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

Hint: Use the principle of conversion of units.
Step 1: Find the dimensions of energy.

We know that dimension of energy =[ML2T2]Let M1,L1,T1 and M2,L2,T2areunitsofmass,length,andtimeingiventwosystems.

M1=1kg,L1=1m,T1=1secM2=αkg,L2=βm,T2=γsec

Step 2: Use the conversion of units.

The magnitude of a physical quantity remains the same, whatever be the system of units of its measurement i.e.,  nu1=n2u2

n2=n1u1u2=n1[M1L12T12][M2L22T22]=5M1M2×L1L22×T1T2-2
=5[1αkg]×[1βm]2×[1γs]2
=5×1α×1β2×1γ2
n2=5γ2αβ2

Thus, a new unit of energy will be γ2αβ2.