User:Krauss/Sandbox

TESTE HIDEN
Equivalent Spring Constant (Series)

Deriving $$k_{eq}$$ in the series case is a little trickier than in the parallel case. Defining the equilibrium position of the block to be $$x_2$$, we'll be looking for equation for the force on the block that looks like:
 * $$F_b = -k_{eq} x_2 .\,$$

To begin, we'll also define the equilibrium position of the point between the two springs to be $$x_1$$.

The force on the block is
 * $$F_b = - k_2 \left( x_2 - x_1 \right). \quad \quad \quad (1) \,$$

Meanwhile, the force on the point between the two springs is
 * $$F_s = - k_1 x_1 + k_2 (x_2 - x_1). \,$$

Now, when the block is pushed so the springs are compressed and the system is allowed to come to equilibrium, the force between the strings must sum to zero, so with $$F_s =0$$ we can solve for $$x_1 \,$$:
 * $$- k_1 x_1 + k_2 (x_2 - x_1) = 0 \,$$
 * $$- k_1 x_1 - k_2 x_1 = -k_2 x_2 \,$$
 * $$\left(k_1 + k_2 \right) x_1 = k_2 x_2 \,$$

so
 * $$x_1 = \frac{k_2}{k_1 + k_2} x_2 . \,$$

Now we just plug this back into (1):


 * $$F_b \,$$
 * $$ = -k_2 x_2 + k_2 x_1 \,$$
 * $$ = -k_2 x_2 + k_2 \left( \frac{k_2}{k_1 + k_2} x_2 \right) \,$$
 * $$ = -k_2 x_2 \left( \frac{k_1 + k_2}{k_1 + k_2} \right) + \frac{k_2^2}{k_1 + k_2} x_2 \,$$
 * $$ = x_2 \frac{-k_1 k_2 - k_2^2 + k_2^2}{k_1 + k_2} \,$$
 * }
 * $$ = -k_2 x_2 \left( \frac{k_1 + k_2}{k_1 + k_2} \right) + \frac{k_2^2}{k_1 + k_2} x_2 \,$$
 * $$ = x_2 \frac{-k_1 k_2 - k_2^2 + k_2^2}{k_1 + k_2} \,$$
 * }
 * $$ = x_2 \frac{-k_1 k_2 - k_2^2 + k_2^2}{k_1 + k_2} \,$$
 * }
 * }

Finally, the force on the block has been found:
 * $$F_b = - \left( \frac{k_1 k_2 }{k_1 + k_2} \right) x_2 .\,$$

So we can define everything in the parenthesis to be
 * $$k_{eq} = \frac{k_1 k_2 }{k_1 + k_2} .\,$$

Which can also be written:
 * $$\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2}. \,$$

= Test2 = ver tb Template_system_formalism/FAQ... Web_template_system_%28formalism%29

On CPAN's Perl code, all pl file is documented at the "__END__" mark. But there are another template language, it look like this: ... perl code ... __END__

=head1 NAME

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=head1 SYNOPSIS ...

Lembrete: Web_template_hook_styles