Resistors and Springs

Take two resistors in series, $A$ and $B$. For this circuit, using Kirchhoff's Laws:

• the current through both resistors is equal
• the voltages of the resistors are added

Let $I$ be the current through the whole circuit. In series, the voltages are added. Using $V = IR$,
$$ V_\text{total} = V_A + V_B = IR_A + IR_B = I(R_A + R_B) = IR_\text{total} $$ Therefore the total resistance is the sum of the two resistors.

Take two springs in series, $1$ and $2$. For this combination:

• the force acting on each spring is equal (can be checked by resolving forces in equilibrium)
• the extension of the springs are added (if each spring extends by 1 cm, the combination will extend by 2 cm)

Using $F=kx$, which can be rearranged to $x=F/k$ you can see that the quantity that is equal, $F$, divided by the spring constant; compare this to $V=IR$, where $I$ is conserved but is multiplied by the resistance. Using a similar method, $$ x_\text{total} = x_1 + x_2 = \frac{F}{k_1} + \frac{F}{k_2} = F(\frac{1}{k_1} + \frac{1}{k_2}) = F(\frac{1}{k_\text{total} }) $$ Therefore the reciprocal of the total spring constant is equal to the sum of the reciprocals of the two spring constants.