Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 10: Problem 29

Three equal masses \(m\) are located at the vertices of an equilateral triangle of side \(L,\) connected by rods of negligible mass. Find expressions for the rotational inertia of this object (a) about an axis through the center of the triangle and perpendicular to its plane and (b) about an axis that passes through one vertex and the midpoint of the opposite side.

Short Answer

Expert verified
The rotational inertia about an axis through the center of the triangle (a) is \(m*L^2/4\), and about an axis passing through one vertex and the midpoint of the opposite side (b) is \(m*L^2/2\).

Step by step solution

01

Understand and apply principles of rotational inertia

Masses at the vertices of the triangle can be seen as point masses. Rotational inertia of a system of the point masses is just the sum of the moment of inertia of each individual point mass. The moment of inertia of a point mass is \(I = m*r^2\), where \(m\) is the mass and \(r\) is the distance of the point mass from the axis of rotation. In this scenario, for (a), the axis of rotation is through the center of the triangle, thus \(r\) would be the distance from each mass to the center of the triangle.
02

Identify requirements for equilateral triangle

Use the formula of the centroid of an equilateral triangle. The centroid (G) of an equilateral triangle with a side length \(L\) lies a distance of \(r = L/(2*\sqrt{3})\) from each vertex.
03

Calculate rotational inertia (a)

After substituting \(r = L/(2*\sqrt{3})\) into equation of moment of inertia, then multiply the result by 3 (because there are 3 masses), the rotational inertia about an axis through the center of the triangle \(I_G\) is given as \(I_G = 3*m*(L/(2*\sqrt{3}))^2 = m*L^2/4\).
04

Identify requirements for axis passing through vertex

For (b), consider that the axis of rotation passes through one vertex and the midpoint of the opposite side. Here, one of the masses is on the axis of rotation and thus contributes nothing to the rotational inertia. The other two masses are at a distance of \(L/2\) from the axis.
05

Calculate rotational inertia (b)

After substituting \(r = L/2\) into equation of moment of inertia, then multiply the result by 2 (because there are 2 masses contribute to the rotational inertia), the rotational inertia about an axis passing through one vertex and the midpoint of the opposite side \(I_V\) is given as \(I_V = 2*m*(L/2)^2 = m*L^2/2\).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

Express each of the following in radians per second: (a) 720 rpm: (b) \(50^{\circ} \mathrm{M} ;\) (c) 1000 rev/s; (d) 1 rev/year (Earth's angular speed in its orbit).

Calculate the rotational inertia of a uniform right circular cone of mass \(M,\) height \(h,\) and base radius \(R\) about its axis.

A \(320-\mathrm{N}\) frictional force acts on the rim of a 1.0 -m-diameter wheel to oppose its rotational motion. Find the torque about the wheel's central axis.

A \(50-\mathrm{kg}\) mass is tied to a massless rope wrapped around a solid cylindrical drum, mounted on a frictionless horizontal axle. When the mass is released, it falls with acceleration \(a=3.7 \mathrm{m} / \mathrm{s}^{2} .\) Find (a) the rope tension and (b) the drum's mass.

An eagle with 2.1 -m wingspan flaps its wings 20 times per minute, each stroke extending from \(45^{\circ}\) above the horizontal to \(45^{\circ}\) below. Downward and upward strokes take the same time. On a given down stroke, what's (a) the average angular velocity of the wing and (b) the average tangential velocity of the wingtip?
See all solutions

Recommended explanations on Physics Textbooks

Electromagnetism

Read Explanation

Nuclear Physics

Read Explanation

Physics of Motion

Read Explanation

Circular Motion and Gravitation

Read Explanation

Magnetism

Read Explanation

Quantum Physics

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.

Sign-up for free