Further Mechanics Flashcards
How to covert between Radians and Degrees ?
degrees -> radians = x π/180
radians -> degrees = x 180/π
What is Angular speed ?
is the angle an object rotates through per second
(angle /time)
-measured in Rads^-1
What is the period (T/s) of rotation ?
is the time taken for one completion revolution
T = 1/f
What is the frequency (f/Hz) of rotation ?
is the number of complete revolutions per second
f = 1/T
(RPM -> Hz = ÷60)
What is Linear Velocity (V/ms^-1) ?
= distance/time
v = 2πr/t = 2πrf
- distance - at the tangent
What is Angular Velocity (w/rads^-1) ?
= angle /time
w = 2π/T = 2πf
- same at any point on a solid rotating object
Relationship between linear and angular velocity ?
w = v/r
(angular velocity = linear velocity/radius)
- moving in a circle has a constant speed but not a constant velocity
What is Centripetal Acceleration (A/ms^-2) ?
A = v^2/r = w^2r
- the direction of acceleration is always TOWARDS THE CENTRE of the circle
- the acceleration is cause by a centripetal force
What is Centripetal Force (F/N) ?
F = mv^2/r = mw^2r
- requirement for circular motion
- the direction of force is always TOWARDS THE CENTRE of the circle
e. g. move out in car around a roundabout due to feeling reaction force.
What is Amplitude ?
the distance between the equilibrium position and the maximum displacement
What is Simple Harmonic Motion (SHM) ?
is the ACCELERATION of an object is directly proportional to its displacement from its equilibrium position and acts in OPPOSITE DIRECTION.
w = angular frequency
Graphs of SHM of displacement, velocity and acceleration against time ?
1) displacement/time = cosine graph
- gradient =velocity
2) velocity /time = sin graph (opposite direction)
- gradient = acceleration
3) acceleration/time graph = opposite cosine graph
What the phase difference of SHM ?
velocity to displacement is π/2 out of phase and acceleration to displacement if π out of phase.
SHM maximum and minimum ?
at AMPLITUDE - max displacement (x=A) - min velocity (V=0) - max acceleration (a=w^2A) at EQUILIBRIUM - min displacement (x=0) - max velocity (V=wA) - min acceleration (a=0)
energy involved in SHM ?
exchanges between potential and kinetic energy as it oscillates.
- at equilibrium the Ek is maximum (velocity = max)
- at amplitude the Ep is Maximum (velocity = min)
graph is two x^2 functions and straight line for total energy
Calculations with SHM
1) displacement -> X=Acos(wt)
2) velocity -> v=±w√(A^2-X^2)
- graph is a circle (velocity/displacement)
- max velocity -> wA
3) acceleration -> a=-w^2X (y=-mx)
- graph is a negative linear (acceleration /displacement)
- max acceleration -> w^2A
What is mechanical energy ?
MECHANICAL ENERGY is the sum of the kinetic and potential energy of an object and that it stays constant for undamped oscillations
Calculations with SHM energy
Ep max = 1/2mw^2A^2 (x=A)
Ek max = 1/2mw^2A^2 (x=0)
Total energy = 1/2mw^2A^2
Mass-Spring system
mass on a spring is a simple harmonic oscillator as the mass pushes and pulls to cause a force on it
F=-kΔx (negative sign because force acts in opposite direction to the displacement)
ASSUMPTION = don’t exceed the elastic limit and must obey hook’s law (F∝X)
Simple Pendulum system
Simple Pendulum is a simple harmonic oscillator
ASSUMPTION = only true for small angular displacement
(Ѳ is small)
Period of Mass-Spring system
T=2π√m/k
(T∝ √m/k) - increase mass increases period
f=1/2π√k/m
(f∝ √k/m) - increase mass decreases frequency
Period of Simple Pendulum system
T=2π√l/g
(T∝ √l/g) - increase length increases period
f=1/2π√g/l
(f∝ √g/l) - increase length decreases frequency
What is free vibrations ?
involves no energy transfer of energy to or from the surroundings
- oscillates freely
- natural frequency (fo)
What is forced vibrations ?
is a frequency driven by an external force
- driving frequency (f)
how the driver frequency affects the driven oscillators ?
FFo
- frequency = f
- amplitude = very small
- path difference = 180
what happens when the driver frequency = natural frequency ?
causes Resonance -efficient transfer of energy -rapid increased amplitude - phase difference is 90 - E∝A^2 Graph (amplitude/driving frequency) - peaks at Fo
What is Damping and what are the types ?
damping is the loses of energy (smaller amplitude) to frictional forces which stops the system oscillating
- reduce sharpness of resonance ( smaller peak in graph)
1) light damping - simple pendulum
2) heavy damping -water
3) critical damping - car suspension
Examples of resonance + how to reduce ?
examples :
- MRI scanner
- radio
- stationary waves in organ pipes
reductions :
- fluid damping on bridges (avoid similar frequencies)
- shock absorbers - return to equilibrium very quickly
