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Chapter 8: Problem 30

A solid ball, a solid disk, and a hoop, all with the same mass and the same radius, are set rolling without slipping up an incline, all with the same initial energy. Which goes farthest up the incline? A. the ball B. the disk C. the hoop D. the hoop and the disk roll to the same height, farther than the ball E. they all roll to the same height

Short Answer

Expert verified
The answer to the exercise is C, the hoop goes farthest up the incline among the three objects given its greater resistance to rotational motion.

Step by step solution

01

Identify Moment of Inertia

First, recall the formulas for the moments of inertia of a ball (\(\frac{2}{5}mr^2\)), a disk (\(\frac{1}{2}mr^2\)), and a hoop (\(mr^2\)). The potential energy at maximum height of the incline for all objects is \(mgh\), where m is mass, g is gravity and h is height. Since energy is conserved and all objects start with the same initial energy, the potential energy at the maximum height is equal to the initial kinetic energy.
02

Form Equations

Construct equations equating the initial kinetic energy with final potential energy for all objects with the original rotational kinetic energy of each object: Ball: \(\frac{2}{5}mvr^2 = mgh\), Disk: \(\frac{1}{2}mvr^2 = mgh\), Hoop: \(mvr^2 = mgh\)
03

Calculate Heights

Solve each equation to obtain values for h (height), substituting v as \( g*r \) as they roll without slipping, Ball: \( h = \frac{2}{5}vr = \frac{2}{5}gr = \frac{2}{5}h_{ball} \),Disk: \( h = \frac{1}{2}vr = \frac{1}{2}gr = \frac{1}{2}h_{disk} \),Hoop: \( h = vr = gr = h_{hoop} \).
04

Compare Heights

Comparing the three heights derived from the equations, we find that the hoop reaches the highest point, followed by the disc and then, the ball. Hence, the hoop goes farthest up the incline.

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Key Concepts

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

Moment of Inertia
The Moment of Inertia is a fundamental concept when discussing rotational motion. It describes how difficult it is to change an object's rotational state. Think of it like mass in linear motion.
For a rotating object, the moment of inertia depends on its mass and how that mass is distributed relative to the rotation axis.
  • Ball: The moment of inertia is \(\frac{2}{5}mr^2\), because the mass is symmetrically distributed.
  • Disk: Slightly larger value \(\frac{1}{2}mr^2\), due to mass being slightly more distributed from the center.
  • Hoop: Highest \(mr^2\), because all mass is concentrated at the rim.
The hoop requires more effort to start or stop spinning because of its greater moment of inertia. This difference matters significantly when these objects roll up an incline.
Rotational Kinetic Energy
Rotational Kinetic Energy is the energy due to an object's rotation and is similar to translational kinetic energy.
While translational kinetic energy relates to an object's center of mass motion, rotational kinetic energy relates to rotation around an axis.
The formula is given by \[ E_{rotational} = \frac{1}{2}I\omega^2 \] where
  • \(I\) is the moment of inertia, representing resistance to changes in rotational velocity.
  • \(\omega\) is the angular velocity, expressing rotation speed.
For rolling objects, both kinetic energies play a role.
Their initial moment of inertia informs how energy is partitioned between linear and rotational forms.
Rolling Motion
When an object rolls on a surface, it involves both rotation and translation. Notably, in rolling without slipping, a relationship between linear and angular velocity exists: \[ v = r\omega \]

91影视

This intertwines kinetic and rotational motion in analyses.
  • Linear motion moves the center of mass uphill.
  • Rotational motion involves spinning around the center of mass.
Each object's moment of inertia influences how much energy splits into these motion forms. For instance, greater inertia means more energy in rotation, as seen in the hoop, which translates to less translational energy during uphill rolling.
Incline Physics
Incline physics examines motion on sloped surfaces. As objects roll uphill, they convert kinetic energy (potential and rotational) into potential energy. The pivotal energy conservation principle is:\[\text{Initial Total Energy} = \text{Potential Energy at Peak}\] Since all models begin with identical initial energies:
  • Object with less kinetic energy transfers more of it to potential energy.
  • This conversion determines how far it rolls uphill.
Hoops, with the higher moment of inertia, retain less translational kinetic energy, allowing them to roll higher compared to disks or balls, confirming observations in the original problem.

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Most popular questions from this chapter

In 1932 Albert Dremel of Racine, Wisconsin, created his rotary tool that has come to be known as a dremel. (a) Suppose a dremel starts from rest and achieves an operating speed of \(35,000 \mathrm{rev} / \mathrm{min}\). If it requires \(1.2 \mathrm{~s}\) for the tool to reach operating speed and it is held at that speed for \(45 \mathrm{~s}\), how many rotations has the bit made? Suppose it requires another \(8.5 \mathrm{~s}\) for the tool to return to rest. (b) What are the angular accelerations for the start-up and the slowdown periods? (c) How many rotations does the tool complete from start to finish?

Find an expression for the moment of inertia of a spherical shell (for example, the peel of an orange) that has a mass \(M\), a radius \(R\) and rotates about an axis which is tangent to the surface.

Suppose a roulette wheel is spinning at \(1 \mathrm{rev} / \mathrm{s}\). How long will it take for the wheel to come to rest if it experiences an angular acceleration of \(-0.02 \mathrm{rad} / \mathrm{s}^{2}\) ? How many rotations will it complete in that time?

\(\bullet\) Bob and Lily are riding on a merry-go-round. Bob rides on a horse at toward the outer edge of a circular platform and Lily rides on a horse toward the center of the circular platform. When the merry-go-round is rotating at a constant angular speed \(\omega\), Bob's speed \(v\) is A. exactly half as much as Lily's. B. larger than Lily's. C. smaller than Lily's. D. the same as Lily's. E. exactly twice as much as Lily's.

In your own words, define rotation and revolution and compare the concepts of rotational motion and orbital motion (revolution) for a planet in our solar system.
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