Chapter 5

Find the centrifugal acceleration at the equator of the planet Jupiter and of the Sun. In each case, express your answer also as a fraction of the surface gravity. (The rotation periods are 10 hours and 27 days, respectively, the radii \(7.1 \times 10^{4} \mathrm{~km}\) and \(7.0 \times 10^{5} \mathrm{~km}\), and the masses \(1.9 \times 10^{27} \mathrm{~kg}\) and \(\left.2.0 \times 10^{30} \mathrm{~kg} .\right)\)

Water in a rotating container of radius \(50 \mathrm{~mm}\) is \(30 \mathrm{~mm}\) lower in the centre than at the edge. Find the angular velocity of the container.

The water in a circular lake of radius \(1 \mathrm{~km}\) in latitude \(60^{\circ}\) is at rest relative to the Earth. Find the depth by which the centre is depressed relative to the shore by the centrifugal force. For comparison, find the height by which the centre is raised by the curvature of the Earth's surface. (Earth radius \(=6400 \mathrm{~km}\).)

Find the velocity relative to an inertial frame (in which the centre of the Earth is at rest) of a point on the Earth's equator. An aircraft is flying above the equator at \(1000 \mathrm{~km} \mathrm{~h}^{-1}\). Assuming that it flies straight and level (i.e., at a constant altitude above the surface) what is its velocity relative to the inertial frame (a) if it flies north, (b) if it flies west, and (c) if it flies east?

The wind speed in colatitude \(\theta\) is \(v\). By considering the forces on a small volume of air, show that the pressure gradient required to balance the horizontal component of the Coriolis force, and thus to maintain a constant wind direction, is \(\mathrm{d} p / \mathrm{d} x=2 \omega \rho v \cos \theta\), where \(\rho\) is the density of the air. Evaluate this gradient in mbar \(\mathrm{km}^{-1}\) for a wind speed of \(50 \mathrm{~km} \mathrm{~h}^{-1}\) in latitude \(30^{\circ} \mathrm{N} .\left(1\right.\) bar \(=10^{5} \mathrm{~Pa}\); density of air \(=\) \(\left.1.3 \mathrm{~kg} \mathrm{~m}^{-3} .\right)\)

An aircraft is flying at \(800 \mathrm{~km} \mathrm{~h}^{-1}\) in latitude \(55^{\circ} \mathrm{N}\). Find the angle through which it must tilt its wings to compensate for the horizontal component of the Coriolis force.

An orbiting space station may be made to rotate to provide an artificial gravity. Given that the radius is \(25 \mathrm{~m}\), find the rotation period required to produce an apparent gravity equal to \(0.7 g\). A man whose normal weight is \(75 \mathrm{~kg}\) weight runs around the station in one direction and then the other (i.e., on a circle on the inside of the cylindrical wall) at \(5 \mathrm{~ms}^{-1}\). Find his apparent weight in each case. What effects will he experience if he climbs a ladder to a higher level ( i.e., closer to the axis), climbing at \(1 \mathrm{~ms}^{-1} ?\)

A beam of particles of charge \(q\) and velocity \(v\) is emitted from a point source, roughly parallel with a magnetic field \(\boldsymbol{B}\), but with a small angular dispersion. Show that the effect of the field is to focus the beam to a point at a distance \(z=2 \pi m v /|q| B\) from the source. Calculate the focal distance for electrons of kinetic energy \(500 \mathrm{eV}\) in a magnetic field of \(0.01 \mathrm{~T}\). (Charge on electron \(=-1.6 \times 10^{-19} \mathrm{C}\), mass \(=9.1 \times 10^{-31} \mathrm{~kg}\), \(\left.1 \mathrm{eV}=1.6 \times 10^{-19} \mathrm{~J} .\right)\)

Write down the equation of motion for a charged particle in uniform, parallel electric and magnetic fields, both in the \(z\)-direction, and solve it, given that the particle starts from the origin with velocity \((v, 0,0)\). A screen is placed at \(x=a\), where \(a \ll m v / q B .\) Show that the locus of points of arrival of particles with given \(m\) and \(q\), but different speeds \(v\), is approximately a parabola. How does this locus depend on \(m\) and \(q ?\)

A beam of particles with velocity \((v, 0,0)\) enters a region containing crossed electric and magnetic fields, as in the example at the end of $$\$S$$ \(5.2.\) Show that if the ratio \(E / B\) is correctly chosen the particles are undeviated, while particles with other speeds follow curved trajectories. Suppose the particles have velocities equal to \(v\) in magnitude, but with a small angular dispersion. Show that if the path length \(l\) is correctly chosen, all such particles are focussed onto a line parallel to the \(z\)-axis. (Thus a slit at that point can be used to select particles with a given speed.) For electrons of velocity \(10^{8} \mathrm{~m} \mathrm{~s}^{-1}\) in a magnetic field of \(0.02 \mathrm{~T}\), find the required electric field, and the correct (smallest possible) choice for \(l\).

