Course:PHYS350/Tutorial 1 2007W

0. Find the speed at which a mass falling from rest at the distance of the Moon would hit the surface of the Earth. The radius of the Earth is 6378.1 km. The distance from the centre of the Earth to the Moon is 385000 km and the acceleration due to gravity at the surface of the Earth is 9.8 ms^-2.

2. A weight of mass $m$ is hung from the end of a spring which provides a restoring force equal to $k$ times its extension. The weight is released from rest with the spring unextended. Find its position as a function of time, assuming negligible damping.

12. The potential energy function of a paritcle of mass $m$ is

$V={\frac {cx}{x^{2}+a^{2}}}$

where $c$ and $a$ are positive constants. Sketch $V$ are a function of $x$. Find the position of stable equilibrium, and the period of small oscillations about it. Given that the particle starts from this point with velocity $v$, find the ranges of values of $v$ for which it (a) oscillates, (b) escapes to $-\infty$, and (c) escapes to $+\infty$.

21. Repeat the calculation of Problem 2 assuming that the system is critically damped. Given that the final position of equilibrium is 0.4 m below the point of release, find how close to the equilibrium position the particle is after one second.

23. Write down the solution for a harmonic oscillator for the case with $\omega _{0}>\gamma$ if the oscillator starts from $x=0$ with velocity $v$. Show that as $\omega _{0}$ is reduced to the critical value $\gamma$, the solution tends to the corresponding solution for the critically damped oscillator.

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The following problem will not count toward your grade for this tutorial. Rather, it will count as half of a reading quiz, so if you attempt it you will be earning points against the final exam. There is nothing to lose.

25. Solve the problem of an oscillator under a simple periodic force (turned on at $t=0$) by the Green's function method, and verify that it reproduces the solution of Section 2.6. [Assume that the damping is less than critical. To do the integral, write

$\sin \omega t={\frac {e^{i\omega t}-e^{-i\omega t}}{2i}}.$