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NOTES 20 - Topic 2 - Oscillations*

NOTES 20 - Topic 2 - Oscillations* - -----------------------------------------------------------------------------. REVIEW of NOTES 18 & 19: • Displacement (x, q ) - the distance of the oscillating object from its rest position; - symbol = x; - unit = q ( angle in radians);.

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NOTES 20 - Topic 2 - Oscillations*

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  1. NOTES 20 - Topic 2 - Oscillations* ------------------------------------------------------------------------------ REVIEW of NOTES 18 & 19: • Displacement (x, q) - the distance of the oscillating object from its rest position; - symbol = x; - unit = q (angle in radians); • Amplitude (xo , qo) - maximum displacement of the oscillating object from its rest position; - symbol = xo - unit = qo (angle in radians); • Period (T) - the time needed for one complete oscillation (B-->C-->B); - symbol = T; - unit = seconds (s); • Frequency (¦) - number of oscillations that occur in one second; - symbol = ¦ ; - unit = oscillations per second = s-1 = Hertz (Hz);

  2. •Angular Frequency (w) - the number of radians moved by an object in SHM in one second: w = 2 p¦ Remember... ¦ = 1 / T ... So... w = 2 p¦ = 2 p / T Ifw = 2 p rad s-1, then the object completes one oscillation per second and has f = 1 Hz; • Angular Speed (w) - the distance, in radians, moved by an object in SHM in one period: v = Dx / Dt = Dq / Dt = 2 p / T ; But...this is the same as... w = 2 p / T ; So... angular speed = angular frequency = w ; Both are measured in radians per second... rad s-1 ;

  3. •Angular Acceleration - the acceleration of an object in true SHM is directly related to its displacement; a is greatest at max θ; α = 0 at θ=0; α = - k x, where k is a constant; Mathematical analysis shows that k = w2; So... α = - w2 x ... this is the mathematical definition of SHM;

  4. •SHM Equations: x = xo cos w t v = - vo sin w t a = - ao cos w t Remember...initial cos or sin is determined by the shape of the displacement-time graph;

  5. 4.2.1 Force and Energy in SHM

  6. z• Force - Hooke’s Law states that the force exerted on or by a spring equals the stretch of the spring multiplied by the spring constant... Fe = -kx ... measured in Newtons; • Angular Force - the force exerted on an object in true SHM is directly related to its displacement: F = - k x ... where k is a constant; Remember Newton’s 2nd Law... F = ma ; So... - k x = ma = m (- w2 x) ... w = √(k / m) Remember from 4.1.3... w = 2 p / T ... Therefore... √(k / m)= 2 p / T ... Finally... T = 2π√(m / k)

  7. • Potential Energy - whenever the spring is stretched, the potential energy is given by this equation: PEe = (1/2) k x2 ... where...k is the spring constant (Nm-1); x is the distance the spring is stretched; PEe measured in Joules (J); - What is the potential energy of a spring with k = 10.0 Nm-1 that is stretched 0.50 m horizontally? Given: Unknown: Equation: • Kinetic Energy- the energy of the mass when it is in motion: KE = (1/2) mv2; measured in Joules (J); - For the spring above, what would be the speed of a 1.00 kg bob, at x=0, when released from a strech of 0.50 m on a frictionless surface? Given: Unknown: Equation:

  8. • Energy Transformation- the total energy of system remains constant and is transferred from PE to KE to PE to KE every time the mass moves through an oscillation; Etot = KE + PE

  9. - According to the Law of Conservation of Energy, the maximum KE is determined by the total PE, such that: KEmax = PEtot (1/2) m v2 = (1/2) k x2 v2 = k x2 / m v = √(k / m) (x) Remember... w = √(k / m) So... vo = ω xo - What is the angular frequency (speed) of the mass on the spring with with the speed determined in the last example when xo = 0.25 m ? Given: Unknown: Equation:

  10. 4.2.2 SHM Energy Equations • Energy Transformation- the total energy of system remains constant and is transferred from PE to KE to PE to KE every time the mass moves through an oscillation; • Total Energy (Etot)- must be the energy supplied by the stretch of the spring (PE): Etot = (1/2) k x2 Remember... ω2 = k / m ... k = m ω2 So... Etot = (1/2) m ω2 x2

  11. • Potential Energy (PE)- the energy supplied by the stretch of the spring (PE): PE = (1/2) k x2 = (1/2) m ω2 x2 • Kinetic Energy (KE)- must be the energy of the moving mass: KE = (1/2) m v2 Remember... vo = ω√(xo2 - x2), So... KE = (1/2) m ω2 (xo2 - x2) When xo = 0 (ie., at equilibrium), KE = (1/2) m ω2 x2

  12. Sample Problem: Given: Unknown: Equation:

  13. Summary of Equations Linear SHM F = ma F = mα = m (- w2x) KE = (1/2) mv2 KE = (1/2) m ω2 (xo2 - x2) PE = (1/2) kx2 PE = (1/2) m ω2 x2

  14. 4.2.3 Natural Frequencies and Forced Oscillations

  15. • Natural Frequency - All objects that oscillate through a SHM, like a swing, have their own natural frequency; - If you give a swing one push, it will move back and forth at its natural frequency (determined by length and arrangement of mass) and eventually stop...this is dampened harmonic motion; - Forces of friction (with the air, inside the rope or chain, etc.) cause the dampening and the amplitude of the SHM decreases until the motion stops;

  16. • Forced Oscillation - If the rider of the swing pushes against the ground at the right time (everyone knows how to do this...it's almost instinctual), the amplitude of the swing will increase while the frequency remains essentially constant...this is a forced oscillation; - The same forced oscillation is caused by a parent pushing the child at the high point of the backswing...amplitude of swing increases up to a point; - The push-with-the-feet and the push-by-a-parent are called the drivers of the forced oscillation; - The change in the amplitude of the SHM during forced oscillation is determined by: 1. the frequency of the driving force; 2. the natural frequency of the SHM; 3. the amplitude of the driving force; 4. phase difference between driver and natural frequency; 5. the dampening forces on the system;

  17. 4.3.4 Forced Frequency and Amplitude 4.3.5 Resonance - The factor above that most affects the amplitude of a forced oscillation is #4 (phase difference between driver and natural frequency); - Even a small driving force, one just large enough to overcome the dampening forces, if applied in phase (ie., at the natural frequency of the SHM), can cause a substantial, even catastrophic, increase in the amplitude of oscillation; - It is possible to break a glass with the human voice if the driving voice attains the exact frequency as the vibrating glass...see Myth Busters video; - When the driver is exerted at the natural frequency of the SHM and the maximum amplitude is attained, this the called resonance; - At the resonant frequency, the object in SHM is said to resonate; - While it is possible to achieve resonance in glasses, it is also possible to achieve resonance in telephone wires (Aolian Harp), buildings, and bridges (see Mechanical Universe - RESONANCE); - Because it is possible to cause resonance to occur in structures, it is obvious that these structures have to be critically dampened to prevent their destruction by natural phenomena;

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