Problem 45 - Calculus

Question

When a car's engine makes less than about 200 revolutions per minute, it stalls. What is the period of the rotation of the engine when it is about to stall?

Step-by-Step Solution

Answer

The period is \( \frac{3}{10} \) seconds.

Step 1: Understanding the Problem

The problem states that a car's engine stalls when it makes fewer than 200 revolutions per minute. We need to find the period of the rotation, which means the time it takes for one complete revolution when the engine is operating at this critical speed of 200 revolutions per minute.

Step 2: Converting to Revolutions per Second

First, convert the engine speed to revolutions per second for easier calculation. Since there are 60 seconds in a minute, we calculate the revolutions per second as follows: \[ \text{Revolutions per second} = \frac{200 \text{ revolutions per minute}}{60 \text{ seconds per minute}} = \frac{10}{3} \text{ revolutions per second} \]

Step 3: Calculating the Period

The period is the reciprocal of the frequency (revolutions per second). Therefore, to find the period (in seconds), we take the reciprocal of \( \frac{10}{3} \) revolutions per second:\[ \text{Period} = \frac{1}{\frac{10}{3}} \text{ seconds per revolution} = \frac{3}{10} \text{ seconds per revolution} \]Thus, the period is \( \frac{3}{10} \) seconds.

Key Concepts

Period of Rotation

In the realm of rotational motion, the period of rotation is an essential concept. It refers to the time it takes for an object, such as a car engine, to complete one full rotation. When considering engines, especially those in cars, understanding the period is crucial for diagnosing performance issues
such as stalling. To determine the period, you first need to know how fast the engine is rotating, often measured in revolutions per minute (RPM). If an engine needs around 200 RPM to avoid stalling, as in our example, the next step would be to convert this into revolutions per second (RPS). This conversion makes it easier to calculate the period. Once you have the speed in RPS, the period is simply the reciprocal of this value. It's a method that hinges on the inverse relationship between frequency, measured in RPS, and the period. Calculating the period gives a tangible sense of how long each individual engine rotation takes, providing critical insights into engine performance at low speeds.

Revolutions per Second

Revolutions per second (RPS) is a unit of speed in rotational motion. It's an expression of how many turns an object makes in a single second.
In real-world mechanics, like car engines, RPS provides a clearer and more practical way to understand engine speed.To convert from RPM to RPS, you divide the RPM number by 60, since there are 60 seconds in a minute. For example, an engine operating at 200 RPM would equate to \( \frac{200}{60} \) RPS, simplifying to approximately \( \frac{10}{3} \) RPS.
Understanding RPS can help in calculations involving the period of rotation and other dynamic analyses. By shifting focus from minutes to seconds, engineers and mechanics can obtain a quick and clear understanding of how efficiently and rapidly an engine is running.

Frequency and Period Relationship

The relationship between frequency and period is fundamental in the study of rotational mechanics. Frequency refers to how often something occurs in a unit of time, and in the context of rotation, it is the number of revolutions per second (RPS).The period, on the other hand, indicates how long it takes to complete one full cycle, or one revolution. Notably, frequency and period are reciprocal: if you know one, you can easily determine the other using the formula:

• Period \( (T) = \frac{1}{\text{Frequency} (f)} \)
• Frequency \( (f) = \frac{1}{\text{Period} (T)} \)

This reciprocal relationship allows for seamless transitions between understanding the speed in terms of time per revolution and revolutions per second. It is especially useful in practical scenarios, like adjusting car engine performance, where both quick revolution counts and specific duration calculations provide valuable information for maintaining optimal engine flow and preventing issues such as stalling.