Question

Moment of Inertia An annular cylinder has an inside radius of \(r_{1}\) and an outside radius of \(r_{2}\) (see figure). Its moment of inertia is \(I=\frac{1}{2} m\left(r_{1}^{2}+r_{2}^{2}\right)\) where \(m\) is the mass. The two radii are increasing at a rate of 2 centimeters per second. Find the rate at which \(I\) is changing at the instant the radii are 6 centimeters and 8 centimeters. (Assume mass is a constant.)

Short answer

The rate at which the moment of inertia is changing when the radii are 6 cm and 8 cm respectively is given by \(\frac{dI}{dt}=m * (6*2 + 8*2)= 28m\)

Step-by-step solution

Differentiate the Moment of Inertia

Differentiate the moment of inertia equation with respect to time \(t\). The derivative yields \(\frac{dI}{dt}=m \left(r_{1}\frac{dr_{1}}{dt}+r_{2}\frac{dr_{2}}{dt}\right)\).

Substitute for \(dr_{1}/dt\) and \(dr_{2}/dt\)

Substitute with the rates at which the radii are increasing, which is 2 cm/s. This will yield \(\frac{dI}{dt}=m \left(r_{1}*2 + r_{2}*2\right)\).

Substitute for \(r_{1}\) and \(r_{2}\)

Substitute with the instantaneous values of radii \(r_{1}=6\) cm and \(r_{2}=8\) cm respectively. This will yield \(\frac{dI}{dt}=m \left(6*2 + 8*2\right)\).

Simplify the Expression

Simplify the above equation. The final expression will give the rate at which moment of inertia changes at the instant when radii are 6 cm and 8 cm respectively.

Key concepts

Calculus

Calculus is a vast field of mathematics that deals with rates of change (differential calculus) and the accumulation of quantities (integral calculus). At its core, calculus helps us understand how things change over time or over space, making it an indispensable tool in many areas of study, from physics to economics. Calculus is fundamentally about limits and the behavior of functions as they approach specific points or infinity. Students often encounter calculus when they need to find the maximum or minimum values of functions or when studying the motion of objects.

In the context of the textbook exercise on moment of inertia, calculus is used to calculate the rate at which the moment of inertia changes over time. By setting up and solving a related rates problem, students can see how differential calculus is applied in a practical physics application.

Related Rates

Related rates problems in calculus involve finding the rate at which one quantity changes in relation to another changing quantity. These problems require understanding relationships between variables and how they change with respect to time. Typically, related rates problems are solved by taking the derivative of an equation with respect to time, which then allows one to determine the rate of change for a particular variable when given the rates of change for other variables.

In our annular cylinder problem, the radii are changing at a constant rate, and we are asked to find how quickly the moment of inertia changes as a result. The methodology involves expressing the quantities that change over time as functions, differentiating using the chain rule, and then substituting known values to solve for the desired rate.

Differentiation

Differentiation is one of the two main operations in calculus and refers to the process of finding the derivative of a function—a measure of how a function's value changes as its input changes. The derivative itself is a function, and it can indicate not just the rate of change, but also whether the function is increasing or decreasing, and where it has peaks or troughs.

In the context of the exercise about moment of inertia, differentiation allows us to find the derivative of the moment of inertia with respect to time, \( \frac{dI}{dt} \), which we need in order to identify the rate at which the moment of inertia is changing. This process uses the chain rule to manage the rates of change of the radii as they affect the overall inertia equation.

The Chain Rule

Given that we are asked to find derivatives with respect to time, and our variables are changing over time, the chain rule becomes an essential tool. It allows us to differentiate composite functions—functions defined by other functions that are themselves functions of time. Here, the chain rule makes it possible to link the rate of change of the radii directly to the rate of change of the moment of inertia.

Physics Applications in Calculus

• The Moment of Inertia:
  Moment of Inertia, denoted by \(I\), is a physical quantity expressing an object's tendency to resist rotational motion. In physics, the use of calculus is prevalent in predicting and understanding this rotational behavior through mathematical formulations.
• Related Rates in Physics:
  In physics problems, related rates can describe various phenomena, including acceleration, force, and in this case, the moment of inertia of a rotating body. These calculations are important for engineers and physicists who design everything from machinery to sports equipment.
• The Practicality of Differentiation:
  Differentiation is used in physics to find rates of change of almost any variable aspect: velocity, force, and even energy. An understanding of differentiation allows us not only to solve textbook exercises but also to design experiments and to predict the physical behavior of objects in motion.

In our specific example, students are seeing a direct application of calculus in physics by finding how a geometric property of an object changes as its dimensions change. It demonstrates a real-world scenario where differentiation is used to calculate a crucial physical property: the moment of inertia of an annular cylinder.