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A Calorie is a Unit of Heat Or Energy and It Equals About 4.2 J Where 1J = 1 kg m^2s^–2. Suppose We Employ a System of Units in Which the Unit of Mass Equals α Kg, the Unit of Length Equals β M, the Unit of Time is γ S. - Physics

A calorie is a unit of heat or energy and it equals about 4.2 J where 1J = 1 kg m2s–2. Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, the unit of time is γ s. Show that a calorie has a magnitude 4.2 α–1 β–2 γin terms of the new units.

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Solution 1

Given that,

1 calorie = 4.2 (1 kg) (1 m2) (1 s–2)

New unit of mass = α kg

Hence, in terms of the new unit, 1 kg =`1/alpha = a^(-1)`

In terms of the new unit of length,

`1m = 1/beta = beta^(-1) or 1m^2 = beta^(-2)`

And, in terms of the new unit of time,

`1s = 1y = y^(-1)`

`1s^2  = y^(-2)`

`1s^(-2) = y^2`

∴1 calorie = 4.2 (1 α–1) (1 β–2) (1 γ2) = 4.2 α–1 β–2 γ2

Solution 2

`1 cal = 4.2 `kg m^2s^(-2)`

SI New System
`n_1 =    4.2` `n_2 = ?`
`M_1 = 1 kg` `M_2 = alpha kg`
`L_1 = 1m` `L_2 = beta m`
`T_1 = 1 s` `T_2 = y second`

Dimensional formula of energy is [`M^(1)L^2T^(-2)`]

Comparing with [`M^(a) L^(b) T^(c)`] we get

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

Now, n_2 = n_1 `[M_1/M_2]^a[L_1/L_2]^b[T_1/T_2]^c`


or `n_2 = 4.2 alpha^(-1) beta^(-2) gamma^2`

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NCERT Class 11 Physics Textbook
Chapter 2 Units and Measurements
Q 3 | Page 35
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