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L 14 - Energy in SHM LECTURE OUTLINE 1. Energy in Simple Harmonic Motion; L 14 - Energy in SHM LECTURE OUTLINE 1. Energy in Simple Harmonic Motion; 2. Simple Harmonic and Uniform Circular Motion; 3. Damped Oscillations; 4. Forced Oscillations and Resonance. 1

Project announcement • You have only 2 lab days to take distance v/s voltage Project announcement • You have only 2 lab days to take distance v/s voltage readings. • Group leaders MUST submit their windmills in the Newton’s lab at 4. 00 pm on Monday 24 th Nov. • No extension of deadline is permitted.

1. Energy in SHM Kinetic energy Potential energy The energy required to extend the 1. Energy in SHM Kinetic energy Potential energy The energy required to extend the spring by a displacement x is stored in the spring and it is called spring potential energy Hence the total SHM energy is given by: x 3

How total energy, kinetic energy, and potential energy depend on the displacement x Energy How total energy, kinetic energy, and potential energy depend on the displacement x Energy diagrams for SHM: Maximum KE is Maximum PE is Proof 1: show that 4

Example 1 A 0. 500 kg object, attached to a vertical spring of k Example 1 A 0. 500 kg object, attached to a vertical spring of k = 0. 200 k. N/m, is initially in equilibrium. The object is then pulled downwards over a distance of 2. 00 cm and released so that it undergoes SHM. When the object has travelled halfway between the equilibrium position and the release point calculate: a) velocity b) net acceleration c) total SHM energy d) kinetic energy e) spring potential energy 5

2. Simple Harmonic and Uniform Circular Motions There is a strong similarity between SHM 2. Simple Harmonic and Uniform Circular Motions There is a strong similarity between SHM and uniform circular motion 6

P has a circular trajectory about O. The shadow of P on the screen P has a circular trajectory about O. The shadow of P on the screen undergoes SHM. 7

3. Damped oscillations SHM is more realistic if it includes friction, air resistance, etc. 3. Damped oscillations SHM is more realistic if it includes friction, air resistance, etc. (damping). The equation of motion becomes: Proof 2: show that the total rate of energy loss is equal to the power dissipated through friction: 8

b is damping coefficient 9 b is damping coefficient 9

Examples of damping • A car’s suspension system; • Shock absorbers on a mountain Examples of damping • A car’s suspension system; • Shock absorbers on a mountain bike; • Friction slowing down a pendulum; • Electromagnetic damping. 10

 11 11

4. Forced Oscillations and Resonance 12 4. Forced Oscillations and Resonance 12

Periodic Driving Force Apply a periodic force to a damped system undergoing SHM at Periodic Driving Force Apply a periodic force to a damped system undergoing SHM at its natural frequency. Consider a periodic force And the equation of motion becomes: 13

 0 0. 5 1. 0 1. 5 Relative driving frequency ω/ω0 2. 0 0 0. 5 1. 0 1. 5 Relative driving frequency ω/ω0 2. 0 14

Web Interactive. Example: of driven simple harmonic oscillator: http: //www. upscale. utoronto. ca/General. Interest/Harrison/Flash/Class. Web Interactive. Example: of driven simple harmonic oscillator: http: //www. upscale. utoronto. ca/General. Interest/Harrison/Flash/Class. Mechanics/Drive n. SHM/Driven. SHM. html 15

Resonance • Resonance (highest amplitude peak) occurs when the driving frequency, ω, equals the Resonance • Resonance (highest amplitude peak) occurs when the driving frequency, ω, equals the natural frequency, ω0; • Amplitude increases as damping decreases; • The amplitude vs. frequency curve (resonance curve) broadens as damping increases; • The shape of the resonance curve depends on the damping coefficient b. 16

Example 2 A free swing is a lightly damped system. If a girl starts Example 2 A free swing is a lightly damped system. If a girl starts swinging from a amplitude A = 50. 0 cm with a frequency 0. 500 Hz, calculate the maximum damping force if the damping coefficient is 58. 3 N s m-1 17

Examples of resonance • Bridge collapse due to turbulent winds; • Collapse of buildings Examples of resonance • Bridge collapse due to turbulent winds; • Collapse of buildings following earthquakes; • Nuclear Magnetic Resonance (NMR); • Magnetic Resonance Imaging (MRI); • Electron Spin Resonance (ESR); • Children on swings; • Singing or playing a musical instrument. 18

Shattering a wine glass: http: //www. acoustics. salford. ac. uk/feschools/waves/wine 1 video. htm Volgograd Shattering a wine glass: http: //www. acoustics. salford. ac. uk/feschools/waves/wine 1 video. htm Volgograd Bridge on May 2010: http: //www. youtube. com/watch? v=4 Pnl. DAHW 9 co&feature=related Tacoma Narrows Bridge on July 1940: http: //www. youtube. com/watch? v=Az 503 VJ 6 k. Hw&feature=related 19

 • • LECTURE CHECK LIST Understand PE and KE and total energy for • • LECTURE CHECK LIST Understand PE and KE and total energy for SHM; Be able to perform calculations to find the amplitude, period, frequency and angular frequency of objects performing SHM in a variety of situations; Be able to perform calculations involving PE, KE, acceleration velocity, displacement and energy; Understand the links between SHM and circular motion Understand the notions of natural frequency of an oscillating system and the driving frequency of a system Be able to define resonance Give some real examples of resonance Understand what is meant by damping and be able to give examples in which damping occurs 20

 READING: Adams and Allday, 3. 32 3. 36 Serway, Chapter 13, Sections 13. READING: Adams and Allday, 3. 32 3. 36 Serway, Chapter 13, Sections 13. 1 13. 4 Answers: Example 1. a) 0. 346 m/s; b) 4. 00 m/s²; c) 4. 00 x 10 ²־ J; d) 3. 00 x 10 ² J; e) 1. 00 x 10 ²־ J. ־ Example 2. 91. 6 N 21