An electric motor consumes 9.00 kJ of electrical energy in 1.00 min. If one-third of this energy goes into heat and other forms of internal energy of the motor, with the rest going to the motor output, how much torque will this engine develop if you run it at 2500 rpm?

Verified step by step guidance

1. Calculate the total energy consumed by the motor in one minute, which is given as 9.00 kJ.
2. Determine the amount of energy converted to useful work by subtracting the energy lost as heat and other forms (one-third of the total energy) from the total energy consumed.
3. Convert the useful energy from kJ to joules (since 1 kJ = 1000 joules) to facilitate further calculations.
4. Calculate the power output of the motor by dividing the useful energy by the time in seconds (since power is energy per unit time and 1 minute = 60 seconds).
5. Use the formula for power in terms of torque and angular velocity, $P=\tau \omega$, where $\tau$ is the torque and $\omega$ is the angular velocity in radians per second. Convert the motor's speed from rpm to radians per second ($\omega =\text{rpm}\times \frac{2\pi }{60}$), and solve for $\tau$ by rearranging the formula to $\tau =\frac{P}{\omega }$.

Key Concepts

Energy Conversion

Energy conversion refers to the process of changing energy from one form to another. In the context of the electric motor, electrical energy is converted into mechanical energy, with some energy lost as heat. Understanding this concept is crucial for calculating the effective output energy available for doing work, which directly impacts the motor's performance.

Torque

Torque is a measure of the rotational force applied by an engine or motor. It is calculated as the product of the force applied and the distance from the pivot point (lever arm). In this scenario, torque is essential for determining how effectively the motor can perform work at a given rotational speed, which is expressed in revolutions per minute (rpm).

Rotational Speed (RPM)

Rotational speed, measured in revolutions per minute (rpm), indicates how fast an object rotates around an axis. In electric motors, rpm is a critical factor that influences the relationship between torque and power output. Understanding this relationship helps in calculating the torque produced by the motor when it operates at a specified speed, which is vital for assessing its performance.