= Solution
For direct conversion, descending masses can drive paddles in an insulated vessel of <water>. Measure the lost gravitational <potential energy> and the <temperature> rise, including the vessel's <heat capacity> and corrections for unwanted <heat transfer>. Repeated reversal of the paddles and fixed baffles inhibit simple bulk co-rotation, directing more of the input towards dissipative fluid motion. <Joule's paddle-wheel experiment> tests a quantitative mechanical-to-thermal balance, rather than merely observing that stirring can warm water.<On the Mechanical Equivalent of Heat by James Prescott Joule>[1850, friction experiments and calorimetric corrections]{ref}
For conversion through <electricity>, mechanical <work> drives a generator, whose <electric current> passes through a resistor in a <calorimeter>. The measured electrical input is $\int VI\,dt$, or $I^2Rt$ for constant <electric current> and <electrical resistance>. <Joule heating> increases the <internal energy> of the resistor and surrounding <water>. Compare the generator's mechanical input with the total thermal gains, including generator and circuit losses if the system boundary contains them. A battery alone demonstrates electrical-to-thermal conversion; without accounting for a mechanically driven source it does not establish the requested mechanical intermediate chain.
Using the given value relative to <water>, the copper requires
$$
Q=(500\,\mathrm g)(0.095\,\mathrm{cal\,g^{-1}K^{-1}})(20-10)\,\mathrm K
=475\,\mathrm{cal}.
$$
With the modern thermochemical <calorie>, $1\,\mathrm{cal}=4.184\,\mathrm J$, so the ideal required <work> is
$$
\boxed{W=1987.4\,\mathrm J\simeq1.99\,\mathrm{kJ}.}
$$
Equivalently it is about $203\,\mathrm{kgf\,m}$ using standard gravity. A nineteenth-century numerical equivalent of heat could give a slightly different rounded result. The stipulated constant <specific heat capacity> is used here; an unprovided vessel <heat capacity> and environmental losses would add to the required input.
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