Course:PHYS350/2007W Midterm 1

1. A ball is dropped from height $h$ and bounces. The coefficient of restitution at each bounce is $e$. Find the velocity immediately after the first bounce, and immediately after the nth bounce. Find how long it takes for the ball to finally come to rest.

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Let's calculate its velocity when it first hits. We have

$mgh={\frac {1}{2}}mv^{2}$

so

$v={\sqrt {2gh}}.$

After it bounces it will have a velocity of $ev_{0}$ so $v_{n}=e^{n}{\sqrt {2gh}}$. How long does each bounce last? The initial velocity is $v_{n}=e^{n}{\sqrt {2gh}}$ up and the velocity just before the next bounce is $v_{n}$ down, so each bounce lasts

$t_{n}={\frac {2v_{n}}{g}}=2e^{n}{\sqrt {\frac {2h}{g}}}.$

Let's sum over the bounces and the initial drop

$t={\sqrt {\frac {2h}{g}}}+2{\sqrt {\frac {2h}{g}}}\sum _{n=1}^{\infty }e^{n}=\left(1+{\frac {2e}{1-e}}\right){\sqrt {\frac {2h}{g}}}={\frac {1+e}{1-e}}{\sqrt {\frac {2h}{g}}}$

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2. A particle of mass $m$ slides without friction on the inside of a spherical shell of radius $R$. What is the Lagrangian for this system including a uniform gravitational field? What are the equations of motion for the mass? What quantities are conserved?

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It is natural to use spherical polar coordinates for this, so we have $z=-R\cos \theta$ so

$V=-mR\cos \theta .$

The kinetic energy is

$T={\frac {1}{2}}mR^{2}\left(\sin ^{2}\theta {\dot {\phi }}^{2}+{\dot {\theta }}^{2}\right)$

so

$L={\frac {1}{2}}mR^{2}\left(\sin ^{2}\theta {\dot {\phi }}^{2}+{\dot {\theta }}^{2}\right)+mR\cos \theta .$

And for the equations of motion,

$mR^{2}{\ddot {\theta }}=-mR\sin \theta +mR^{2}\sin \theta \cos \theta$

and

${\frac {d}{dt}}\left(mR^{2}\sin ^{2}\theta {\dot {\phi }}\right)=0$

where the quantity in parenthesis is the conserved angular momentum. The energy $T+V$ is also conserved.