Problem 13 - Applied Physics

Question

What power (in hp) is developed by an engine with torque 524 lb ft (a) at \(30 \overline{0} 0 \mathrm{rpm} ?\) (b) at 6000 rpm?

Step-by-Step Solution

Answer

299.45 hp at 3000 rpm; 598.90 hp at 6000 rpm.

Step 1: Convert RPM to Angular Velocity

The angular velocity (ω) in radians per second can be found using the formula: \[ \omega = \frac{2\pi \times \text{RPM}}{60} \]. For part (a), with \(3000\) RPM, we have: \[ \omega = \frac{2\pi \times 3000}{60} = 314.16 \text{ rad/s} \]. Similarly, for part (b) with \(6000\) RPM, \[ \omega = \frac{2\pi \times 6000}{60} = 628.32 \text{ rad/s} \].

Step 2: Calculate Power Using Torque and Angular Velocity

The power (P) developed by an engine can be calculated using the formula: \[ P = \tau \times \omega \]. Where \(\tau\) is the torque, and \(\omega\) is the angular velocity from Step 1. For part (a), the torque \(\tau = 524\) lb-ft, then: \[ P = 524 \times 314.16 = 164696.64 \text{ ft-lb/s} \]. For part (b), \[ P = 524 \times 628.32 = 329393.28 \text{ ft-lb/s} \].

Step 3: Convert Power to Horsepower

To convert the power from ft-lb/s to horsepower, use the conversion factor: \[ 1 \text{ horsepower} = 550 \text{ ft-lb/s} \]. Calculate for part (a): \[ \text{Horsepower} = \frac{164696.64}{550} \approx 299.45 \text{ hp} \]. For part (b): \[ \text{Horsepower} = \frac{329393.28}{550} \approx 598.90 \text{ hp} \].

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or revolves relative to another point. It's commonly denoted by the symbol \( \omega \) and is measured in radians per second.
In engineering mechanics, angular velocity is crucial in understanding the rotational movement of machinery parts and engines. To convert revolutions per minute (RPM) to angular velocity, the formula used is:

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

This conversion involves the geometric constant \( \pi \) (approximately 3.14159) to switch from revolution-based measures to radians, which are standard in physics and engineering for measuring angles.
Always ensure that the calculated angular velocity is in radians per second as it simplifies the integration with other units in further calculations.

Torque

Torque is a force that causes rotation and is often described as rotational force or twisting power. It determines how effectively a force can cause an object to rotate around an axis.
The torque formula is \( \tau = rF \sin \theta \), where:

• \( \tau \) is the torque.
• \( r \) is the distance from the pivot or axis to the point where the force is applied.
• \( F \) is the force applied.
• \( \theta \) is the angle between the force vector and the lever arm vector.

In our calculations, torque usually comes given, such as the 524 lb-ft in our exercise. Torque is fundamental in calculating power through rotational movements, allowing engineers to design systems that can efficiently perform tasks like moving vehicles or turning gears.

Horsepower Conversion

Horsepower is a unit of measurement for power, which describes the rate at which work is done or energy is transferred. Converting power measured in foot-pounds per second to horsepower makes understanding engine outputs easier.
The conversion factor for this is derived from the fact that 1 horsepower is defined as 550 foot-pounds per second. Therefore, converting requires the formula:

• \( \text{Horsepower} = \frac{P}{550} \)

where \( P \) is the power in ft-lb/s calculated earlier using torque and angular velocity. Horsepower conversion is essential in automotive and machinery industries where it serves to standardize the performance metrics of motors and engines. Calculating horsepower allows engineers and car enthusiasts alike to compare different engines easily.

Engineering Mechanics

Engineering mechanics involves studying forces and their effects on bodies at rest or in motion. It is a foundational aspect of all engineering fields that deals with the behavior of systems under external forces.
In the context of our problem, engineering mechanics applies to understanding how rotating parts of an engine produce power. The core components include:

• Analyzing forces that act on bodies in motion (Dynamics).
• Understanding equilibrium in stationary systems (Statics).
• Studying the strength and motion of materials (Mechanics of Materials).

By combining these areas, engineers can predict the behavior of physical systems under various conditions, leading to designs that are efficient, safe, and economically viable. Knowledge in engineering mechanics not only allows solving exercises like the one given but also prepares future engineers for tackling complex real-world engineering problems.