2 [Einstein] conjectured that the infrared characteristic frequencies of a solid might be determined by the same forces between atoms as determine the solid's ordinary elastic behavior. The relevant quantities are these. quantity characteristic frequency v compressibility k dimensional formula L°M°T-1 L¹M-¹T² L-³M°T° L°M'T® number of atoms per cubic cm N mass of an atom m Show that there is one dimensionless product. Conclude that, in any complete relationship among quantities with these dimensional formulas, k is a constant times v-2N-1/3m-1. This conclusion played an important role in the early study of quantum phenomena.
2 [Einstein] conjectured that the infrared characteristic frequencies of a solid might be determined by the same forces between atoms as determine the solid's ordinary elastic behavior. The relevant quantities are these. quantity characteristic frequency v compressibility k dimensional formula L°M°T-1 L¹M-¹T² L-³M°T° L°M'T® number of atoms per cubic cm N mass of an atom m Show that there is one dimensionless product. Conclude that, in any complete relationship among quantities with these dimensional formulas, k is a constant times v-2N-1/3m-1. This conclusion played an important role in the early study of quantum phenomena.
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Please do Exercise 2 and 3 and please show step by step and explain.
![2 [Einstein] conjectured that the infrared characteristic frequencies of a solid might
be determined by the same forces between atoms as determine the solid's ordinary
elastic behavior. The relevant quantities are these.
quantity
characteristic frequency v
compressibility k
dimensional
formula
LOM°T-1
L¹M-¹T²
L-³ M°T°
number of atoms per cubic cm N
mass of an atom m
LºM¹Tº
Show that there is one dimensionless product. Conclude that, in any complete
relationship among quantities with these dimensional formulas, k is a constant
times v-2N-1/³m-¹. This conclusion played an important role in the early study
of quantum phenomena.
3 [Giordano, Wells, Wilde] The torque produced by an engine has dimensional
formula L²M¹T-². We may first guess that it depends on the engine's rotation
rate (with dimensional formula LºMºT−¹), and the volume of air displaced (with
dimensional formula L³ M°T°).
(a) Try to find a complete set of dimensionless products. What goes wrong?
(b) Adjust the guess by adding the density of the air (with dimensional formula
L-³M¹T°). Now find a complete set of dimensionless products.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F283ba010-5696-4bdf-b24c-d80823daa132%2Fwr9vkb_processed.png&w=3840&q=75)
Transcribed Image Text:2 [Einstein] conjectured that the infrared characteristic frequencies of a solid might
be determined by the same forces between atoms as determine the solid's ordinary
elastic behavior. The relevant quantities are these.
quantity
characteristic frequency v
compressibility k
dimensional
formula
LOM°T-1
L¹M-¹T²
L-³ M°T°
number of atoms per cubic cm N
mass of an atom m
LºM¹Tº
Show that there is one dimensionless product. Conclude that, in any complete
relationship among quantities with these dimensional formulas, k is a constant
times v-2N-1/³m-¹. This conclusion played an important role in the early study
of quantum phenomena.
3 [Giordano, Wells, Wilde] The torque produced by an engine has dimensional
formula L²M¹T-². We may first guess that it depends on the engine's rotation
rate (with dimensional formula LºMºT−¹), and the volume of air displaced (with
dimensional formula L³ M°T°).
(a) Try to find a complete set of dimensionless products. What goes wrong?
(b) Adjust the guess by adding the density of the air (with dimensional formula
L-³M¹T°). Now find a complete set of dimensionless products.
Expert Solution
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As per guidelines we are advised to answer only one question so I am answering question 2
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