Further Mechanics

Question 1

How to covert between Radians and Degrees ?

Answer 1

degrees -> radians = x π/180

radians -> degrees = x 180/π

Question 2

What is Angular speed ?

Answer 2

is the angle an object rotates through per second
(angle /time)
-measured in Rads^-1

Question 3

What is the period (T/s) of rotation ?

Answer 3

is the time taken for one completion revolution

T = 1/f

Question 4

What is the frequency (f/Hz) of rotation ?

Answer 4

is the number of complete revolutions per second
f = 1/T
(RPM -> Hz = ÷60)

Question 5

What is Linear Velocity (V/ms^-1) ?

Answer 5

= distance/time
v = 2πr/t = 2πrf
- distance - at the tangent

Question 6

What is Angular Velocity (w/rads^-1) ?

Answer 6

= angle /time
w = 2π/T = 2πf
- same at any point on a solid rotating object

Question 7

Relationship between linear and angular velocity ?

Answer 7

w = v/r
(angular velocity = linear velocity/radius)
- moving in a circle has a constant speed but not a constant velocity

Question 8

What is Centripetal Acceleration (A/ms^-2) ?

Answer 8

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

Question 9

What is Centripetal Force (F/N) ?

Answer 9

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.

Question 10

What is Amplitude ?

Answer 10

the distance between the equilibrium position and the maximum displacement

Question 11

What is Simple Harmonic Motion (SHM) ?

Answer 11

is the ACCELERATION of an object is directly proportional to its displacement from its equilibrium position and acts in OPPOSITE DIRECTION.
w = angular frequency

Question 12

Graphs of SHM of displacement, velocity and acceleration against time ?

Answer 12

1) displacement/time = cosine graph
- gradient =velocity
2) velocity /time = sin graph (opposite direction)
- gradient = acceleration
3) acceleration/time graph = opposite cosine graph

Question 13

What the phase difference of SHM ?

Answer 13

velocity to displacement is π/2 out of phase and acceleration to displacement if π out of phase.

Question 14

SHM maximum and minimum ?

Answer 14

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)

Question 15

energy involved in SHM ?

Answer 15

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

Question 16

Calculations with SHM

Answer 16

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

Question 17

What is mechanical energy ?

Answer 17

MECHANICAL ENERGY is the sum of the kinetic and potential energy of an object and that it stays constant for undamped oscillations

Question 18

Calculations with SHM energy

Answer 18

Ep max = 1/2mw^2A^2 (x=A)
Ek max = 1/2mw^2A^2 (x=0)
Total energy = 1/2mw^2A^2

Question 19

Mass-Spring system

Answer 19

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)

Question 20

Simple Pendulum system

Answer 20

Simple Pendulum is a simple harmonic oscillator
ASSUMPTION = only true for small angular displacement
(Ѳ is small)

Question 21

Period of Mass-Spring system

Answer 21

T=2π√m/k
(T∝ √m/k) - increase mass increases period
f=1/2π√k/m
(f∝ √k/m) - increase mass decreases frequency

Question 22

Period of Simple Pendulum system

Answer 22

T=2π√l/g
(T∝ √l/g) - increase length increases period
f=1/2π√g/l
(f∝ √g/l) - increase length decreases frequency

Question 23

What is free vibrations ?

Answer 23

involves no energy transfer of energy to or from the surroundings

• oscillates freely
• natural frequency (fo)

Question 24

What is forced vibrations ?

Answer 24

is a frequency driven by an external force

- driving frequency (f)

Question 25

how the driver frequency affects the driven oscillators ?

Answer 25

FFo

• frequency = f
• amplitude = very small
• path difference = 180

Question 26

what happens when the driver frequency = natural frequency ?

Answer 26

causes Resonance 
-efficient transfer of energy 
-rapid increased amplitude 
- phase difference is 90
- E∝A^2
Graph (amplitude/driving frequency) - peaks at Fo

Question 27

What is Damping and what are the types ?

Answer 27

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

Question 28

Examples of resonance + how to reduce ?

Answer 28

examples :
- MRI scanner
- radio
- stationary waves in organ pipes
reductions :
- fluid damping on bridges (avoid similar frequencies)
- shock absorbers - return to equilibrium very quickly