Problem 18 - Applied Physics

Question

Find the power developed by an engine with a torque of \(12000 \mathrm{~N} \mathrm{~m}\) applied at \(20 \overline{0} 0\) rpm.

Step-by-Step Solution

Answer

The power developed is approximately 2,513,280 watts.

Step 1: Convert RPM to Radians per Second

The engine speed is given in revolutions per minute (RPM), so we need to convert this to radians per second. Use the conversion: \[ \omega = \text{RPM} \times \frac{2\pi}{60} \]. Substitute the given RPM value: \[ \omega = 2000 \times \frac{2\pi}{60} \approx 209.44 \text{ radians/second} \].

Step 2: Calculate the Power

The power generated by the engine can be calculated using the formula: \( P = \tau \cdot \omega \), where \( \tau \) is the torque and \( \omega \) is the angular velocity. Substitute the given values: \[ P = 12000 \times 209.44 \approx 2,513,280 \text{ watts} \].

Key Concepts

Torque

Torque is the measure of rotational force applied to an object. Imagine twisting a door handle or using a wrench to tighten a bolt. The strength of that twisting or turning action is torque.

• Torque is denoted by the Greek letter \( \tau \).
• The unit of measurement for torque is Newton-meters (Nm).
• It is calculated as the product of force applied and the distance from the rotational axis.

An important aspect of torque is that it quantifies how much a force acting on an object causes that object to rotate. In the context of engines, higher torque means the engine can turn more forcefully, impacting its ability to do work.

Angular Velocity

Angular velocity describes how quickly an object rotates or revolves relative to another point. It represents the rate of rotation about an axis.

• Angular velocity is denoted by the Greek letter \( \omega \).
• Its standard unit of measure is radians per second (rad/s).
• It is calculated as the angle rotated per unit of time.

Understanding angular velocity helps us quantify rotational motion. For instance, in engines, angular velocity determines how fast the engine crankshaft spins. Combined with torque, it helps in calculating the engine's power output. The faster it spins, the greater the angular velocity, affecting how efficiently power is transferred.

RPM Conversion

RPM, or Revolutions Per Minute, is a way to express how many complete turns a rotating object makes in one minute. It's like counting how many times a clock hand moves around the clock face in a minute.

To work with equations in physics that use either radians or seconds, it is often necessary to convert RPM to radians per second. The conversion formula is: \[ \omega = \text{RPM} \times \frac{2\pi}{60} \]

• Here, \(2\pi\) accounts for the circumference of a circle, as there are \(2\pi\) radians in a circle.
• The division by 60 converts minutes into seconds.

Converting RPM to radians per second is crucial in power calculations for engines and other rotating systems. It ensures compatibility with standard units of measurement in physics for more precise calculations.