The author of a scientific paper claims to have done the following experiment 3600 times. The subject wrote down a number, then a die was thrown and the number shown on the die compared with the prediction. He claims that the results were as shown.
Let \(\mathbf{r}\) denote the position vector of a particle relative to a point \(O\) on the earth's surface. In a certain approximation the effects of the earth's rotation are described by the equation \[\ddot{\mathbf{r}}+2\mathbf{\omega} \wedge \dot{\mathbf{r}} = \mathbf{g},\] where \(\mathbf{g}\) is the acceleration due to gravity, pointing vertically downwards, and \(\mathbf{\omega}\) is another constant vector (pointing in the direction of the earth's axis of rotation and equal in magnitude to its angular velocity). If the particle is projected from \(O\), with velocity \(\mathbf{v}\), at time \(t = 0\), show that \[\dot{\mathbf{r}}+2\mathbf{\omega} \wedge \mathbf{r} = \mathbf{g}t+\mathbf{v}.\] Deduce that \[\mathbf{r} = \frac{1}{2}\mathbf{g}t^2 + \mathbf{v}t - \frac{1}{3}\mathbf{\omega} \wedge \mathbf{g}t^3 - \mathbf{\omega} \wedge \mathbf{v}t^2\] if terms of order \(\omega^2\) may be neglected. The flight ends when the particle hits the horizontal plane through \(O\). Continuing to neglect terms of order \(\omega^2\) show that the time of flight is \[2g^{-2}\mathbf{g}.\mathbf{v}(1 + 2g^{-2}\mathbf{\omega}.(\mathbf{v} \wedge \mathbf{g})).\]
A heavy particle is projected horizontally with velocity \(V\) along the smooth inner surface of a sphere of radius \(a\). Its initial depth below the centre is \(d\) and in the subsequent motion it never leaves the surface of the sphere. Show that, if \(u\) is the horizontal component of its velocity when the radius to the particle makes an angle \(\theta\) with the downward vertical, \[au\sin\theta = V(a^2 - d^2)^{\frac{1}{2}}.\] Calculate the maximum and minimum heights attained by the particle and determine whether it moves upwards or downwards initially.
A hollow cylinder of radius \(a\) rolls without slipping on the inside of a cylinder of radius \(b(b > a)\). The axes are always horizontal. If \(\theta\) is the angle between the vertical and the line-of-centres of the cylinders (in a plane perpendicular to the axes), obtain the equation of motion \[\ddot{\theta} = -\omega^2\sin\theta,\] where \(\omega^2(b-a) = g\). If the coefficient of limiting friction is \(\mu\), show that two classes of motion are possible: (i) where \(\dot{\theta}^2 \leq \omega^2[1-(1+4\mu^2)^{-\frac{1}{2}}]\), and \(\theta\) oscillates about zero; (ii) where \(\dot{\theta}^2 \geq \omega^2[(1+16\mu^2)^{\frac{1}{2}}/2\mu-1]\), and \(\theta\) increases or decreases monotonically.
A particle of unit mass orbits the sun under an inverse square law of gravity. Interplanetary gas imposes a resistive force which is \(-k\) times the velocity, in magnitude and direction. Use the equation of motion in polar coordinates to show that the angular momentum decreases exponentially with time. If the resistive force is neglected show that the particle can move in a circular orbit, say with angular frequency \(\omega\). If \(k \ll \omega\), so that \(k^2\) can be neglected in comparison with \(\omega^2\), show that the radius of the orbit decreases by a fraction \(4\pi k/\omega\) per revolution, and that the tangential velocity increases by a fraction \(2\pi k/\omega\). Comment on the fact that as a result of the resistive force the velocity actually increases.
Show that \((\mathbf{l} \wedge \mathbf{m}).\mathbf{n} = (\mathbf{n} \wedge \mathbf{l}).\mathbf{m} = (\mathbf{m} \wedge \mathbf{n}).\mathbf{l}\). Hence, or otherwise, show that \[|\mathbf{l} \wedge \mathbf{m}|^2 = |\mathbf{l}|^2|\mathbf{m}|^2-(\mathbf{l}.\mathbf{m})^2.\] If the point \(P\) has position vector \(\mathbf{r}\) given by \[\mathbf{r} = \mathbf{a} + s\mathbf{u}\] show that \(P\) lies on a line if \(s\) is allowed to vary, and explain the geometrical significance of \(\mathbf{a}\) and \(\mathbf{u}\). Suppose two lines are given by equations \[\mathbf{r}_i = \mathbf{a}_i+s_i\mathbf{u}_i, \quad i = 1, 2.\] By considering \(|(\mathbf{r}_1-\mathbf{r}_2) \wedge (\mathbf{u}_1 \wedge \mathbf{u}_2)|^2\), determine necessary and sufficient conditions for the lines to meet, and if they do not meet, find the shortest distance between them in the two cases \(\mathbf{u}_1 \wedge \mathbf{u}_2 = \mathbf{0}\) and \(\mathbf{u}_1 \wedge \mathbf{u}_2 \neq \mathbf{0}\).