The angular velocity of the electron in the lowest Bohr orbit of the hydrogen atom is approximately \(4 \times 10^{16} \mathrm{~s}^{-1}\). What is the largest magnetic field which may be regarded as small in this case, in the sense of \(\S 5.5 ?\) Determine the Larmor frequency in a field of \(2 \mathrm{~T}\).

The orbit of an electron (charge \(-e)\) around a nucleus (charge \(Z e\) ) is a circle of radius \(a\) in a plane perpendicular to a uniform magnetic field \(\boldsymbol{B}\). By writing the equation of motion in a frame rotating with the electron, show that the angular velocity \(\omega\) is given by one of the roots of the equation \(m \omega^{2}-e B \omega-Z e^{2} / 4 \pi \epsilon_{0} a^{3}=0\) Verify that for small values of \(B\), this agrees with \(\S 5.5\). Evaluate the two roots if \(B=10^{5} \mathrm{~T}, Z=1\) and \(a=5.3 \times 10^{-11} \mathrm{~m}\). (Note, however, that in reality \(a\) would be changed by the field.)

A projectile is launched due north from a point in colatitude \(\theta\) at an angle \(\pi / 4\) to the horizontal, and aimed at a target whose distance is \(y\) (small compared to Earth's radius \(R\) ). Show that if no allowance is made for the effects of the Coriolis force, the projectile will miss its target by a distance \(x=\omega\left(\frac{2 y^{3}}{g}\right)^{1 / 2}\left(\cos \theta-\frac{1}{3} \sin \theta\right)\) Evaluate this distance if \(\theta=45^{\circ}\) and \(y=40 \mathrm{~km}\). Why is it that the deviation is to the east near the north pole, but to the west both on the equator and near the south pole? (Neglect atmospheric resistance.)

Find the equations of motion for a particle in a frame rotating with variable angular velocity \(\boldsymbol{\omega}\), and show that there is another apparent force of the form \(-m \dot{\boldsymbol{\omega}} \wedge \boldsymbol{r}\). Discuss the physical origin of this force.

Find the equation of motion for a particle in a uniformly accelerated frame, with acceleration \(\boldsymbol{a} .\) Show that for a particle moving in a uniform gravitational field, and subject to other forces, the gravitational field may be eliminated by a suitable choice of \(\boldsymbol{a}\).

The co-ordinates \((x, y, z)\) of a particle with respect to a uniformly rotating frame may be related to those with respect to a fixed inertial frame, \(\left(x^{*}, y^{*}, z^{*}\right)\), by the transformation $$ \left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{ccc} \cos \omega t & \sin \omega t & 0 \\ -\sin \omega t & \cos \omega t & 0 \\ 0 & 0 & 1 \end{array}\right]\left[\begin{array}{c} x^{*} \\ y^{*} \\ z^{*} \end{array}\right] $$ (Here, we use matrix notation: this stands for three separate equations, \(x=\cos \omega t \cdot x^{*}+\sin \omega t \cdot y^{*}\) etc.) Write down the inverse relation giving \(\left(x^{*}, y^{*}, z^{*}\right)\) in terms of \((x, y, z)\). By differentiating with respect to \(t\), rederive the relation (5.15) between \(\mathrm{d}^{2} \boldsymbol{r} / \mathrm{d} t^{2}\) and \(\ddot{\boldsymbol{r}}\). [Hint: Note that \(\ddot{\boldsymbol{r}}=(\ddot{x}, \ddot{y}, \ddot{z})\), while \(\mathrm{d}^{2} \boldsymbol{r} / \mathrm{d} t^{2}\) is the vector obtained by applying the above transformation \(\left.\operatorname{to}\left(\ddot{x}^{*}, \ddot{y}^{*}, \ddot{z}^{*}\right) .\right]\)

Another way of deriving the equation of motion (5.16) is to use Lagrange's equations. Express the kinetic energy \(\frac{1}{2} m(\mathrm{~d} \boldsymbol{r} / \mathrm{d} t)^{2}\) in terms of \((x, y, z)\), and show that Lagrange's equations (3.44) reproduce (5.16) for the case where the force is conservative.