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?

Verified step by step guidance

1. Identify the configuration of the resistors in the circuit. The resistors are connected in a combination of series and parallel arrangements.
2. Recognize that the resistors labeled 14.0 Ω and 15.0 Ω are in series with each other. Calculate their equivalent resistance using the formula for series resistors: $R_{\mathrm{eq}}=R_1+R_2$.
3. Next, observe that the equivalent resistance of the 14.0 Ω and 15.0 Ω resistors is in parallel with the 13.0 Ω resistor. Calculate the equivalent resistance of this parallel combination using the formula: $R_{\mathrm{eq}}=\frac{1}{\frac{1}{R_1}}$.
4. The equivalent resistance from the previous step is in series with the resistor X (R). Use the series formula again to find the total resistance between points a and b: $R_{\mathrm{total}}=R_{\mathrm{eq}}+R$.
5. Set the total resistance equal to the ohmmeter reading of 2.00 Ω and solve for the unknown resistor X (R) using algebraic manipulation: $R=2.00-R_{\mathrm{eq}}$.

Key Concepts

Series and Parallel Resistors

Resistors can be connected in series or parallel, affecting the total resistance differently. In series, resistances add up directly, while in parallel, the reciprocal of the total resistance is the sum of the reciprocals of individual resistances. Understanding these configurations is crucial for calculating the equivalent resistance in complex circuits.

Ohm's Law

Ohm's Law states that the current through a conductor between two points is directly proportional to the voltage across the two points, with the constant of proportionality being the resistance. This fundamental principle is essential for analyzing electrical circuits and determining unknown resistances when voltage and current are known.

Equivalent Resistance Calculation

To find the equivalent resistance between two points in a circuit, one must consider the arrangement of resistors. For resistors in parallel, use the formula 1/Req = 1/R1 + 1/R2 + ..., and for series, Req = R1 + R2 + .... This calculation helps in determining the resistance of unknown components when the total resistance is given.