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A new system of units is proposed in whi...

A new system of units is proposed in which unit of mass is `alpha` kg, unit of length `beta`m and unit of time `gamma`s. How much will 5J measure in this new system?

Text Solution

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[Energy] `=[ML^(2)T^(-2)]`
Let `M_(1),L_(1),T_(2) and M_(2), L_(2), T_(2)` are units of mass, length and time in given two system,
`:.M_(1)=kg, L_(1)=1m, T_(1)=1s`
`M_(2)= alpha kg, L_(2)= beta m,T_(2) = gammas`
The magnitude of a physical quantity remains the same, whatever be the system of units of its measurement i.e.
`n_(1)u_(1)=n_(2)u_(2)`
`n_(2)=n_(1)(u_(1))/(u_(2))=n_(1)([M_(1)L_(1)^(2)T_(1)^(2)])/([M_(2)L_(2)^(2)T_(2)^(-2)])=5`
`[(M_(1))/(M_(2))]xx[(L_(1))/(L_(2))]^(2)xx[(T_(1))/(T_(2))]^(-2)`
`=5[(1)/(alpha)]xx[(1)/(beta)]xx[(1)/(gamma)]^(-2)`
`=5xx(1)/(alpha)xx(1)/(beta^(2))xx(1)/(gamma^(-2))`
`n_(2)=(5gamma^(2))/(alpha beta^(2))`
Thus, new unit of energy will be `(gamma^(2))/(alpha beta^(2))`
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