CORE 6: FURTHER MECHANICS & THERMAL PHYSICS
6.1 | CIRCULAR MOTION |
6.2 | SHM |
6.3 | DAMPING & RESONANCE |
6.4 | THERMAL ENERGY TRANSFER |
6.5 | IDEAL GASES |
6.6 | KINETIC THEORY |
# angles all measured in RADIANS
# Speed is constant, velocity changes towards the centre, acceleration is towards
the centre or centripetal.
# Be familiar with equations:
ω = v / r = 2πf, T = 2π / ω T = 2π r / v
a = v^{2} / r = r ω^{2}
# This centripetal acceleration is caused by a resultant
force towards the centre called the centripetal force, F
F = m v^{2} / r = mrω^{2}
# SHM : an object whose acceleration is proportional to its displacement from a
fixed point AND in the opposite direction to the displacement.
a = − ω^{2} x
MAKE SURE YOU KNOW YOUR GRAPHS : a/x, x/t, v/t, a/t:
# velocity is always the gradient of the displacement / time graph
# acceleration is always the gradient of the velocity / time graph
# Useful equations:
for displacement, x = A cos ω t = A cos 2π f t NB. set calculator for radians
for velocity, v = ± ω √(A^{2} - x^{2}) = ± 2π f √(A^{2} - x^{2}), maximum velocity = ω A
for acceleration, a = - ( 2π f )^{2} x , maximum acceleration = ω^{2 }A
# variation of kinetic and potential energy of SHM with time:
E_{k} / t , and E_{p} / t graphs:
# E_{k} / x and E_{p} / x equations and graphs:
kinetic energy = ½ m v ^{2} , v = 2π f √(A^{2} - x^{2})
kinetic energy = ½ m ω^{2} (A^{2} - x^{2}) ^{ } = ½ m [2π f √(A^{2} - x^{2}) ]^{2 }
total energy = kinetic energy E_{k} + potential energy E_{p }
total energy = maximum kinetic energy = ½ m ^{2} ω^{2 }A^{2} = ½ m (2π f A)^{2 }
You will be asked to solve problems using the simple pendulum equation:
You will also be asked to solve problems using the mass-spring equation:
where k is the spring constant in N / m.
# Resonance occurs when the driving frequency of an object = its natural
frequency, then large amplitude oscillations occur and rate of energy transfer (power)
from the source is maximized.
# Critical damping occurs when an oscillator returns to its mean position in the
shortest time without overshoot.
# Amplitude = maximum displacement of a particle from its mean position.
Be familiar with the Amplitude against driving frequency graph showing resonance.
f_{0} is the natural frequency of the object.
NB Increased damping causes a broader peak of lower amplitude as shown above.
# The internal energy of a system is increased when energy is supplied either by heating it, OR doing work on it.
# Total internal energy = molecular KE + molecular PE
# molecular KE depends on temperature, as temperature goes up average molecular speed goes up.
Specific heat capacity (c) heat needed to raise temp.of 1 kg of object by 1 K.
Q = mc Dθ unit of SHC = J kg^{-1} K^{-1}
where Q is the heat supplied or removed, m is the mass, and Dθ is the temperature change
# make sure you are familiar with the continuous flow method
# During a change of state, at constant temperature, PEs of particles are changing but KEs of particles are constant.
Specific latent heat of fusion is heat needed to change 1kg of solid into its liquid without change in temp.
Specific latent heat of vaporisation is heat needed to change 1kg of liquid into its vapour without change in temp.
Q
= m
l
unit of SLH = J kg^{-1
}
where Q is the heat supplied or removed, m is the mass which is changing state, and l is the specific latent heat.
5. IDEAL GASES
Ideal gas : one which obeys the gas laws at all temps. and pressures.
Real gas only obeys gas laws at low pressures & at high temps.
# Ideal gas equation: pV = nRT (molar equation)
n IS THE NUMBER OF MOLES (THE AMOUNT) OF THE GAS
pressures are in N m^{-2 }, volumes are in m^{3 }, temperatures
are in K
R IS THE UNIVERSAL MOLAR GAS CONSTANT (J mol^{-1} K^{-1} )
and, pV = NkT (molecular equation)
for N molecules, where k is the Boltzmann Constant.
ABSOLUTE ZERO IS THE LOWEST ATTAINABLE TEMP. 0 K
0 °C = 273 K, 100 °C = 373 K (make sure you know how to do conversions).
# N_{A} = Avogadro’s Constant = 6 x 10^{23} . This is the number of particles present in 0.012 kg of carbon-12.
1 mole = 6 x 10^{23} particles
# MOLAR MASS (M_{m} = the
mass of 1 mole of a gas)
# Brownian motion as evidence for existence of atoms
# the Gas Laws (Boyles, Charles', Pressure Laws) come from experiment, whereas kinetic theory equations are theoretical
PRESSURE EQUATIONS pV = 1/3 Nm`c^{2 }p = 1/3 r`c^{2 } ^{ }
`c^{2 }= the mean of the squares of the speeds of the gas molecules
c_{rms} = the square
root of the mean of the squares of the molecular speeds
RANDOMNESS : speed aspect, direction aspect _{ }
INTERNAL ENERGY = molecular ke + molecular pe
The Boltzmann Constant k = R/N_{A, }this is the gas constant per molecule. Also,_{ } pV = NkT
_{ } ½ m`c^{2} = 3/2 kT = 3 RT / 2N_{A },_{ }c_{rms} = Ö3kT/m = Ö3RT/M_{m}
m = the mass of a single molecule