Problem 20 - Applied Physics

Question

Find the power developed by an engine with torque \(1250 \mathrm{~N} \mathrm{~m}\) applied at \(5000 \mathrm{rpm}\)

Step-by-Step Solution

Answer

The power developed is 654.5 kW.

Step 1: Understand the Formula

The power developed by an engine can be calculated using the formula: \( P = \tau \cdot \omega \), where \( P \) is the power, \( \tau \) is the torque, and \( \omega \) is the angular velocity. However, \( \omega \) must be in radians per second (rad/s), not in revolutions per minute (rpm).

Step 2: Convert RPM to Radians per Second

To convert revolutions per minute to radians per second, we use the conversion: \( \omega = \frac{2\pi \times \text{RPM}}{60} \). In this problem, \( \omega = \frac{2\pi \times 5000}{60} \approx 523.6 \) rad/s.

Step 3: Calculate Power

Substituting the known values into the power formula, we have: \( P = 1250 \times 523.6 = 654500 \) Watts. Therefore, the power developed by the engine is 654.5 kW (since 1 kW = 1000 W).

Key Concepts

Understanding Torque

Torque is a fundamental concept in physics and engineering, often referred to as the rotational equivalent of linear force. It is the measure of how much a force acting on an object causes that object to rotate. Imagine opening a door; the torque is higher if you push at the edge farthest from the hinge, compared to pushing closer to the hinge. This is because torque is the product of the force applied and the distance from the pivot point (hinge).In mathematical terms, torque \( \tau \) is calculated as:

• \( \tau = r \times F \)

where \( r \) is the distance to the pivot point and \( F \) is the force applied. In engines, it represents the twisting force that the engine generates to rotate around its axis. High torque means the engine can produce more rotation per unit of force, leading to powerful performance. This concept is crucial in understanding engine performance or any system involving rotational mechanics.

Converting Angular Velocity

Angular velocity is an essential factor when calculating power in any rotating system, such as an engine. It defines how quickly an object rotates or spins around a central point, usually in terms of revolutions per minute (rpm).However, to compute power using the formula given in physics \( P = \tau \cdot \omega \), angular velocity \( \omega \) must be expressed in radians per second (rad/s).To convert rpm to rad/s, the formula used is:

• \( \omega = \frac{2\pi \times \text{RPM}}{60} \)

This conversion accounts for the fact that one complete revolution (360 degrees) is \( 2\pi \) radians, and there are 60 seconds in a minute. Proper conversion is crucial in ensuring accurate power calculations. Once the conversion is made, it becomes straightforward to apply the values in the power formula and proceed with the calculations.

Evaluating Engine Performance

The performance of an engine is generally assessed by its ability to produce power. Power is essentially the rate at which work is done or energy is transferred. In the context of engines, a higher power output means the engine can do more work in a shorter amount of time. The power produced by an engine depends on two main factors:

• Torque: The twisting force the engine produces.
• Angular velocity: How fast the engine spins or rotates.

Together, these factors interact to deliver a performance characteristic suitable for different applications. For example, in vehicles, higher engine power can equate to faster acceleration and greater speed. This makes understanding how to calculate power crucial for designing and analyzing engine capabilities, especially when optimizing for specific tasks or conditions.

Using Radians per Second

Radians per second (rad/s) is a standard unit in physics to measure rotational speed. Unlike revolutions per minute (rpm), rad/s directly correlates to angular movements in mathematics, making it the preferred unit for precise scientific calculations.Understanding radians is pivotal when crossing into rotational dynamics. A radian is the angle created when the radius of a circle is laid along its circumference. One full revolution around a circle is \( 2\pi \) radians, illustrating the bridge between linear and angular calculations.For students or professionals dealing with rotational systems, converting rotational speed to rad/s allows one to precisely implement equations that predict real-world behaviors, thus ensuring better design and analysis results. Accurately converting units helps in achieving more reliable results when solving physics problems, especially those related to power in engines or machinery.