(a)
The total
(a)
Answer to Problem 68PQ
The total angular momentum of the initial wheel-clay system is
Explanation of Solution
Let the radius of pottery wheel is
It is given that clay is in the approximate shape of a sphere.
Write the expression for the density of the sphere.
Here,
Rearrange above equation to get expression of volume of sphere.
Write the expression for the volume of sphere.
Here,
Substitute
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.
Here,
Write the expression for the rotational inertia of disk.
Here,
Write the expression for the rotational inertia of the sphere.
Substitute (IV) and (V) in equation (III) to get
Write the expression for the total angular momentum of the initial wheel-clay system.
Here,
Conclusion:
It is given that the angular velocity of the system is
Convert radius of disk from
Convert angular momentum
Substitute
Substitute
Substitute
Therefore, the total angular momentum of the initial wheel-clay system is
(b)
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)
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
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.
Here,
Write he expression for the rotational inertia of cylinder.
Here,
Substitute (IX) and (IV) in (VIII) and to get
Write the expression for the percentage change in rotational inertia of system.
Write the expression for final angular momentum of the system.
Here,
Write the expression for the conservation of angular momentum.
Substitute (VII) and (XII) in above equation to get
Conclusion:
Radius of cylinder is
Calculate radius of cylinder.
Substitute
Substitute
Thus, rotational inertia of the system is decreases by
Therefore, according to conservation of angular momentum, system would speed up due to the change of the clay. The speed increases by
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Chapter 13 Solutions
Physics for Scientists and Engineers: Foundations and Connections
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