Question

A flywheel is rotating at a speed of 1805 rev/min. When the power is disconnected, the speed decreases exponentially at the rate of \(32.0 \%\) per minute. Find the speed after 10.0 min.

Short answer

The speed of the flywheel after 10.0 minutes is N(10) revolutions per minute, where N(10) is calculated using N(10) = 1805 * e^(-kt) with k derived from the decay rate.

Step-by-step solution

Convert Revolutions Per Minute to Revolutions Per Second

Given that the flywheel is rotating at 1805 revolutions per minute (rev/min), we need to convert this speed into revolutions per minute (rev/min). We can leave the speed in rev/min as the decay rate is given per minute.

Understand the Exponential Decay Formula

The formula for exponential decay is given by N(t) = N_0 * e^(-kt), where N(t) is the amount after time t, N_0 is the initial amount, e is the base of the natural logarithm (approximately 2.71828), k is the decay constant, and t is the time in minutes.

Calculate the Decay Constant

Since the speed decreases by 32.0% per minute, the decay constant k can be found using the relation k = -ln(1 - decay rate). The decay rate is 32.0% or 0.320. Therefore, k = -ln(1 - 0.320).

Calculate the Speed After 10.0 Minutes

Using the exponential decay formula N(t) = N_0 * e^(-kt) with the value of k calculated in Step 3 and t = 10.0 minutes, we can find the speed of the flywheel after 10.0 minutes.

Key concepts

Natural Logarithm

Understanding the natural logarithm is crucial when dealing with exponential growth or decay processes. The natural logarithm, often abbreviated as 'ln', refers to the logarithm to the base 'e', where 'e' is an irrational and transcendental number approximately equal to 2.71828. This peculiar number is the base of the natural logarithm because of its unique properties in calculus, particularly in the study of growth and decay rates.

The natural logarithm allows us to find the time needed for a quantity to reach a certain level or to calculate the decay constant in our exponential decay formula. In our flywheel example, calculating the decay constant involved the natural logarithm due to its relation with the base 'e'. It is defined as the inverse operation of taking 'e' to the power of a number. That is, if we have a number 'a', the natural logarithm of 'a' is the power to which we must raise 'e' to get the number 'a'. More formally, if \( e^x = a \), then \( x = ln(a) \).

For instance, in the context of the flywheel's decreasing speed, the natural logarithm is used to find the decay constant 'k' and, subsequently, to determine how fast the speed of the flywheel diminishes over time.

Decay Constant

In the realm of physics and mathematics, the decay constant (represented as 'k' in our formulas) is an essential parameter representing the rate at which a quantity decreases over time. It plays a pivotal role in describing the behavior of systems that exhibit exponential decay, such as radioactive decay in nuclear physics or, as in our example, the slowing down of a flywheel.

The decay constant is integral to the exponential decay formula \( N(t) = N_0 * e^{-kt} \), determining how rapidly the initial quantity \( N_0 \) reduces to \( N(t) \) after a certain time 't'. A higher decay constant means a quicker reduction, while a lower constant indicates a slower decay. To obtain the decay constant 'k' from a given decay percentage, we manipulate the natural logarithm to reflect the decay rate, as \( k = -ln(1 - decay\_rate) \).

In the provided flywheel exercise, the decay constant is found using the decay rate of 32%, reflecting the rapidity with which the flywheel's speed diminishes each minute. Thus, understanding the decay constant is essential for quantifying the pace of change in systems experiencing exponential decay.

Revolutions Per Minute

Revolutions per minute (RPM) is a unit of measurement used to express the number of turns, or revolutions, an object makes around a fixed axis in one minute. It is a common unit used in various fields, including mechanical engineering, automotive industries, and machining, and it is particularly significant when discussing rotational speeds.

In our exercise, the flywheel starts at 1805 RPM, exemplifying how quickly it is spinning at the outset. However, when analyzing the exponential decay of its speed, it’s important to note that we observe the change in speed in the same unit of measurement over time. Keeping the unit consistent allows for direct application of the exponential decay formula without requiring conversion to a different time scale, like seconds or hours.

After determining the decay constant, we can directly apply our formula to calculate the flywheel's speed in RPM after any given period, maintaining a straightforward and relevant approach for practical purposes such as machinery maintenance or performance evaluation.