A calorie is a unit of heat or energy and it equals about 4.2 J where 1J = 1 kg m^{2}s^{–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} γ^{2 }in terms of the new units.

#### Solution 1

Given that,

1 calorie = 4.2 (1 kg) (1 m^{2}) (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 γ^{2}) = 4.2 α^{–1} β^{–2} γ^{2}

#### 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`