   Chapter 8, Problem 50P

Chapter
Section
Textbook Problem

Four objects—a hoop, a solid cylinder, a solid sphere, and a thin, spherical shell—each have a mass of 4.80 kg and a radius of 0.230 m. (a) Find the moment of inertia for each object as it rotates about the axes shown in Table 8.1. (b) Suppose each object is rolled down a ramp. Rank the translational speed of each object from highest to lowest, (c) Rank the objects’ rotational kinetic energies from highest to lowest as the objects roll down the ramp.

(a)

To determine
The moment of inertia of the each of the object it rotates.

Explanation

Given Info: mass of the hoop mh is 4.80kg   and radius of the hoop rh is 0.230m2

Formula to calculate the moment of inertia of the hoop,

Ih=mhrh2

• Ih is the moment of inertia of the hoop,
• mh is the mass of the hoop,
• rh is the radius of the hoop,

Substitute 4.80kg for mh and 0.230m2 for rh to find Ih ,

Ih=(4.80kg)(0.230m2)2=(4.80kg)(0.0529m2)=0.2539kgm20.254kgm2

The moment of inertia of the hoop is 0.254kgm2

Formula to calculate the moment of inertia of the solid cylinder,

Isc=12mscrsc2

• Isc is the moment of inertia of the solid cylinder,
• msc is the mass of the solid cylinder,
• rsc is the radius of the solid cylinder,

Substitute 4.80kg for msc and 0.230m2 for rsc to find Isc ,

Isc=12[(4.80kg)(0.230m2)2]=12[(4.80kg)(0.0529m2)]=0.1179kgm20.127kgm2

The moment of inertia of the solid cylinder is 0.127kgm2 .

Formula to calculate the moment of inertia of the solid sphere,

• Iss is the moment of inertia of the solid sphere,
• mss is the mass of the solid sphere,
• rss is the radius of the solid sphere,

Substitute 4

(b)

To determine
The translational speed of each of the object from highest to lowest when it rolled down the ramp.

(c)

To determine
The rotational kinetic energy of each of the object from highest to lowest when it rolled down the ramp.

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