CORE 6: FURTHER MECHANICS & THERMAL PHYSICS

 

CONTENTS

6.1 CIRCULAR MOTION
6.2 SHM
6.3 DAMPING & RESONANCE
6.4 THERMAL ENERGY TRANSFER
6.5  IDEAL GASES
6.6  KINETIC THEORY

 

  1. CIRCULAR MOTION

                  

                         

    # 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  =  v2 / r   =  r ω2

 

    # This centripetal acceleration is caused by a resultant

        force towards the centre called the centripetal force, F

        

                    F = m v2 / r   =  mrω2

 

RETURN TO CONTENTS 

2. SIMPLE HARMONIC MOTION

      #  SHM : an object whose acceleration is proportional to its displacement from a

           fixed point AND in the opposite direction to the displacement.

 

            a   =  −  ω2 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 √(A2 - x2)   =   ± 2π f  √(A2 - x2),   maximum velocity =  ω A

 

       for acceleration,  a = - ( 2π f )2 x ,   maximum acceleration  =  ω2 A

 

       #  variation of kinetic and potential energy of SHM with time:

 

            Ek / t ,  and   Ep / t  graphs:

     #  Ek / x  and Ep / x  equations and graphs:

 

           kinetic energy  =   ½ m v 2  ,         v  =  2π f  √(A2 - x2)

 

           kinetic energy  =   ½ m ω2 (A2 - x2)     =   ½ m  [2π f  √(A2 - x2) ]2

 

      total energy  =  kinetic energy Ek +  potential energy Ep  

 

      total energy  =  maximum kinetic energy  =   ˝ m 2 ω2 A2    =   ½ m (2π f A)  

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. 

3. DAMPING AND RESONANCE

       # 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.

            f0 is the natural frequency of the object. 

           

 NB Increased damping causes a broader peak of lower amplitude as shown above.

 RETURN TO CONTENTS

4.  THERMAL ENERGY TRANSFER

     # 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.

          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.

   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

    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 m3 , 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.

0 °C = 273 K,  100 °C = 373 K  (make sure you know how to do conversions).

 

NA = Avogadro’s Constant = 6 x 1023 . This is the number of particles present in 0.012 kg of carbon-12.

    1 mole = 6 x 1023 particles

 

#  MOLAR MASS (Mm = the mass of 1 mole of a gas) and MOLECULAR MASS (mass of 1 molecule)

#  work done (when a gas expands or contracts),   W  =  p ΔV

6.  KINETIC THEORY

    #  Brownian motion as evidence for existence of atoms

    #  the Gas Laws (Boyles, Charles', Pressure Laws) come from experiment, whereas kinetic theory equations are theoretical

        `c2 = the mean of the squares of the speeds of the gas molecules

 

                                    

        

   crms  = the square root of the mean of the squares of the molecular speeds

        INTERNAL ENERGY = molecular ke + molecular pe

 

        The Boltzmann Constant k = R/NA,  this is the gas constant per molecule.  Also,   pV = NkT

             ˝ m`c2  = 3/2 kT   =  3 RT / 2NA ,                                    crms = Ö3kT/m = Ö3RT/Mm

         m = the mass of a single molecule