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Question

A calorie is a unit of heat or energy and it equals about 4.2J where 1J=1Kg2m2/s2. Suppose we employ a system of units in which the units of mass equals αkg, the units of length equals βm, the unit of time is γs. Show that a calorie has a magnitude 4.2α-1β-2γ2in terms of the new units.


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Solution

Step 1: Given

1J=1Kg2/s2

Unit of mass: αkg

Unit of length: βm

Unit of time: γs

Step 2: State the dimensional formula of Joule

The units of Joules are kg2m2/s2

The formula for unit conversion, GivenunitNewunit=(M1M2)x(L1L2)y(T1T2)z

Hence, the dimensional formula is [M1L2T-2]

M1=1kgL1=1mT1=1s

And,

x=1y=2z=-2

Step 3: Convert using the formula of unit conversion

GivenunitNewunit=(M1M2)x(L1L2)y(T1T2)zCalorieNewunit=4.2(1kgαkg)1(1mβm)2(1sγs)-2Calorie=4.2α-1β-2γ2

Hence, proved.


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