Question

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 γ² in terms of the new units.

Answer 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

Answer 2

`n_2=n_1u_1/u_2=n_1([M_1^aL_1^bT_1^c])/([M_2^aL_2^bT_2^c])`

= n₁ `[M_1/M_2]^a[L_1/L_2]^b[T_1/T_2]^c`

1 cal = 4.2 kg m² s^-2 ∴ a = 1, b = 2, c = -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"`

Now, n₂ `=4.2[(1kg)/(alphakg)]^1[(1m)/(betam)]^2[(1s)/(gammas)]^(-2)`

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

∴ 1 cal = `4.2 alpha^(-1) beta^(-2) gamma^2` in new system