SHM (test) Flashcards
what does 1 rad roughly equal to
57 degrees
Angular speed
the angle of an object rotates through per second
Defined as angle / time
units = rad s^-1
equation for angular speed
w = theta / t
arc lenth equation
arc lenth = r x theta
equation for linear velcoty, v
v = arc lenth / t
v = r x theta / t
v / r == theta / t
equation for angular speed using linear speed
w = v / r
frequency
f = 1 / t
w = 2pi / t
w = 2pi x f
acceleration
the rate of change of velocity
centipetal acceleration
objects travelling in a circle are accelerating since their velocity is changing due to the fact that the the direction the object is travelling in is changing
therefore an object would be accelerating even though its not going any faster
equation for l
l = v x change in t
change in velocity equation
change in v = Vb - Va
l / r == change in v / Va == vhange in v / v
v x change in t / r == change in v / v == change in v / change in t === v x v / r ===
v^2 / r
a = vv / r as a = vhange in v / change in t
intergrate w into a = vv / r
a == ww x r
centripetal force
newtons 1st law states that an objects velocity will not change unless there is an external force acting upon it
since objects moving in a circle have acceleration, there must be a force causing the acceleration
centipetal force equation
sub f = m x a
F = m vv / r
F = m ww r
SHM can be defined as
An oscillation in which the acceleration of an object is directly proportional to its displacement from its equilibrium position and is directed towrads the equilibrium
Simple harmonic motion
An object moving with SHM oscillates to and from either side of an equilibrium position
This equilibrium position is the midpoint of the objects motion
Distance of object from equilibrium is called its displacement
restoring force
the force pulling or pushing the object back towards the equilibrium position
size of the force depends of the displacement
the restoring force makes the object accelerate towards the equilibrium
SHM rule
a is directly proportional to -x
also force is DP TO x
the minus sign shows that the acceleration is always opposing displacement
Displacement ( graph )
avries with cosine and sine wave
max value of A (amplitude)
velocity ( graph)
gradient of the displacement-time graph
max vale of wA
angular frequency, w
same as angular speed
w = 2pi f
acceleration (graph)
gradient is velocity-time graph
max value of wwA
2 notable points for the graphs
when gradient of displacemnt time graph = 0, the velocity
when gradient of velocity-time graph is at maximum, the acceleration is at maximum
Phase difference
measures how much one wave lags behind another wave
2 wave in phase, phase difference = 0 or 2pi
if two waves are exactly out of phase, (anti-phase), they have a phase difference of pi or 180
Phase differnec for DVA graphs
velocity is pi / 2 out of phase with displacement, i.e. velocity is a quarter of a cycle ahead
acceleration is a quarter of a cycle ahead of velocity meaning acceleration and displacement are in antiphase
frequency
number of cycles per second (Hz)
period
time taken to complete one cycle
amplitude
magnitude of max displacement
kinetic and potencial energy
-As the object moves twoard the equilibrium, the restoring force doies work and transfers some Ep to Ek
-When object is moving back away the Ek transferred back to Ep
-At equilibrium the Ep is said to be zero and Ek at maximum, therefore the velocity is at maximum
-at maximum displacement (the amplitude) on botn sides of the equilibrium, the objects Ep is max and Ek is zero, so velocity = 0
mechanical energy
sum of the potencial energy and kinetic energy and stays constant as long as motion isnt damped
displacement equation
x = r cos( theta )
ball is equal to horizontal component of the ball position
combine equation for x of any object doing SHM
x = r cos( theta ) = A cos( wt)
x = A cos( wt )
acceleration equation
a of an object in SHM is equal to the horizontal componet
a = ww r
a = -ww r cos( theta )
minus sign as acceleration is always acting towrads the centre of the circle
r cos ( theta ) ==to the horizontal component of the ball so acceleration of object is -
a = -ww x
max acceleration equation
objects max acceleration occurs when its at its max magnitude of displacement , i.e. when x = + or - A
a (max) = ww A
Velocity equation
v = +- w root A^2 -x^2
velocity is positive when object is moving in positive direction ( right )
and negative when moving left
max velocity equation
max speed is when object is passing through the equilibrium
V (max) = wA
mass on a spring
mass on a spring is a simple harmonic oscillator (SHO), when mass is pushed or pulled either side of the equilibrium position, theres a restoring force
Hookes law
give size and direction of restoring force
F = k x change in length
equation for time period for a mass on a spring
T = 2pi x root m / k
equation for time period for a simple pendulum
T = 2pi x root l / g
free vibrations
involve no transfer of energy to or from the surroundings
Natural frequency
the frequency of an object oscillating freely
Resonance
when an object driven by a periodic external force at a frequency close to its natural frequency begins to to oscillate with a rapidly increasingly amplitude
Resonant frequency
A frequency at which a stationary wave is formed because an exact number of waves are produced in the time it takes for a wave to get to the end of the vibrating medium and back again
Forced vibrations
happens when theres a external driving force
a system can be forced to vibrate by a periodic external force
The frequency of this force is called is called the driving frequency
Damping
when any oscillation loses energy to its surroundings, usually down to frictional forces like air resistance
systems are sometimes deliberately damped to stop them oscillating or to minimise the effect to resonance
overdamping
system with even heavier damping are damped
they take longer to return to equilibrium than a critically damped system
critical damping
critical damping reduces the damping ( i.e. stops the system oscillating) in the shortest possible time
Damping and resonance peaks
Lightly damped system have a very sharp resoanace peak
Their amplitude only increases framatically when the driving frequency is very close to the natural frequency
Heavily damped systems have a much flatter responce
Their amplitude doesnt increase very much near the natural frequency and they arent as sensative to the driving frequency
