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.
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`
`=4.2[(1kg)/(alphakg)]^1[(1m)/(betam)]^2[(1s)/(gammas)]^(-2)`
or `n_2 = 4.2 alpha^(-1) beta^(-2) gamma^2`
