Question

A calorie is a unit of heat or energy and it is equal to about 4.2 J where 1 J = 1 kg m² s⁻². Suppose, we employ a system of units in which 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 of 4.2 β⁻²α⁻¹ γ² in terms of the new units.

Answer 1

Dear Student,

◆ Answer -

1 cal = 4.2 β⁻²α⁻¹γ² xy²z⁻²

◆ Explaination -

Let x, y and z be units of mass, length and time in new system of units.

x = α kg, 1 kg = x/α

y = β m, 1 m = y/β

z = γ s, 1 s = z/γ

Now we know,

1 cal = 4.2 J

1 cal = 4.2 kgm²s⁻²

Substitute values,

1 cal = 4.2 × (x/α) × (y/β)² × (z/γ)⁻²

z

1 cal = 4.2 β⁻²α⁻¹γ² xy²z⁻²

Therefore, magnitude of 1 cal in new system of units is 4.2 β⁻²α⁻¹γ².

Hope this helps you...

Answer 2

Solution 1⤵️

1 calorie = 4.2 (1kg) (1m^2) (1s^-2) ...(given)

Hence in terms of new unit.,

1 kg = 1 = a^-1

α

In terms of the new unit of length.,

1 m = 1 = β^-1 or 1 m^2 = β^-2

β

And, in terms of new unit of time,

1 s = 1 y = y^-1

1 s^2 = y^-2

1 s^-2 = y^2

∴ 1 calorie = 4.2 ( 1α^-1 ) (1β^-2)

( 1γ^2) = 4.2 α^-1 β^-2γ^2✅

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Solution 2⤵️

1 calorie = 4.2 (1kg) (1m^2) (1s^-2) ...(given)

• S.I unit • New system

• n1 = 4.2 •n2 = ?

•m1 = 1 kg • m2 = α kg

•L1 = 1m • L2 = β m

•T1 = 1s • T2 = y second

Dimensional formula of Energy is [M^1 L^2 T^-2]

Comparing with [M^a L^b T^c] we get.,

a= 1 , b = 2 , c = -2

Now, n_2 = n_1

[ m1 / m2] ^a [ L1 / L2 ]^b [ T1 / T2]^c

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

or

[n2 = 4.2 α^-1 β^-2 γ^2 ]✅