Physics for Scientists and Engineers: Foundations and Connections
Physics for Scientists and Engineers: Foundations and Connections
15th Edition
ISBN: 9781305289963
Author: Debora M. Katz
Publisher: Cengage Custom Learning
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Chapter 13, Problem 68PQ

(a)

To determine

The total angular momentum of the initial wheel-clay system using estimated values of masses of clay and wheel, the radius of the wheel, and the density of the clay.

(a)

Expert Solution
Check Mark

Answer to Problem 68PQ

The total angular momentum of the initial wheel-clay system is 5.05kgm2/s .

Explanation of Solution

Let the radius of pottery wheel is 7in , the  approximate mass of the pottery wheel is 25.0kg , and density of clay is 1500kg/m3 . Estimated mass of a clay vase is 2.50kg.

It is given that clay is in the approximate shape of a sphere.

Write the expression for the density of the sphere.

  ρ=MsphereVsphere

Here, ρ is the density of the sphere, Msphere is the mass of the sphere and Vsphere is the volume of the sphere.

Rearrange above equation to get expression of volume of sphere.

  Vsphere=Msphereρ                                                                                                          (I)

Write the expression for the volume of sphere.

  Vsphere=43πRsphere3

Here, Rsphere is the radius of sphere.

Substitute 43πRsphere3 for Vsphere in equation (I) to modify equation (I).

  43πRsphere3=MsphereρRsphere=3Msphere4ρπ3                                                                                              (II)

The wheel is in form of a disk. Thus, consider wheel as disk to find rotational inertia.

Write the expression for the total rotational inertia of wheel-clay system.

  Itotal,i=Idisk+Isphere                                                                                                   (III)

Here, Itotal,i is the initial rotational inertia of wheel-clay system, Idisk is the rotational inertia of the disk and Isphere is the rotational inertia of disk.

Write the expression for the rotational inertia of disk.

  Idisk=12MdiskRdisk2                                                                                                  (IV)

Here, Mdisk is the mass of the disk and Rdisk is the radius of the disk.

Write the expression for the rotational inertia of the sphere.

  Isphere=25MsphereRsphere2                                                                                              (V)

Substitute (IV) and (V) in equation (III) to get Itotal,i .

  Itotal,i=12MdiskRdisk2+25MsphereRsphere2                                                                        (VI)

Write the expression for the total angular momentum of the initial wheel-clay system.

  Li=Itotal,iωi                                                                                                            (VII)

Here, Li is the total angular momentum of the initial wheel-clay system and ωi is the initial angular momentum of the system.

Conclusion:

It is given that the angular velocity of the system is 120rpm.

Convert radius of disk from 7in to meter.

  Rdisk=(7in.)(1m39.37in.)=0.178m

Convert angular momentum 120rpm into rads .

  ωi=(120revmin)(2πrad1rev)(1min60s)=12.56rads

Substitute 2.50kg for Msphere and 1500kg/m3 for ρ in equation (II) to get Rsphere .

  Rsphere=3(2.50kg)4(1500kg/m3)(3.14)3=0.074m

Substitute 0.178m for Rdisk, 25.0kg for Mdisk, 2.50kg for Msphere and 0.074m for Rsphere in equation (VI) to get Itotal,i .

  Itotal,i=12(25.0kg)(0.178m)2+25(2.50kg)(0.074m)2=0.401kgm2

Substitute 0.401kgm2 for Itotal,i and 12.56rads for ωi in equation (VII) to get Li .

  Li=(0.401kgm2)(12.56rads)=5.05kgm2/s

Therefore, the total angular momentum of the initial wheel-clay system is 5.05kgm2/s .

(b)

To determine

Whether the system need to be sped up or slowed down, or should nothing be done to maintain a constant angular speed, if shape of the clay changed into a tall, cylindrical vase.

(b)

Expert Solution
Check Mark

Answer to Problem 68PQ

According to conservation of angular momentum, system would speed up due to the change of shape of the clay. The speed increases by 1% .

Explanation of Solution

The shape of clay changed from sphere to cylinder. It is given that radius of the cylindrical vase is one fourth of radius of sphere.

Since shape of clay is cylinder, calculate rotational inertia of cylinder to find rotational inertia of clay.

Write the equation for the total rotational inertia of the wheel-clay system after the shape is changed.

  Itotal,f=Idisk+Icylinder                                                                                             (VIII)

Here, Itotal, f is the rotational inertia of the system after shape of sphere clay changed to cylindrical shape and Icylidner is the rotational inertia of cylinder about its own axis.

Write he expression for the rotational inertia of cylinder.

  Icylinder=12McylidnerRcylinder2                                                                                         (IX)

Here, Mcylinder is the mass of clay and Rcylinder is the radius of cylindrical vase.

Substitute (IX) and (IV) in (VIII) and to get Itotal,f .

  Itotal,f=12MdiskRdisk2+12McylinderRcylinder2                                                                       (X)

Write the expression for the percentage change in rotational inertia of system.

  %change=Itotal,fItotal,iItotal,i×100                                                                                (XI)

Write the expression for final angular momentum of the system.

  Lf=Itotal,fωf                                                                                                         (XII)

Here, Lf is the final angular momentum and ωf is the final angular velocity.

Write the expression for the conservation of angular momentum.

  Li=Lf

Substitute (VII) and (XII) in above equation to get ωf .

  Itotal,iωi=Itotal,fωf                                                                                                 (XIII)

Conclusion:

Radius of cylinder is 14th of sphere.

Calculate radius of cylinder.

  Rcylinder=14(0.074m)=0.018m

Substitute 0.178m for Rdisk, 25.0kg for Mdisk, 2.50kg for Mcylinder and 0.018m for Rcylinder in equation (X) to get Itotal,f .

  Itotal,f=12(25.0kg)(0.178m)2+25(2.50kg)(0.018m)2=0.397kgm2

Substitute 0.397kgm2 for Itotal,f and 0.401kgm2 for Itotal,i in (XI) to get %change.

  %change=0.397kgm20.401kgm20.401kgm2×100=1.0%

Thus, rotational inertia of the system is decreases by 1.0% so that from conservation of angular momentum given in equation (XIII), angular momentum of the system increases by 1.0% .

Therefore, according to conservation of angular momentum, system would speed up due to the change of the clay. The speed increases by 1% .

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Chapter 13 Solutions

Physics for Scientists and Engineers: Foundations and Connections

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