Question

An engine delivers 175 hp to an aircraft propeller at 2400 rev/min. (a) How much torque does the aircraft engine provide? (b) How much work does the engine do in one revolution of the propeller?

Short answer

(a) Torque is 519.2 Nm. (b) Work done in one revolution is 3261.2 J.

Step-by-step solution

Convert Horsepower to Watts

The engine delivers 175 horsepower (hp). We first need to convert this to watts because horsepower is not a standard SI unit. The conversion factor is: \[1 \text{ hp} = 746 \text{ W}\]Thus, \[175 \text{ hp} = 175 \times 746 = 130550 \text{ W}\]

Convert Revolutions per Minute to Radians per Second

The engine operates at 2400 revolutions per minute. We need to convert this to radians per second since the SI unit for angular velocity is radians/second. 1 revolution = \( 2\pi \) radians, and 1 minute = 60 seconds, so: \[2400 \frac{\text{rev}}{\text{min}} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} = 2400 \times \frac{2\pi}{60} \approx 251.33 \text{ rad/s}\]

Calculate Torque

Torque \( \tau \) is related to power \( P \) and angular velocity \( \omega \) by the formula: \[P = \tau \times \omega\]Rearranging for torque, \[\tau = \frac{P}{\omega}\]Substitute the known values: \[\tau = \frac{130550}{251.33} \approx 519.2 \text{ Nm}\]

Compute Work Done Per Revolution

The work done in one revolution can be found by using the torque and the angle in radians for one revolution. Work done \( W \) is given by: \[W = \tau \times \theta\]where \( \theta \) is the angle of one complete revolution which is \( 2\pi \) radians. So, \[W = 519.2 \times 2\pi \approx 3261.2 \text{ J}\]

Key concepts

Power Conversion

Power conversion is essential to understand when dealing with engines and machinery. Power is initially given in horsepower (hp) when discussing engines, but for calculations, we usually need to convert this to watts, which is the SI unit for power. This is because the majority of scientific equations and constants are based on SI units.
To convert from horsepower to watts, use this simple conversion factor: 1 hp equals 746 watts. For example, if an engine delivers 175 hp, this is equivalent to 175 multiplied by 746, resulting in 130,550 watts. This conversion allows us to use this value seamlessly in further calculations such as torque or work done.

Angular Velocity

Angular velocity describes how fast an object rotates or spins in terms of the angle turned per unit of time. In mechanics, angular velocity is often given in revolutions per minute (rev/min) or rpm. However, in most calculations, especially those using SI units, angular velocity needs to be converted into radians per second (rad/s).
There are some key conversions to keep in mind:

• 1 revolution is equal to \( 2\pi \) radians because the circumference of a circle is \( 2\pi \).
• 1 minute is 60 seconds.

So, to convert 2400 rev/min to rad/s, multiply 2400 by \( \frac{2\pi}{60} \), arriving at approximately 251.33 rad/s. Understanding these units is crucial for correctly applying formulas in physics.

SI Units

SI units, or the International System of Units, provide a standard set of units based on the metric system. These units cover parameters like length, mass, time, and electric current, ensuring consistency in scientific and engineering calculations worldwide.
For example:

• Power is measured in watts (W).
• Torque is measured in newton meters (Nm).
• Angular velocity is in radians per second (rad/s).
• Work and energy are measured in joules (J).

Employing SI units allows for uniformity and accuracy in calculations, making international collaboration and communication in the scientific community much more straightforward.

Work Done

The concept of work done plays a significant role when discussing energy transformation in mechanical systems. In physics, work is defined as the force exerted over a distance. It is calculated as the product of torque and the angle turned, both of which must be in SI units for calculation.
The formula for work done in a rotational system is given by: \[ W = \tau \times \theta \] where \( \tau \) is the torque in newton meters and \( \theta \) is the angle turned in radians.
For example, if a propeller completes one full revolution, the angle \( \theta \) is \( 2\pi \) radians. If the torque provided is 519.2 Nm, then the work done in one revolution is \( 519.2 \times 2\pi = 3261.2 \) joules. This value reflects how much energy the engine expends to make that complete turn.