Periodic Motion
Defination
Vibration/Oscillation repeats itself over and over
Parameters
Oscillation/Vibration: Motion that repeats itself with no net displacement
Equilibrium position/Rest position: Point that the object oscillates around
Displacement: How far the mass is from the equilibrium point (x)
Maximum displacement: How far the mass moves from the equilibrium position (x max occurs at A)
Amplitude (A): Distance from the equilibrium point to the maximum displacement
Cycle: A complete to and fro motion
Period (T): Time taken to complete one cycle (units: s)
Frequency (f): Number of cycles completed in one second (units: Hz/s^-1)
Simple Harmonic Motion
Any motion which the restoring force is proportional to displacement
Restoring force: Force that pushes/pulls the mass back to equilibrium
Hooke's Law
Restoring force of spring= F(spring) = F(elastic) = -kx (applied force is in the opposite direction of spring force)
k: "Stiffness" of a spring (units: N/m)
At maximum displacement, U(spring) is maximum, KE is minimum @ At equilibrium position, U(spring) is minimum, KE is maximum
U(spring) = kx^2 / 2, KE = mv^2 /2
x(t)=Acos(wt) [position function over time]
v(t)=d/dt [x(t)]=-wAsin(wt) [velocity function over time]
a(t)= d/dt [v(t)]=-w^2 Acos(wt) [acceleration function over time]
Pendumlum
Motion of a pendulum
When the pendulum is pulled back, weight x component gets larger while y component gets smaller
The greater the displacement, the larger the restoring force
For small displacement, pendulum motion is simple harmonic
Energy of a pendulum
At maximum displacement, v=0, a=largest, E=U @ At equilibrium, v=largest, a=0, E=KE
Energy is conserved
Parameters
Period: Time taken for the pendulum to swing from one side to the other and back again
Frequency: Number of complete cycles in one second
Physical Pendulum
(d^2 (0))/(dt)^2 = torque/I = -Mgl(0)/I
T=2(pi)sqrt(L/g)
Depends on the length and free fall acceleration
For small amplitudes, period does not depend on the amplitude
Damped Harmonic Motion
Amplitude of any real oscillating spring slowly decreases
Damping is due to friction and air
Forced Vibrations/Resonance
When the system is in motion and left alone, it will vibrate to its natural frequency (f0)
When outside force is constantly applied, it creates forced vibrations
Amplitude of forced vibration depends on difference between f and f0