Transcript T - Helios
Simple Harmonic Motion Physics 202 Professor Lee Carkner Lecture 3 PAL #2 Archimedes a) Iron ball removed from boat Boat is lighter and so displaces less water b) Iron ball thrown overboard While sinking iron ball displaced water equal to its volume c) Cork ball thrown overboard Both ball and boat still floating and so displaced amount of water is the same Simple Harmonic Motion A particle that moves between 2 extremes in a fixed period of time Examples: mass on a spring pendulum SHM Snapshots Key Quantities Frequency (f) -Unit=hertz (Hz) = 1 oscillation per second = s-1 Period (T) -T=1/f Angular frequency (w) -- w = 2pf = 2p/T Unit = We use angular frequency because the motion cycles Equation of Motion What is the position (x) of the mass at time (t)? The displacement from the origin of a particle undergoing simple harmonic motion is: x(t) = xmcos(wt + f) Amplitude (xm) -Phase angle (f) -Remember that (wt+f) is in radians SHM Formula Reference SHM in Action Consider SHM with f=0: x = xmcos(wt) t=0, wt=0, cos (0) = 1 t=1/2T, wt=p, cos (p) = -1 t=T, wt=2p, cos (2p) = 1 Min Rest Max 10m SHM Monster Phase The value of f relative to 2p indicates the offset as a function of one period It is phase shifted by 1/2 period Amplitude, Period and Phase Velocity If we differentiate the equation for displacement w.r.t. time, we get velocity: v(t)=-wxmsin(wt + f) Since the particle moves from +xm to -xm the velocity must be negative (and then positive in the other direction) High frequency (many cycles per second) means larger velocity Acceleration If we differentiate the equation for velocity w.r.t. time, we get acceleration a(t)=-w2xmcos(wt + f) Making a substitution yields: a(t)=-w2x(t) Min Rest Max 10m SHM Monster Displacement, Velocity and Acceleration Consider SMH with f=0: x = xmcos(wt) v = -wxmsin(wt) a = -w2xmcos(wt) Mass is momentarily at rest, but being pulled hard in the other direction Mass coasts through the middle at high speed Derivatives of SHM Equation Force Remember that: a=-w2x But, F=ma so, Since m and w are constant we can write the expression for force as: F=-kx This is Hooke’s Law Simple harmonic motion is motion where force is proportional to displacement but opposite in sign Why is the sign negative? Linear Oscillator Example: a mass on a spring We can thus find the angular frequency and the period as a function of m and k k ω m m T2π k Linear Oscillator Application of the Linear Oscillator: Mass in Free Fall However, for a linear oscillator the mass depends only on the period and the spring constant: m/k=(T/2p)2