Three square roots

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Scientific programming for physicists · Scientific programming for physicists

Scientific Computing for Physicists, Ch Letting the computer do advanced algebra: operators, series and differentiated programs

Problem

Three square roots, one lesson about assumptions. In each part, say what a computer algebra system such as SymPy returns. ((a)) An LC circuit has inductance

L

and capacitance

C

. What does

\sqrt{L/C}\,\sqrt{C/L}

return with

L

and

C

as bare symbols, and what does it return once both are declared positive? Why is positive the right declaration here? ((b)) Write

\beta=v/c

for a velocity component

v

that may take either sign. State what

\sqrt{\beta^2}

returns when

\beta

carries no assumption, and what it returns once

\beta

is declared real, and once it is declared positive. Which declaration matches

v

, and which matches the speed of light

c

? ((c)) Evaluate the Lorentz factor

1/\sqrt{1-\beta^2}

for

v=-1.80\times10^{8}\ \mathrm{m\,s^{-1}}

and

c=3.00\times10^{8}\ \mathrm{m\,s^{-1}}

to three significant figures, and say why the result is unchanged for

v=+1.80\times10^{8}\ \mathrm{m\,s^{-1}}

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