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A calorie is a unit of heat or energy and it equals about 4.2 J where 1 J = 1 kg m2 s-2. Suppose, we employ a system of units in which unit of mass equal α kg, the unit of length equals β m, and the unit of time is γ s. Show that a calory has a magnitude of 4.2 α-1 β-2 γ2 in terms of the new units.

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1 cal. = 4.2 = kg m2 s-2

S.I. system New system
n1 = 4.2 n2 = ?
M1 = 1 kg M2 = α kg
L1 = 1 m L2 = β m
T1 = 1 seconds T2 = γ second

Dimensional formulae of energy is [M1L2T-2]

Comparing with [Ma Lb Tc], we get

a =1, b = 2 c = -2

Now, n2 =n1 [M1/M2]a [L1/L2]b [T1/T2]c

= 4.2 [1 kg/α kg]1 [1 m/β m]2 [1 s/γ s]-2

= 4.2 α-1 β-2 γ2.

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