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etotheipi

Just for fun!

Feel free to have a go at any of the problems.

A spherical ball of radius ##a## and centre ##C## rolls on the rough outer surface of a fixed sphere of radius ##b## and centre ##O##. Show that the radial spin ##\boldsymbol{\omega} \cdot \mathbf{c}## is conserved where ##\mathbf{c} = \mathbf{c}(t)## is a unit vector in the direction ##\overrightarrow{OC}##. Show further that$$7(a+b) \mathbf{c} \times \ddot{\mathbf{c}} + 2an \dot{\mathbf{c}} + 5g \mathbf{c} \times \mathbf{k} = \mathbf{0}$$where ##n := \boldsymbol{\omega} \cdot \mathbf{c}## and ##\mathbf{k}## is a vertical unit vector. Compare this to the equation of motion of a spinning top and deduce that the ball can roll on the surface without ever falling off!

A relativistic particle moves in a potential ##V(x) = \frac{1}{2}m \Omega^2 x^2##. The Lagrangian is $$L(x, \dot{x}) = mc^2\left[ 1- \left(1- \frac{\dot{x}^2}{c^2} \right)^{1/2} \right] - V(x)$$Show that the period of oscillations of amplitude ##a## is ##\tau = \frac{2\pi}{\Omega} \left( 1 + \frac{3}{16} \epsilon^2 + \mathcal{O}(\epsilon^4) \right)## where ##\epsilon := \Omega a /c## is small. [

A light elastic string is stretched between two points ##A## and ##B## a distance ##(n+1)a## apart, and ##n## particles of mass ##m## are attached to the string at regular intervals. Each of the ##n+1## sections of string has spring constant ##k##. If the system performs longitudinal oscillations along the line ##\overline{AB}## then show that the normal frequencies ##\omega## satisfy$$\Delta_n = \begin{vmatrix}

2\cos{\theta} & -1 & 0 & \dots & 0 & 0 \\

-1 & 2\cos{\theta} & -1 & \dots & 0 & 0 \\

0 & -1 & 2\cos{\theta} & \dots & 0 & 0 \\

\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\

0 & 0 & 0 & \dots & 2\cos{\theta} & -1 \\

0 & 0 & 0 & \dots & -1 & 2\cos{\theta}\end{vmatrix} = 0$$where ##\cos{\theta} = 1- m\omega^2 / 2k##, and by expanding the determinant along the top row hence show that$$\Delta_n = 2\cos{\theta} \Delta_{n-1} - \Delta_{n-2}$$Solve this recursion relation for ##\Delta_n## and deduce the normal frequencies of the system.

A vertical beam of light of power ##P## and radius ##a## is incident on the base of a glass hemisphere of radius ##R##, weight ##W## and refractive index ##n##. Given that ##a \ll R##, determine the power ##P## of the beam such that the hemisphere hovers in static equilibrium. Neglect all terms in ##(a/R)^3## or higher order.

A particle moves in a potential ##V## with helical symmetry; ##V## is invariant under a rotation by angle ##\alpha## about the axis ##Oz## composed with a translation ##c\alpha## in the ##z## direction. What is the conserved Noether charge corresponding to this symmetry?

Calculate the magnitude of the magnetic field produced by a square loop of side length ##a## and carrying a current ##I## at a point a distance ##x## along the axis of the loop from the centre of the loop.

Feel free to have a go at any of the problems.

**Problem 1**A spherical ball of radius ##a## and centre ##C## rolls on the rough outer surface of a fixed sphere of radius ##b## and centre ##O##. Show that the radial spin ##\boldsymbol{\omega} \cdot \mathbf{c}## is conserved where ##\mathbf{c} = \mathbf{c}(t)## is a unit vector in the direction ##\overrightarrow{OC}##. Show further that$$7(a+b) \mathbf{c} \times \ddot{\mathbf{c}} + 2an \dot{\mathbf{c}} + 5g \mathbf{c} \times \mathbf{k} = \mathbf{0}$$where ##n := \boldsymbol{\omega} \cdot \mathbf{c}## and ##\mathbf{k}## is a vertical unit vector. Compare this to the equation of motion of a spinning top and deduce that the ball can roll on the surface without ever falling off!

**Problem 2**A relativistic particle moves in a potential ##V(x) = \frac{1}{2}m \Omega^2 x^2##. The Lagrangian is $$L(x, \dot{x}) = mc^2\left[ 1- \left(1- \frac{\dot{x}^2}{c^2} \right)^{1/2} \right] - V(x)$$Show that the period of oscillations of amplitude ##a## is ##\tau = \frac{2\pi}{\Omega} \left( 1 + \frac{3}{16} \epsilon^2 + \mathcal{O}(\epsilon^4) \right)## where ##\epsilon := \Omega a /c## is small. [

**Hint**: ##h := \frac{\partial L}{\partial \dot{x}} \dot{x} - L## is a constant of motion!]**Problem 3**A light elastic string is stretched between two points ##A## and ##B## a distance ##(n+1)a## apart, and ##n## particles of mass ##m## are attached to the string at regular intervals. Each of the ##n+1## sections of string has spring constant ##k##. If the system performs longitudinal oscillations along the line ##\overline{AB}## then show that the normal frequencies ##\omega## satisfy$$\Delta_n = \begin{vmatrix}

2\cos{\theta} & -1 & 0 & \dots & 0 & 0 \\

-1 & 2\cos{\theta} & -1 & \dots & 0 & 0 \\

0 & -1 & 2\cos{\theta} & \dots & 0 & 0 \\

\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\

0 & 0 & 0 & \dots & 2\cos{\theta} & -1 \\

0 & 0 & 0 & \dots & -1 & 2\cos{\theta}\end{vmatrix} = 0$$where ##\cos{\theta} = 1- m\omega^2 / 2k##, and by expanding the determinant along the top row hence show that$$\Delta_n = 2\cos{\theta} \Delta_{n-1} - \Delta_{n-2}$$Solve this recursion relation for ##\Delta_n## and deduce the normal frequencies of the system.

**Problem 4**A vertical beam of light of power ##P## and radius ##a## is incident on the base of a glass hemisphere of radius ##R##, weight ##W## and refractive index ##n##. Given that ##a \ll R##, determine the power ##P## of the beam such that the hemisphere hovers in static equilibrium. Neglect all terms in ##(a/R)^3## or higher order.

**Problem 5**A particle moves in a potential ##V## with helical symmetry; ##V## is invariant under a rotation by angle ##\alpha## about the axis ##Oz## composed with a translation ##c\alpha## in the ##z## direction. What is the conserved Noether charge corresponding to this symmetry?

**Problem 6**(solved by @kshitij)Calculate the magnitude of the magnetic field produced by a square loop of side length ##a## and carrying a current ##I## at a point a distance ##x## along the axis of the loop from the centre of the loop.

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