/*! 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 72 If the body's center of mass wer... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 72

If the body's center of mass were not placed on the rotational axis of the turntable, how would the person's measured moment of inertia compare to the moment of inertia for rotation about the center of mass? A. The measured moment of inertia would be too large. B. The measured moment of incrtia would be too small. C. The two moments of inertia would be the same. D. It depends on where the body's center of mass is placed relative to the center of the turntable.

Short Answer

Expert verified
A. The measured moment of inertia would be too large.

Step by step solution

01

Understand the concept of Moment of Inertia

The moment of inertia measures an object's resistance to rotational changes about an axis. It depends on the distribution of mass relative to that axis. If mass is further from the axis, the moment of inertia increases.
02

Analyze the Center of Mass Position

Normally, the moment of inertia is least when rotating about the center of mass. If the center of mass is not on the rotation axis, additional mass distribution around the axis affects the moment of inertia.
03

Apply the Parallel Axis Theorem

According to the Parallel Axis Theorem, if an object's center of mass is not on the rotational axis, the moment of inertia about this new axis is: \[ I = I_{cm} + Md^2 \] where \( I_{cm} \) is the moment of inertia about the center of mass, \( M \) is the mass of the body, and \( d \) is the distance from the center of mass to the new axis.
04

Compare Moments of Inertia

From the previous step, if \( d > 0 \), then \( Md^2 > 0 \). Thus, the moment of inertia, \( I \), for rotation about the axis not passing through the center of mass is greater than \( I_{cm} \).
05

Answer the Exercise Question

With the understanding that when the center of mass is not on the axis, additional inertia results, the answer to the exercise is: A. The measured moment of inertia would be too large.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Center of Mass
The center of mass is a point representing the average position of the entire mass of a body. In simple terms, it's like the "balance point" of an object.
This point is crucial in physics because it gives insight into how an object will react when forces are applied to it.
When an object rotates, the axis that passes through its center of mass usually results in the least resistance to change in rotational motion.
  • Imagine spinning a pizza dough: if you thrown it with your hand near its center, it spins more smoothly compared to when you hit it on the edge.
  • In dynamic systems, maintaining the center of mass on the rotational axis optimizes mechanical performance.
Rotational Dynamics
Rotational dynamics deals with the motion and forces that cause objects to rotate. Just as linear dynamics involves forces causing straight-line movements, rotational dynamics involves torque and angular momentum.
The moment of inertia in this context acts like mass does in linear dynamics but for rotational motion.
This means the larger the moment of inertia, the harder it is to get something spinning or to change its rotational speed.
  • Torque is the rotational equivalent of force and is what makes an object start rotating or stop.
  • Angular momentum is conserved in a closed system, meaning it remains constant if no external torque acts on the system.
  • This concept is key when analyzing systems like gyroscopes and wheels.
Parallel Axis Theorem
The Parallel Axis Theorem helps you calculate the moment of inertia of a body about any axis, given you know the moment of inertia through the center of mass.
According to this theorem, the moment of inertia around a new axis parallel to the one through the center of mass can be expressed as:
\[ I = I_{cm} + Md^2 \]
where:
  • \( I_{cm} \) is the moment of inertia about the center of mass
  • \( M \) is the total mass of the body
  • \( d \) is the perpendicular distance between the center of mass axis and new axis
This shows that the more distant the rotational axis is from the center of mass, the greater the increase in the moment of inertia.
Mass Distribution
Mass distribution refers to how mass is spread out in an object. It greatly affects an object's stability and how it will rotate.
Moving more mass farther from the axis of rotation increases the object's moment of inertia. It becomes harder to spin, leading to slower acceleration than if the mass were closer to the axis.
  • Think about figure skaters: they spin faster when they pull their arms closer to their body because their moment of inertia decreases.
  • When designing objects meant to rotate, understanding mass distribution helps in creating efficient and balanced systems.
Physics Problem-Solving
Physics problem-solving requires a methodical approach to break down complex situations into understandable parts.
Key steps include:
  • Identifying known variables like mass, forces, and their locations.
  • Understanding the principles involved, whether it鈥檚 energy conservation or moments of inertia.
  • Applying relevant formulas along with logical reasoning, such as using the Parallel Axis Theorem when dealing with off-center axes.
Equipped with these strategies, one can tackle most physics problems methodically and effectively. Practice honing these skills with varied exercises to strengthen proficiency.

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 small block on a frictionless horizontal surface has a mass of \(0.0250 \mathrm{~kg}\). It is attached to a mascless cord passing through a hole in the surface. (Sce Figure \(10.58 .\) ) The block is originally revolving at a distance of \(0.300 \mathrm{~m}\) from the hole with an angular speed of \(1.75 \mathrm{rad} / \mathrm{s}\). The cord is then pulled from below, shortening the radius of the circle in which the block revolves to \(0.150 \mathrm{~m}\). You may treat the block as a particle. (a) Is angular momentum conserved? Why or why not? (b) What is the new angular speed? (c) Find the change in kinctic encrgy of the block. (d) How much work was done in pulling the cord?

A thin. light string is wrapped around the rim of a \(4.00 \mathrm{~kg}\) solid uniform disk that is \(30.0 \mathrm{~cm}\) in diameter. A person pulls on the string with a constant force of \(100.0 \mathrm{~N}\) tangent to the disk, as shown in Figure \(10.53 .\) The disk is not attached to arything and is free to move and hum. (a) Find the angular acceleration of the disk about its center of mass and the linear acceleration of its center of mass. (b) If the disk is replaced by a hollow thin-walled cylinder of the same mass and diameter, what will be the accclerations in part (a)?

Two people are carrying a uniform wooden board that is \(3.00 \mathrm{~m}\) long and weighs \(160 \mathrm{~N}\). If one person applies an upward force equal to \(60 \mathrm{~N}\) at one end, at what point and with what force does the other person lift? Start with a free-body diagram of the board.

II Under some circumstances, a star can collapse into an extremely dense object made mostly of neutrons and called a newtron star. The density of a neutron star is roughly \(10^{14}\) times as great as that of ordinary solid matter. Suppose we represent the star as a uniform. solid, rigid sphere, both before and after the collapse. The star's initial radius was \(7.0 \times 10^{5} \mathrm{~km}\) (comparable to our sun); its final radius is \(16 \mathrm{~km}\). If the original star rotated once in 30 days, find the angular speed of the neutron star.

On an old-fashioned rotating piano stool, a woman sits holding a pair of dumbbells at a distance of \(0.60 \mathrm{~m}\) from the axis of rotation of the stool. She is given an angular velocity of \(3.00 \mathrm{rad} / \mathrm{s}\), after which she pulls the dumbbells in until they are only \(0.20 \mathrm{~m}\) distant from the axis. The woman's moment of inertia about the axis of rotation is \(5.00 \mathrm{~kg} \cdot \mathrm{m}^{2}\) and may be considered constant. Each dumbbell has a mass of \(5.00 \mathrm{~kg}\) and may be considered a point mass. Ignore friction. (a) What is the initial angular momentum of the system? (b) What is the angular velocity of the system after the dumbbells are pulled in toward the axis? (c) Compute the kinetic energy of the system before and after the dumbbells are pulled in. Account for the difference, if any.
See all solutions

Recommended explanations on Physics Textbooks

Electrostatics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Fields in Physics

Read Explanation

Oscillations

Read Explanation

Quantum Physics

Read Explanation

Engineering 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.