Question

A machine part has a resistor X protruding from an opening in the side. This resistor is connected to three other resistors, as shown in Fig. E26.2. An ohmmeter connected across a and b reads 2.00 Ω. What is the resistance of X?

Short answer

The resistance of X is 7.5 $\Omega$.

Step-by-step solution

Definition of resistance

The resistance that is measured by the ohmmeter represents the equivalent resistance of the total resistors because these four resistors are connected in parallel.

Therefore, use an equation that represents the connection of the resistors in parallel as:

$$\begin{gathered} \frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\frac{1}{R_x} \\ \frac{1}{R_x}=\frac{1}{R_{eq}}-\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3} \end{gathered}$$

Resistance of X

Substitute to get R_x as:

$$\begin{gathered} \frac{1}{R_x}=\frac{1}{R_{eq}}-\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3} \\ =\frac{1}{2\Omega }-\frac{1}{15\Omega }-\frac{1}{5\Omega }-\frac{1}{10\Omega } \\ =\frac{2}{15} \\ {R}_x=\frac{15}{2}\Omega \end{gathered}$$

The resistance of X is 7.5$\Omega$ .