/*! 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

91影视

Log out

Learning Materials

Features

Discover

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 \(\theta\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A \(55-\mathrm{g}\) firecracker is at rest at the origin when it explodes into three pieces. The first, with mass \(7 \mathrm{~g}\), moves along the \(x\)-axis at \(33 \mathrm{~m} / \mathrm{s}\). The second, with mass \(15 \mathrm{~g}\), moves along the \(\mathrm{y}\)-axis at 26 \(\mathrm{m} / \mathrm{s}\). Find the velocity of the third piece.

Mass \(m\) moving at speed \(v\), approaches mass \(2 m\), moving in the opposite direction at speed \(2 v\). The two collide elastically head-on. Show that mass \(2 m\) stops while mass \(m\) rebounds at speed \(3 v\).

A proton (mass \(1 \mathrm{u}\) ) collides elastically with a stationary deuteron (mass \(2 \mathrm{u}\) ). If the proton is deflected \(37^{\circ}\) from its original direction, what fraction of its kinetic energy does it transfer to the deuteron?

Why does a liquid-carrying tanker usually have a baffled tank, where the tank is divided into several small parts with holes that let the liquid flow through them?

Two objects moving in opposite directions with the same speed \(v\) undergo a totally inelastic collision, and two-thirds of the initial kinetic energy is lost. Find the ratio of their masses.
See all solutions

Recommended explanations on Physics Textbooks

Thermodynamics

Read Explanation

Waves Physics

Read Explanation

Electrostatics

Read Explanation

Rotational Dynamics

Read Explanation

Dynamics

Read Explanation

Radiation

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.