Topic outline

  • General

  • Topic 1

    • Centre of Mass/ Centre of Momentum Frame Assignment

      The message of the principle of relativity, whether it is Newtonian or Einsteinian relativity, is that all inertial frames are equivalent as far as the laws of mechanics are concerned. However, in practice, many problems can be solved more readily by a judicious choice of frame. In collision problems the best choice is usually the centre of momentum frame. This is the frame in which the total momentum of the system is zero.

      Consider the following two collisions between a pair of cars.

      Collision (i): A car travelling at 60 mph (96 km h- 1) collides with a stationary car. Collision

      (ii): Two cars each travelling at 30 mph (48 km h- 1) collide head on. It is clear that the single moving car in (i) has more kinetic energy than the two moving cars in (ii). But the need to conserve momentum in collision (i) means that not all the kinetic energy is available to cause damage, whereas in collision (ii) all the kinetic energy can go into causing damage and momentum will still be conserved. So the message is clear: to determine the energy that can be made available we have to transfer to the centre of momentum frame. Bearing this in mind which of the above collisions, assuming they are both inelastic, causes the most damage?

      Solution : In collision (i) not all the \(\frac{1}{2}mv^{2}\) of kinetic energy is available to cause damage since conservation of momentum implies both cars must be moving after the collision. In collision (ii) the cars come to rest after the collision so the kinetic energy available is \(2 \frac{1}{2}m (v /2)^{2} = \frac{1}{4}mv^{2}\). The amount of damage cannot depend on the frame from which we view the event. So in (i) look at the collision in the centre of mass frame: it is exactly the same as collision (ii)! Thus the damage must be equal.

    • Forbidden processes Assignment
      (a) The photoelectric effect is the absorption of a photon by an electron: It IS responsible for ejecting electrons from an illuminated metal surface and for the ionisation of atoms. However photoelectric absorption by isolated electrons is never observed. Why not?
      (b) How are energy and momentum conserved in the ionisation of an atom, which can take place in isolation?
      (c) The production of an electron-positron pair from a photon can occur when the photon energy exceeds the combined rest energy of the pair. The process is observed when gamma rays interact with matter. Pair production by isolated gamma rays is never observed. Explain why this is.
      Solution : (a) Consider the putative absorption of the photon by the lone electron. The process is shown in the laboratory frame in figure

      (a) Consider the putative absorption of the photon by the lone electron. The process is shown in the laboratory frame in figure

      and transformed to the CM frame in figure
      In the CM frame it is clear that momentum is conserved but relativistic energy is not since, comparing the energy before and after, hv'+ \( \gamma \)(mc2 > mc2 as \( \gamma \)> 1.
      (b) When an isolated atom absorbs a photon its rest mass changes. This is not possible for the electron since, being elementary, it has no excited states.
      (c) In the laboratory frame the putative process is shown in the figure. In the final state the total momentum is zero in the centre of mass frame. However, for the initial state, there is no frame in which the momentum is zero because a single photon cannot have a rest frame. It is therefore obvious that momentum cannot be conserved so the process cannot occur.

  • Topic 2

    • Galileo's legendary experiment Assignment

      Galileo is credited with establishing that all bodies fall with the same acceleration under gravity. He did this not by dropping bodies from the leaning tower of Pisa, as legend has it, but by rolling balls down an inclined plane. This has the advantage of diluting gravity which makes it easier to measure the time of fall. However, Galileo was fortunate in the shapes of bodies he chose to compare.

      (a) Show that a sphere of lead and a sphere of wood rolling down an inclined plane cover the same distance in the same time. [Hint: Write down the conservation of energy for a rolling ball and differentiate it with respect to time to find the acceleration.]

      (b) Does the size of the balls make any difference?

      (c) Does the shape of the rolling object make any difference?

      (d) Would Galileo have made his discovery if he had compared the rolling motion of spheres and cylinders?


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