What are the conditions for simple harmonic motion?
Acceleration must be proportional to its displacement from the equilibrium point
It must act towards the equilibrium point
(frictionless)
What is the constant of proportionality linking acceleration and x?
-w^2 or -k/m
F resultant equation for SHM
Fres = -kx
Acceleration for SHM
a = (-k/m)x = -(w^2)x
What is x as a function?
x = Acos(wt) = Asin(wt)
w = s^-1
A = m, amplitude
x = m
t = s
Equation for max speed in SHM
Max speed = wA
w = s^-1
A - m
How to calculate maximum acceleration?
Max acceleration = (w^2)A
graphs of x(t), v(t), a(t)
v(t) equation
x(t) = Acos(wt)
(dx/dt) = -wAsin(wt)
a(t) equation
(dv/dt) = (d2x/dt2) = -(w^2)Asin(wt)
When displacement is at the max
v = 0, a is at max
acceleration still occurs as there is still force acting on the object
if x is positive ...
if x is negative ...
extension
compression
(if there is no friction) in SHM the object will continuously go backwards and forwards
Is one oscillation completed in the same amount of time regardless of amplitude?
Yes, though in small oscillations there is less to travel there is also less speed and the opposite for large oscillations
Time period equation for SHM system
T = 2pi/w =2pi(square root(m/k))
Frequency equation for SHM system
1/T = w/2pi
angular frequency equation
w = 2pif
small angle approximation for sinx
sin x ~ x (radians)
small angle approximation for cos x
1 - ( x^2 /2 )
What happens when v = 0/ v is greatest
when v = 0 the acceleration is at its greatest
when v is at the greqatest the acceleration = 0 , as at this point
when v = 0 x is the greatest and so acceleration is greatest
when v is the greatest x = 0 so acceleration = 0
What happens at the max/min displacement?
The object is stationary but accelerating
largest possible acceleration at extreme points
as F is proportional to x so at extreme points the force is at its greatest
What is time period irrespecitve of?
amplitude
What happens to the time period if you increase the mass?
increas time period so greater oscillation
What happens when there are 3 springs in series?
spring constant is k/3 so time period is greater so there is a slower/greater oscillation
when they are in parallel it is 3k so time period is less so oscillation is smaller
Why SHM?
Approximately same for small displacements for almost anything, because any general f(x) is approximately linear for small x
As any complex mostion, too complex to analyse can be decomposed into sine and cosinr SHM oscillations which can be analysed
The graph for potential energy and kinetic energy against displacement, for a SHM system?
Equation for total energy in SHM
Etotal = KE + EPE
Since no external object does work on it, we'd expect the system's energy to be unchanging
When is max KE and max EPE?
When is min KE and min EPE?
max KE when v is greatest
max EPE when x is greatest
0 KE when v is 0
0 EPE when x is 0
equation for max EPE in SHM
PEmax = 1/2 x k x A^2
Total energy equation for SHM
1/2 x k x A^2
What does zero PE imply?
max KE
Does EPE and KE vary during an oscillation?
Between zero and minimum values
What is the equation for the time period of a simple harmonic pendulum?
T = 2pi(square root(l/g))
g = ms^-2
l - m
T - s
Free vibrations
The frequency of a system tends to vibarte at in a free vibration, called the natural frequency
Damping
occurs when an opposing force dissipates energy to the surroundings
Critical damping
reduced the amplitude to zero in the quickest time
overdamping
when the damping force is too strong and it returns to equilibrium slowly without oscillation
underdamping
when the damping force is too weak and it oscillates with exponentially decreasing amplitude
what happens to a vibration with gretaer damping?
for a vibration with gretaer damping, the amplitude is lower at all frequencies due to the greater energy losses from the system
the resonant peak is broader because of the damping
Forced vibrations
a driving force causes the system to vibrate at a different frequency
for higher driving frequencies, the phase difference between the driver and the oscillations rise to pi radians
for lower frequencies, the oscillations are in phase with the driving force
when resonance occurs which is when it most efficiently transfers energy to the system, the phase difference will be pi/2 radians
What are some implications of resonance in real life?
Implications of resonance include that soilders must break stop when crossing bridges and vehicles must be designed so there are no unwanted vibrations
What is meant by forced vibrations?
A system made to oscillate by a periodic external force
Epe equation
0.5ke^2 = 0.5Fe
A limitation that applies to the pendulum when the time period equation for a pendulum is used
Amplitude is smaller