Problem 8
Question
Find the moment of inertia with respect to the \(y\) -axis of a thin sheet of constant density \(\delta=1\) bounded by the curve \(y=\left(\sin ^{2} x\right) / x^{2}\) and the interval \(\pi \leq x \leq 2 \pi\) of the \(x\) -axis.
Step-by-Step Solution
Verified Answer
The moment of inertia is \( \frac{\pi}{2} \).
1Step 1: Understand the Moment of Inertia Formula
The moment of inertia about the y-axis can be calculated with the formula \( I_y = ext{density} imes ext{integral} \), which translates to \( I_y = \int x^2 imes f(x) \, dx \). Given the density \( \delta = 1 \), this simplifies to \( I_y = \int x^2 imes f(x) \, dx \), where \( f(x) = \frac{\sin^2(x)}{x^2} \).
2Step 2: Substitute the Function in the Formula
Substitute the function \( f(x) = \frac{\sin^2(x)}{x^2} \) into the moment of inertia integral formula: \[I_y = \int_{\pi}^{2\pi} x^2 \cdot \frac{\sin^2(x)}{x^2} \, dx = \int_{\pi}^{2\pi} \sin^2(x) \, dx\]
3Step 3: Simplify and Integrate
Using trigonometric identities, \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \), the integral becomes: \[I_y = \int_{\pi}^{2\pi} \frac{1 - \cos(2x)}{2} \, dx = \frac{1}{2}\int_{\pi}^{2\pi}(1 - \cos(2x)) \, dx\]This can be split into two integrals: \[\frac{1}{2}\left[ \int_{\pi}^{2\pi}1 \, dx - \int_{\pi}^{2\pi}\cos(2x) \, dx \right]\]
4Step 4: Evaluate the Integrals
Evaluate each integral separately: \[\int_{\pi}^{2\pi}1 \, dx = [x]_{\pi}^{2\pi} = 2\pi - \pi = \pi\]For the second integral, note that \( \int \cos(2x) \, dx = \frac{1}{2} \sin(2x)\):\[\int_{\pi}^{2\pi}\cos(2x) \, dx = \frac{1}{2} \left[ \sin(2x) \right]_{\pi}^{2\pi} = \frac{1}{2}(0 - 0) = 0\]
5Step 5: Compute Final Moment of Inertia
Substitute the evaluated integrals back into the expression for \( I_y \):\[I_y = \frac{1}{2} [\pi - 0] = \frac{\pi}{2}\]
Key Concepts
Understanding Integral CalculusUtilizing Trigonometric IdentitiesApproach to Calculus Problem SolvingRole of Density in Physics
Understanding Integral Calculus
Integral calculus is an essential branch of calculus that helps us calculate the area under a curve, among other things. In this context, we are using integral calculus to determine the moment of inertia, a physical property that measures how an object's mass is distributed with respect to a particular axis.
By integrating, we are essentially summing up infinitesimally small contributions to get a total value. In the problem, we integrate over a specific interval, \([\pi, 2\pi]\), which defines the area of interest on the curve. The basic integral used is \[ I_y = \int x^2 \cdot f(x) \, dx, \] where \(f(x) = \frac{\sin^2(x)}{x^2}\). This means we are looking at how the given function contributes to the total moment of inertia across this specified interval.
By integrating, we are essentially summing up infinitesimally small contributions to get a total value. In the problem, we integrate over a specific interval, \([\pi, 2\pi]\), which defines the area of interest on the curve. The basic integral used is \[ I_y = \int x^2 \cdot f(x) \, dx, \] where \(f(x) = \frac{\sin^2(x)}{x^2}\). This means we are looking at how the given function contributes to the total moment of inertia across this specified interval.
Utilizing Trigonometric Identities
Trigonometric identities help in transforming complex trigonometric expressions into simpler forms that are easier to integrate. One of the identities crucial to solving this exercise is \(\sin^2(x) = \frac{1 - \cos(2x)}{2}\).
This identity allows us to avoid a more complex integration by breaking \(\sin^2(x)\) into parts that are easier to work with. Such simplifications are common in calculus, making seemingly complex integrals more tractable. Once transformed using this identity, the integral becomes \[ I_y = \frac{1}{2}\left[ \int_{\pi}^{2\pi}(1) \, dx - \int_{\pi}^{2\pi}(\cos(2x)) \, dx \right] \] which is significantly easier to evaluate.
This identity allows us to avoid a more complex integration by breaking \(\sin^2(x)\) into parts that are easier to work with. Such simplifications are common in calculus, making seemingly complex integrals more tractable. Once transformed using this identity, the integral becomes \[ I_y = \frac{1}{2}\left[ \int_{\pi}^{2\pi}(1) \, dx - \int_{\pi}^{2\pi}(\cos(2x)) \, dx \right] \] which is significantly easier to evaluate.
Approach to Calculus Problem Solving
Calculus problem-solving often involves breaking down complex problems into more manageable steps. The process usually begins with understanding the problem, such as knowing what needs to be found—in this case, the moment of inertia.
The next step involves setting up the correct equations or expressions, like substituting the given function into the moment of inertia formula. After simplifying these equations using mathematical tools such as identities or formulas, the final steps involve carrying out the calculations accurately to find the solution.
The next step involves setting up the correct equations or expressions, like substituting the given function into the moment of inertia formula. After simplifying these equations using mathematical tools such as identities or formulas, the final steps involve carrying out the calculations accurately to find the solution.
- First, identify the mathematical tools required, like integration techniques or identities.
- Break up the integral into parts if necessary, to make calculations simpler.
- Perform each calculation carefully, paying particular attention to limits of integration.
- Finally, combine all calculated components to solve for the desired result, double-checking for errors along the way.
Role of Density in Physics
The concept of density in physics describes how much mass is contained in a given volume. For this exercise, density is used to measure how it affects the moment of inertia. In our example, the density \(\delta = 1\), simplifies the computations as it does not change the value of the integrand.
Generally, in physics, varying density would mean the whole quantity to be integrated could become more complex. The mass distribution due to the density impacts how easily an object can rotate around an axis.
Generally, in physics, varying density would mean the whole quantity to be integrated could become more complex. The mass distribution due to the density impacts how easily an object can rotate around an axis.
- Uniform density simplifies mathematical problems as seen in this example.
- Diverse density distributions require more complex integrations and adjustments for accurate calculations.
- In practical applications, understanding how density affects physical properties is crucial for engineers and physicists.
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