(a) 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. Convert the angular speed from revolutions per minute (rev/min) to radians per second (rad/s). Use the conversion factor where 1 rev = 2\pi rad and 1 min = 60 s.
2. Calculate the torque using the formula \( \tau = \frac{P}{\omega} \), where \( \tau \) is the torque, \( P \) is the power in watts, and \( \omega \) is the angular speed in radians per second.
3. Substitute the power (150 kW or 150,000 watts) and the converted angular speed into the torque formula.
4. Perform the division to solve for the torque.
5. Ensure the units are consistent and correct throughout the calculation, and express the final torque in newton-meters (Nm).

Key Concepts

Torque

Torque is a measure of the rotational force applied to an object, typically expressed in Newton-meters (Nm). It is calculated as the product of the force applied and the distance from the pivot point (lever arm). In the context of motors, torque is crucial for understanding how effectively a motor can perform work at a given speed.

Power

Power is the rate at which work is done or energy is transferred, measured in watts (W). In this scenario, the motor's output power of 150 kW indicates how much energy it can convert into mechanical work per unit time. The relationship between power, torque, and angular speed is fundamental in determining the performance of rotating machinery.

Angular Speed

Angular speed refers to how quickly an object rotates around an axis, typically measured in revolutions per minute (rev/min) or radians per second. In this question, the motor's angular speed of 4000 rev/min indicates how many complete rotations it makes in one minute. This concept is essential for calculating torque, as it directly influences the relationship between power and torque in rotational systems.