/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 87 Find the center of mass of a uni... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 87

Find the center of mass of a uniform slice of pizza with radius \(R\) and angular width \(\theta\).

Short Answer

Expert verified
The center of mass of the pizza slice is located at a radial distance of \( \frac{2R}{3} \) from the origin (center of the complete pizza), independent of the angular width \(\theta\) of the slice.

Step by step solution

01

Calculation of Area

First, calculate the area of the pizza slice. The area of a slice (sector) of a circle, is equal to half the square of the radius multiplied by the angle subtended by the sector at the center (in radians). Thus the formula is \( area = \frac{1}{2}R^2\theta \). As the mass is distributed uniformly, we can say that the mass \(dm\) of a small slice of angle \(d\theta\) is \( dm = \frac{1}{2}\rho R^2 d\theta \), where \( \rho \) is the mass per unit area.
02

Calculation of Slice Center at Small Angle

At small angle \(d\theta\), the center of the slice can be assumed to be at a distance of \(2R/3\) from the origin (center of the complete pizza). This is because the center of mass of a sector of a circle at small angles is at \(2R/3\) from the origin, a standard result derived using calculus.
03

Integration Over All Slices

To find the overall center of mass of the pizza slice (sector), integrate over all such slices from 0 to \(\theta\). The total mass \(m\) of the sector is \(\frac{1}{2}\rho R^2 \theta\). The location of the center of mass is given by: \( r_{cm} = \frac{1}{m} \int_{0}^{\theta} \frac{2R}{3} dm = \frac{1}{m} \int_{0}^{\theta} \frac{2R}{3} \rho R^2 d\theta \). After performing the integration, we find that \( r_{cm}= \frac{2R}{3} \), which shows that the location of the center of mass is constant and does not depend on \(\theta\).

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