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What is oscillatory motion?. Oscillatory motion occurs when a force acting on a body is proportional to the displacement of the body from equilibrium. F x The Force acts towards the equilibrium position causing a periodic back and forth motion. What are some examples?. Pendulum
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What is oscillatory motion? • Oscillatory motion occurs when a force acting on a body is proportional to the displacement of the body from equilibrium. Fx • The Force acts towards the equilibrium position causing a periodic back and forth motion.
What are some examples? • Pendulum • Spring-mass system • Vibrations on a stringed instrument • Molecules in a solid • Electromagnetic waves • AC current • Many other examples…
What do these examples have in common? • Time-period, T. This is the time it takes for one oscillation. • Amplitude, A. This is the maximum displacement from equilibrium. • Period and Amplitude are scalers.
Forces • Consider a mass with two springs attached at opposite ends… • We want to find an equation for the motion. • How should we start? • Free-body diagram!!
Free body diagram FSpring2 FGravity FSpring1
Fnet = ma • Fnet = Fg + Fs1 + Fs2 = ma • Fnet = Fhorizontal + Fvertical • Let us assume the mass does not move up and down Fvertical = 0 • So, Fnet = Fhorizontal = FS-horizontal(1+2) • Thus, ma = m(d2x/dt2) = -kx
ma = m(d2x/dt2) = -kx • Let k/m = (d2x/dt2) + x = 0 • This is the second order differential equation for a harmonic oscillator. It is your friend. It has a unique solution…
Simple Harmonic motion • The displacement for a simple harmonic oscillator in one dimension is… x(t) = Acos(t + • is the angular frequency. It is constant. • is the phase constant. It depends on the initial conditions. • What is the velocity? • What is the acceleration?
Velocity: differentiate x with respect to t. dx/dt = v(t) = -Asin(t + ) • Acceleration: differentiate v with respect to t.dv/dt = a(t) = -Acos(t + )
X(t)=Acos(t + ) x Xo A t T
a(t) = -Acos(t + ) a t A t ao T
Using data • The accelerometer will give us all the information we need to confirm our analysis • We can measure all the parameters of this particular system and use them to predict the results of the accelerometer.
What can we measure without the accelerometer? • The mass, m • Hooke’s constant, k • That’s all! • T = 2/= 2m/k)1/2(Recall = k/m) • Everything else depends on the initial conditions. What does this tell us? • The time period, T, is independent of the initial conditions!
Energy • The system operates at a particular frequency, v, regardless of the energy of the system. v = 1/T = 2(k/m)1/2 • The energy of the system is proportional to the square of the amplitude. E = (1/2)kA2
Proof of E=(1/2)kA2 • Kinetic Energy = (1/2)mv2 V = SIN(t + ) (1/2)MASIN2 (t + ) • Elastic potential energy U=(1/2)kx2 x = Acos(t + ) U (1/2)kA2cos2./(t + )
E = K + U = (1/2)kA2[sin2 (t + ) + cos2 (t + )] = (1/2)kA2
Damping • Simple harmonic motion is really a simplified case of oscillatory motion where there is no friction (remember our FBD) • For small to medium data sets this will not affect our results noticeably.
for the rest of class... • We are going to find k and m and compare to the results of the accelerometer
Some cool oscillatory motion websites • http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=236 • http://www.kettering.edu/~drussell/Demos/SHO/mass.html • http://farside.ph.utexas.edu/teaching/301/lectures/node136.html • http://www.physics.uoguelph.ca/tutorials/shm/Q.shm.html
