Compute the torque developed by an industrial motor whose output is 150 kW at an angular speed of 4000 rev/min.

Verified step by step guidance

1. First, understand the relationship between power, torque, and angular speed. The formula to use is: \( P = \tau \cdot \omega \), where \( P \) is power, \( \tau \) is torque, and \( \omega \) is angular speed.
2. Convert the given power from kilowatts to watts. Since 1 kW = 1000 W, multiply 150 kW by 1000 to get the power in watts.
3. Convert the angular speed from revolutions per minute (rev/min) to radians per second (rad/s). Use the conversion factor: \( 1 \text{ rev} = 2\pi \text{ rad} \) and \( 1 \text{ min} = 60 \text{ s} \). Therefore, \( \omega = 4000 \text{ rev/min} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} \).
4. Rearrange the formula \( P = \tau \cdot \omega \) to solve for torque: \( \tau = \frac{P}{\omega} \).
5. Substitute the values of power (in watts) and angular speed (in rad/s) into the equation to find the torque \( \tau \).

Key Concepts

Torque

Torque is a measure of the rotational force applied to an object, causing it to rotate around an axis. It is calculated as the product of force and the distance from the axis of rotation, and is expressed in Newton-meters (Nm). In the context of motors, torque determines the motor's ability to perform work by rotating a shaft.

Power

Power in physics is the rate at which work is done or energy is transferred. It is measured in watts (W) and is calculated as the product of torque and angular speed for rotating systems. In this problem, the motor's power output is given as 150 kW, which is crucial for determining the torque.

Angular Speed

Angular speed refers to how quickly an object rotates or revolves relative to another point, typically measured in revolutions per minute (rev/min) or radians per second (rad/s). It is essential for calculating torque when combined with power, as it provides the rate of rotation needed to determine the motor's performance.