Problem 30 - University Physics with Modern Physics

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?

Step-by-Step Solution

Answer

(a) The torque is approximately 519.28 Nm. (b) The work done in one revolution is approximately 3261 J.

Step 1: Convert horsepower to watts

The power output of the engine is given as 175 horsepower (hp). We need to convert this to watts (W) because the standard unit of power in physics is watts. Using the conversion factor, 1 hp = 746 W, we can calculate:\[P = 175 \text{ hp} \times 746 \frac{W}{hp} = 130550 \text{ W}\]

Step 2: Convert revolutions per minute to radians per second

Since angular velocity \( \omega \) is often needed in radians per second (rad/s), we must convert 2400 revolutions per minute (rev/min) appropriately. Use the conversion factor:\[\omega = 2400 \frac{rev}{min} \times \frac{2 \pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}}\]This gives:\[\omega \approx 251.33 \text{ rad/s}\]

Step 3: Find the torque provided by the engine

The torque \( \tau \) provided by the engine can be found using the relationship between power, torque, and angular velocity:\[P = \tau \cdot \omega\]Solving for torque gives:\[\tau = \frac{P}{\omega} = \frac{130550 \text{ W}}{251.33 \text{ rad/s}} \approx 519.28 \text{ Nm}\]

Step 4: Calculate work done in one revolution

Work done in one complete revolution can be calculated using the torque and the angular displacement in radians for one revolution (which is \(2\pi\) radians). Work \(W\) can be described by:\[W = \tau \times \theta\]Thus the work done in one revolution is:\[W = 519.28 \text{ Nm} \times 2\pi \text{ rad} \approx 3261 \text{ J}\]

Key Concepts

Power Conversion

Power conversion is an essential concept in physics and engineering that involves changing the units of power from one system to another. In this exercise, we began with power measured in horsepower (hp), a common unit used in the automotive and aviation industries. However, to perform scientific calculations, it's usually best to convert to watts (W), the standard unit of power in the International System of Units (SI).

• 1 horsepower is equivalent to 746 watts.
• Using this conversion, we can determine the power output of engines or motors in terms that are universally understood in physics and engineering fields.

Converting power to watts helps streamline calculations involving torque and angular velocity, both critical to analyzing mechanical systems. Understanding power conversion ensures accuracy and consistency when working with different systems and measures globally. It emphasizes the importance of universal standards in scientific communication.

Angular Velocity

Angular velocity is a measure of the rate of rotation. It specifies how fast an object rotates or revolves relative to another point or axis. For engines and motors, angular velocity is often given in revolutions per minute (rev/min). However, like many rotational quantities, it's usually converted to radians per second (rad/s) during calculations.

• A full circle equals 2π radians.
• To convert rev/min to rad/s, multiply by \(2π \frac{ rad}{rev}\) and divide by 60 to switch from minutes to seconds.

The angular velocity helps us understand how quickly a system rotates and is crucial for computing torque. In essence, angular velocity bridges the gap between physical rotation and mathematical representations. When computing aspects like torque in our exercise, having the angular velocity in rad/s simplifies the process and aligns with scientific standards.

Work Done

Work done is a vital concept when discussing energy transfer in mechanical systems. In rotational dynamics, work is performed when torque causes an object to rotate through some angle.
The formula for work done by a torque is:
\[W = \tau \times \theta\]
where \(\tau\) is the torque and \(\theta\) is the angular displacement.
In this exercise:

• The work done in one revolution is found by multiplying the torque by the angle of one revolution, which is \(2\pi\) radians.
• This tells us the amount of energy transferred through one complete cycle of the propeller.

Understanding work in this context helps illustrate how mechanical engines convert energy into movement. It's a straightforward demonstration of principles that underpin much of mechanical engineering and physics, showing practical applications in real-world systems.

Radian Conversion

Radians are a unit of angular measure used extensively in mathematics and physics. They offer a direct relationship between the linear distance traveled and the radius of the circle in which the motion occurs.
In rotation:

• One complete revolution is equivalent to \(2\pi\) radians, not 360 degrees.
• Using radians in calculations often simplifies the mathematics involved in rotational dynamics.

In the exercise, converting revolutions to radians was key for accurate torque calculations. This conversion forms the backbone for linking real-world rotations to mathematical models. By working within the radians framework, one can seamlessly compute other variables like angular velocity and work done. This demonstrates the efficiency and simplicity of using radians over degrees in scientific computations.