω ω max 1 ) ( 1 1 mv A m ka E tot ω

39 Quick Quiz Four springs have been compressed from their equilibrium position at x 0 cm. When released, they will start to oscillate. Rank in order, from highest to lowest, the maximum speeds of the oscillators. A. c > b > a d B. c > b > a > d C. d > a > b > c D. a d > b > c E. b > c > a d

40 Quick Quiz Suppose you double the amplitude of the motion, what happens to the maximum speed? a) Doubles b) Four times larger c) Doesn t change

41 Quick Quiz Suppose you double the amplitude of the motion, what happens to the maximum acceleration? a) Doubles b) Four times larger c) Doesn t change

42 Quick Quiz Suppose you double the amplitude of the motion, what happens to the the total energy? a) Doubles b) Four times larger c) Doesn t change

43 Energy Since we know the total energy of a SHM, we can calculate the or displacement velocity at any point in time: E Tot 1/kA K+U 1/mv + 1/kx So, if x0, all E is in kinetic, and v is at max. If xa, all E is in potential, and v is zero.

44 Example A 100g block is attached to a spring with spring constant 0 N/m. When the block is5cm from the equilibrium position, it moves with a speed of 1.5m/s. a) What is the total energy of the system? b) What is the amplitude of the oscillations?

45 Example A 500g block on a spring is pulled 0cm and released. The motion has a period of 0.8s. What is the velocity when the block is 15.4cm from the equilibrium position?

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