SHM, Circular Motion Flashcards
[def] Period, T
Time taken for one complete circuit
UNIT: s
T=1/f
[def] Frequency, f
Number of cycles per unit second
UNIT: Hz
f= 1/T
[def] Radian
PUT CALCULATOR IN RAD MODE
a unit of measurement equal to (180/pi) degrees ~ 57.3 degrees
Equivalent to the angle subtended at the center of a circle by an arc of equal length to the radius
UNIT: rad
[def] Angular velocity, ω
The rate of change of ϴ
ie the angle swept out by the radius per second
UNIT: rad s-1
ω includs word ANGULAR (instead of linear speed = v)
Angular velocity equations (x2) simple ones!
ω = ϴ /t
ω = 2πf
centripetal force
resultant force towards the center, acting on a body moving at constant speed in a circle
Types of centripetal force (x4)
weight
tension
normal contact force
friction
Objects in circular motion are _________ because…
Objects in circular motion are ACCELERATING because their direction is always changing
(Centripetal) acceleration directed towards the center
Linear speed or velocity of an object in circular motion, v
TANGENTIAL to centripetal force
Satellites in orbit explained (x3 points)
continually falling towards earth
weight = centripetal force
Curvature of the earth = never closer to earth
[def] simple harmonic motion
SHM occurs when an object moves such that:
- Its acceleration is always directed towards a fixed point
- Acceleration is proportional to its distance from the fixed point
[acceleration is equal to negative ddisplacement]
SHM acceleration equation
a = -(ω^2)x
SHM Period, T, of an oscillating body
Time taken to complete one cycle
UNIT: s
T= 1/f = 2π/ω
SHM Amplitude, A
The maximum value of the object’s displacemend from the equilibrium position
UNIT: m
SHM T and A relationship
Time period (T) INDEPENDENT of amplitude (A)
SHM pendulum: middle of swing:
v?
a?
x?
v= max a = zero x = zero
SHM pendulum: top of one swing:
v?
a?
x?
v: zero
x: max (+ve)
a: max (-ve)
if x -ve, then a +ve
[def] phase
the phase of an oscillation is the angle (ωt + ε) in the equation x= A cos(ωt + ε)
ε = phase constant
Phase constant, ε
How much the graph is shifted.
Can calculate if other variables are known
switch from sin graphs to cos graphs
cos–> (+π) –> sin
^ v
^ v
cos
Displacement and velocity SHM equations
[given]
v= -Aωsin(ωt + ε)
x= A cos(ωt + ε)
sin and cos can be switched depending on starting point
velocity SHM equation WITH AMPLITUDE
not given in data booklet
(v^2) = (ω^2)*(A squared - x squared)
SHM total vibrational energy
Total vibrational energy=½ (m * ω^2 * A^2)
This value is constant, irrelevant to T
SHM KE equation
KE= ½ m v^2 KE= ½ m (A^2 * ω^2 * sin^2(ωt))
SHM Ep equation (potential energy)
Ep = ½ k * x^2 Ep = (½ m * ω^2)*(A^2 * cos^2(ωt))
SHM KE GPE graph: Amplitude
Max GPE = Min KE, same total energy
[Single parabola, GPE y=x^2, KE = sad face]
see notes
http://tap.iop.org/vibration/shm/305/page_46596.html
SHM KE GPE graph: Total energy
Max GPE = Min KE, same total energy
[like cos graph sin graph because the shapes repeat]
see notes
http://www.cyberphysics.co.uk/topics/shm/shmEnergy.html
[def] Free Oscillations [aka natural oscillations]
When an oscillatory system is displaced and released
- -> no external driving force once in motion
- -> frequency of free oscillations = natural frequency
[def] Damping
Amplitude of free oscillations is reduced because of resistive forces
[def] Critical Damping
and one EG
When the resistive forces on the system are just large enough to prevent oscillations occurring at all when the system is displaced and released
EG Vehicle suspensions: aim to quickly return to equilibrium
Damping: X3 types
Light Damping: amplitude gradually reduced
Over Damping: returns to equilibrium without oscillation
Critical Damping: returns to equilibrium ASAP without oscillation
Damping: impact on natural frequency
Increased damping = decreased natural frequency
THINK GRAPH:
https://i.stack.imgur.com/c0FQ1.jpg
Light damping graph equation
A-t graph
x = Ao * e^(-λt) * cos(ωt + ε)
[def] forced oscillations
a sinusoidally varying DRIVING FORCE is applied, causing the system to oscillate with the frequency of the applied force
[when a periodic force is applied]
[def] resonance
Occurs when the periodic force equals the natural frequency
This makes the amplitude of the resulting oscillations very large
Resonance
- + + +
- BAD on millennium bridge
+ GOOD for MRIs (magnetic resonance imaging)
+ GOOD for playgroud swings
+ GOOD for microwave cooking
