Question 1

(a) A free particle of mass $m$ has momentum $(p_1,p_2,p_3)$ and total energy $E$. State why the quantity

$$p_1^2+p_2^2+p_3^2-{E^2\over c^2}$$

takes the same value in all inertial frames. What does this quantity equal?

[5 marks]

(b) From the viewpoint of a frame rotating with angular velocity $\hbox{\boldmath$\omega$ }$, a particle at position ${\bf x}$ appears to be acted on by an inertial force of the form

$${\bf F}=-m\,\hbox{\boldmath $\omega$}\times(\hbox{\boldmath $\omega$} \times {\bf x}).$$

Describe the nature of this force in terms of its magnitude and direction.

[5 marks]

(c) Write down the expression for the Coriolis force on a particle with position ${\bf x}$ and velocity ${\bf v}$ in a frame that is rotating with angular velocity $\hbox{\boldmath$\omega$ }$.

[5 marks]

(d) Imagine that a hole has been drilled straight through the centre of the Earth. Show that, if the Earth had uniform density, a particle dropped into this hole would execute simple harmonic motion.

[5 marks]

(e) Draw a diagram illustrating a set of principal axes of rotation for a uniform rectangular plate rotating about its centre.

[5 marks]

[Hints can be found here, but do not follow this link until you have attempted the question on your own.]