Exercise 1.3

Transcript

Exercise 1.3 A calorie is a unit of heat (energy in transit) and it equals about 4.2 J, where 1 J = 1 kg m² s⁻². Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, and the unit of time is γ s. Show that a calorie has a magnitude 4.2 α⁻¹ β⁻² γ² in terms of the new units. First we find Dimensional formula 1 Joule is energy, whose dimensional formula is [𝑴𝑳^(𝟐)𝑻^(−𝟐)] Now, 1 calorie = 4.2 J = 4.2 kg m² s⁻² Let’s write both units Joule (SI Unit) 4.2 J = 1 kg m^(2) s^(-2) So, we write n_(1)=4.2 M_(1)=𝑘𝑔 L_(1)=𝑚 T_(1)=𝑠 Now, u_(1)=𝑀_(1)𝐿_(1)^(2)𝑇_(1)^(−2) New units So, we write n_(2)=? M_(2)=𝛼 𝑘𝑔 L_(2)=β 𝑚 T_(2)=γ 𝑠 Now, u_(2)=𝑀_(2)𝐿_(2)^(2)𝑇_(2)^(−2) Using formula 𝒏_(𝟏) × 𝒖_(𝟏)=𝒏_(𝟐) × 𝒖_(𝟐) 𝑛_(1) × 𝑀_(1)𝐿_(1)^(2)𝑇_(1)^(−2)=𝑛_(2) ×𝑀_(2)𝐿_(2)^(2)𝑇_(2)^(−2) 𝑛_(1) ×(𝑀_(1)𝐿_(1)^(2)𝑇_(1)^(−2))/(𝑀_(2)𝐿_(2)^(2)𝑇_(2)^(−2))=𝑛_(2) 𝒏_(𝟐)= 𝒏_(𝟏) × ((𝑴_(𝟏))/(𝑴_(𝟐)))× ((𝑳_(𝟏))/(𝑳_(𝟐)))^(𝟐)× ((𝑻_(𝟏))/(𝑻_(𝟐)))^(−𝟐) Putting values 𝑛_(2)=4.2 × ((1 𝑘𝑔)/(𝛼 𝑘𝑔))× ((1 𝑚)/(β 𝑚))^(2)× ((1 𝑠)/(γ 𝑠))^(−2) 𝑛_(2)=4.2 ×(1)/(𝛼) ×(1)/(β^(2)) ×(1)/(𝑦^(−2)) 𝒏_(𝟐)=𝟒.𝟐𝜶^(−𝟏)𝜷^(−𝟐) × 𝜸^(𝟐) Hence proved