The energy table

The bubble diagram translates into an energy table: each column is a type of energy (a bubble), each row is a state. It’s the definitive computational tool, the one in which the qualitative diagram becomes a system of numbers ready to be solved.

	$E_{\text{cin}}$	$E_{\text{pot,g}}$	$E_{\text{pot,m}}$	$E_{\text{term}}$	$\boldsymbol{E_{\textbf{tot}}}$
A	$\frac{1}{2}mv_A^2$	$mgh_A$	$\frac{1}{2}K(\ell_A-\ell_0)^2$	$mc_sT_A$	$\Sigma$
B	$\frac{1}{2}mv_B^2$	$mgh_B$	$\frac{1}{2}K(\ell_B-\ell_0)^2$	$mc_sT_B$	$\Sigma$

The table is filled in row by row, writing the value (or symbolic expression) of each energy in state A and state B. The last column, the total energy, is the sum of all the others in that row. Comparing the two sums is what solves the problem.

Principle — Conservation of energy

If no arrows enter or leave the system (or if their sum is zero), then: $\ev{E_{\text{tot}}^A = E_{\text{tot}}^B}$ The total energy of the system is conserved.

When total energy is conserved, the two cells of the last column are equal: setting them equal gives the equation linking state A to state B, from which the sought unknown (a velocity, a height, a spring compression) is found. It’s the operational, row-by-row translation of the principle of conservation of energy.

The number of columns in the table depends on the problem: if there's no spring, no $E_\text{pot,m}$ column; if there's no friction, no $E_\text{term}$. The interaction diagram tells us which columns are needed!

Links

Topics: Lavoro ed energia Concepts: Conservazione dell’energia meccanica Skills: Uso della tabella energetica Methods: Tabella energetica a stati

Related exercises: Esercizio svolto — mela e molla · Problema — Blocco su piano inclinato liscio · Problema — Molla compressa da un blocco