Question

A new system of units is proposed in which unit of mass is α kg, unit of length β m and unit of time γ s. How much will 5 J measure in this new system?

Answer 1

For solving this problem, dimensions of physical quantity will remain the same whatever the system of units of its measurement.

Let the physical quantity be Q = n₁u₁ = n₂u₂

Let M₁, L₁, T₁ and M₂, L₂, T₂ be units of mass, length and time in the given two systems.

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

We know that dimension of energy `[U] = [ML^2T^-2]`

According to the problem, M₁ = 1 kg, L₁ = 1 m, T₁ = 1 s

M₂ = α kg, L₂ = β m, T₂ = γ s

Substituting the values, we get

`n_2 = 5 [M_1/M_2] xx [L_1/L_2]^2 xx [T_1/T_2]^-2`

= `5[1/α  kg] xx [1/β  m]^2 xx [1/γ  s]^-2`

= `5 xx 1/α xx 1/β^2 xx 1/γ^-2`

= `(5γ^2)/(αβ^2) J`

This is the required value of energy in the new system of units.