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Year 2 of the mathematics and statistics course of the University of Oxford

Probability, statistics and options II course of the University of Oxford

Past exam of the Probability, statistics and options II course of the University of Oxford

2008

The centre-of-<mass> coordinates are
$$
a=h\sin\theta\cos\phi,\qquad b=h\sin\theta\sin\phi,\qquad c=z+h\cos\theta.
$$
Differentiation and cancellation of cross terms gives
$$
\dot a^2+\dot b^2+\dot c^2=\dot z^2+h^2(\dot\theta^2+\dot\phi^2\sin^2\theta)-2h\sin\theta\,\dot z\dot\theta.
$$
Insert this into the supplied <kinetic energy> and subtract the gravitational potential $mgc$. Thus
$$
\boxed{L=\frac{A+mh^2}{2}(\dot\theta^2+\dot\phi^2\sin^2\theta)+\frac C2(\dot\psi+\dot\phi\cos\theta)^2+\frac m2(\dot z^2-2h\sin\theta\dot z\dot\theta)-mg(z+h\cos\theta).}
$$

Solution

Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / H / 2 / b / i / Solution

Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / H / 2 / b / i / Solution

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