Simple Harmonic Motion. Simple Harmonic Motion

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1 Siple Haronic Motion Basic oscillations are siple haronic otion (SHM) where the position as a function of tie is given by a sign or cosine Eaples of actual otion: asses on springs vibration of atos in a crystal lattice vibration of strings of usical instruents pendula (sall aplitude) bobbing of floating objects More general eaples: pressure in a sound wave electric field strength is a radio wave Many otions are described very well by SHM for sall aplitudes Phy1 - Spring 3 1 Siple Haronic Motion SHM is the solution to all probles which are described by differential equation of the for: = ω dt d The second derivative of with respect to tie = (soe negative constant)() The solution to this equation can be epressed as:? is the angular frequency (radians/s), and it coes fro the differential equation the aplitude ( ) and the phase constant (f) are not deterined by the differential equation, and ust be fied fro the initial conditions = cos( ω t + φ) Phy1 - Spring 3 1

2 Siple Haronic Motion Eaple: A kg ass on a frictionless air track is connected to a spring with spring constant 8N/. It is displaced fro its equilibriu position and released. What is the period of its oscillation? Start fro F=a get the equation into the for a=-c, where C is soe positive constant set C=ω use the relationships between angular frequency and period or frequency: f = ω / π T = 1 / f = π / ω Phy1 - Spring 3 3 Siple Haronic Motion Note that in the previous eaple, there was insufficient inforation given to deterine the aplitude and phase constant. Eaple : A kg ass on a frictionless air track is connected to a spring with spring constant 8N/. It is displaced fro its equilibriu position by 1 c and released. What is the aplitude of its oscillation? What is its aiu speed? Use steps in previous eaple to deterine w. Use relationship between position and velocity for SHM: = cos( ω t + φ) v = ω sin( ωt + φ) v = ω Phy1 - Spring 3 4

3 Siple Haronic Motion Note that in the previous eaple, there was still insufficient inforation given to deterine the phase, but the aplitude and aiu velocity could be deterined without knowing the phase constant. Eaple 3: A kg ass on a frictionless air track is connected to a spring with spring constant 8N/. At tie t=3s, it is oving forward through its equilibriu position with a speed of.75/s. What is its equation of otion? Where is it at t=5s? What is its velocity at t=5s? What is its acceleration at t=5s? When will it be at its aiu displaceent? First get the equation of otion, then the other questions are easy to answer. Phy1 - Spring 3 5 Siple Haronic Motion Solution to previous eaple: Displaceent v at t=3s Velocity at t=3s (c) v (c)/s at t=5s -8-1 v at t=5s = ( 37.5c)cos((.rad/s) 1.9rad) v = ( 75c/s) sin((.rad/s) 1.9rad) Phy1 - Spring 3 6 3

4 Displaceent Siple Haronic Motion Useful facts about cosines and sines: = cos( ω t + φ) The period T is the tie between aia. The tie between zero crossings is 5 4 half of the period. 3 T The aiu peak-to-peak difference is twice the aplitude. ω t +φ (c) is the phase. It has a value of T/,π,- π,4 π, at aiu displaceent, π,- π,3 π,-3 π,5 π, at iniu displaceent, π/,- π /,3 π/,-3 π/,5 π/,-5 π/ at zero displaceent Phy1 - Spring 3 7 Siple haronic otion More useful facts about sines and cosines: cos( ω t + φ) = sin( ωt + φ + π / ) cos( ω t + φ) = cos( ωt + φ + π ) d dt d dt ( cos( ω t + φ) ) = ω sin( ωt + φ) ( cos( ω t + φ) ) = ω cos( ωt + φ) Phy1 - Spring 3 8 4

5 Consider a vertical spring: Siple Haronic Motion F s = k W = g relaed = New g k equilibriu / F = Fs W = k + g = k( ) k a = ( ) = cos( ω t + φ) Phy1 - Spring 3 9 Oscillates about new equilibriu point with sae frequency as for the horizontal case Springs and asses Eaple: When a g ass is attached to a spring, the new equilibriu point is 6. c below the original relaed point of the spring. If it is lowered another 4.c and released What is its period of oscillation? What is its aiu kinetic energy? What is its acceleration at the lowest point of its cycle? Note that the inforation about the equilibriu point can be used to deterine the spring constant What happens to the period if two springs are used in parallel? Phy1 - Spring 3 1 5

6 Torsion Oscillator A torsion spring is the rotational equivalent of a linear spring: F = k Force = -(spring constant) (linear displaceent) τ = κθ Torque = -(tortional spring constant) (angular displaceent) τ κ = Iα α = I d θ κ = θ dt I θ θ θ = θ cos( ωt + φ) κ ω = I Phy1 - Spring 3 11 Torsion Oscillator Eaple: A torsion oscillator is ade by suspending a unifor brass rod fro its center by a copper wire. The rod is 1 c long, and has a ass of g. It is found to oscillate with a period of 3.5 seconds. What is the torsion spring constant of the wire? (Recall that the rotational inertia of a solid rod about its center is I=(1/1)ML.) Phy1 - Spring 3 1 6

7 Siple Pendulu A pendulu is considered siple if it can be approiated by a point-like ass at the end of a assless string or rod: F T sinθ = a g sinθ cosθ a = a = g tanθ = a g L T sin θ T cosθ g θ L a g L looks a lot like a = k ω = g L Phy1 - Spring 3 13 Sall angle approiations Very useful approiations for sall angles: tanθ sin θ θ for θ <<1radian theta(deg) theta(rad) sin(theta) tan(theta) % diff % % % Phy1 - Spring

8 Physical Pendulu Mass is distributed (not all localized in bob ) pivot g h center of ass τ = ghsin θ Torque due to gravity = -(weight)(distance pivot-c..)sin(angle fro vertical) τ = Iα α gh = sin θ I gh θ I ω = gh I Phy1 - Spring 3 15 Physical Pendulu Eaple: A unifor disk of radius c is supported on a horizontal ais through a hole which is halfway between the center and the edge. What is the period of sall oscillations? What is the length of a siple pendulu with the sae period? Phy1 - Spring

9 Displaceent Velocity Oscillations: characteristics The velocity is zero when the displaceent is at its iniu or aiu (c) v(c)/s The speed is aiu as the oscillator passes through its equilibriu point Energy goes back and forth between kinetic energy and elastic potential energy, but the su (total echanical energy) is constant Phy1 - Spring