Problem 50 - University Physics with Modern Physics

Question

A dc motor with its rotor and field coils connected in series has an internal resistance of 3.2 \Omega. When the motor is running at full load on a \(120 - \mathrm { V }\) line, the emf in the rotor is 105\(\mathrm { V }\) . (a) What is the current drawn by the motor from the line? (b) What is the power delivered to the motor? (c) What is the mechanical power developed by the motor?

Step-by-Step Solution

Answer

(a) 4.69 A; (b) 562.5 W; (c) 492.19 W.

Step 1: Identify the Known Values

We are given:- Internal resistance of the motor, \( R = 3.2 \, \Omega \).- Supply voltage, \( V = 120 \, \mathrm{V} \).- Emf in the rotor, \( E = 105 \, \mathrm{V} \).

Step 2: Calculate the Current Drawn By the Motor

Using Ohm's Law, we know that the total voltage across the motor's resistance is the supply voltage minus the rotor emf:\[V - E = I \times R\]Substitute the known values:\[120 - 105 = I \times 3.2\]Solving for \( I \):\[I = \frac{15}{3.2} \approx 4.6875 \, \mathrm{A}\]

Step 3: Calculate the Power Delivered to the Motor

Power delivered to the motor can be calculated using:\[P = V \times I\]Substitute the known values:\[P = 120 \times 4.6875 \approx 562.5 \, \mathrm{W}\]

Step 4: Calculate the Mechanical Power Developed by the Motor

The mechanical power developed is the power associated with the rotor emf:\[P_{mech} = E \times I\]Substitute the known values:\[P_{mech} = 105 \times 4.6875 \approx 492.1875 \, \mathrm{W}\]

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in electrical engineering that relates voltage, current, and resistance in an electrical circuit. It can be expressed with the formula:\[ V = I \times R \]where:

• \(V\) represents voltage in volts (V)
• \(I\) is the current in amperes (A)
• \(R\) is the resistance in ohms (\(\Omega\))

In the context of a DC motor exercise, Ohm's Law helps calculate the current drawn from the power supply. By rearranging the formula, the current \(I\) can be found as:\[ I = \frac{V - E}{R} \]Here, \(V\) is the supply voltage, \(E\) is the electromotive force (emf) induced in the motor, and \(R\) is the internal resistance.
Understanding these relationships is crucial because it allows us to determine how much current flows in the circuit when a particular voltage is applied. It also helps identify potential issues like increased resistance that may affect motor performance.

Power Calculation

Power calculation in a DC motor involves understanding how much energy is consumed and how much is converted into useful mechanical energy. Power can be calculated using the formula:\[ P = V \times I \]This equation calculates the total power delivered to the motor. The total power input depends on the supply voltage \(V\) and the current \(I\).
However, not all supplied power is converted into mechanical power. In any electrical device, some energy is lost as heat due to resistance. Thus, the mechanical power developed by the motor can be determined using the rotor's electromotive force (emf):\[ P_{mech} = E \times I \]where \(E\) is the emf in volts.
These calculations are critical to ensure the motor operates efficiently.

• High losses may indicate maintenance needs or inefficiencies in design.
• Knowing both total and mechanical power allows engineers to optimize motor design and energy consumption.

Internal Resistance

Internal resistance in a DC motor refers to the opposition to current flow within the motor itself. It includes the resistance of wires and components like rotor and field coils. This internal resistance is an integral factor when analyzing circuit performance.
In our DC motor task, internal resistance \(R\) affects the voltage drop across the motor, calculated as:\[ V - E = I \times R \]A high internal resistance can lead to significant power loss as heat, reducing the efficiency of energy conversion from electrical to mechanical.
Key aspects of internal resistance to consider include:

• It can increase with temperature as materials expand and change their conductive properties.
• Aging can also lead to higher resistance over time due to wear and oxidation.
• Regular assessments of internal resistance help maintain motor efficiency and performance.

Understanding and managing internal resistance is key to ensuring the prolonged and optimal performance of a DC motor.