The standard value of acceleration due to gravity is taken as 10 m/s ² . In a newly designed system of units, the unit of length is chosen to be 1 km . If the numerical value of g in this new system is found to be 129600 , what must be the new unit of time?

Step-by-step solution

Using the principle of dimensional homogeneity, the magnitude of a physical quantity remains the same in different systems of units: n₁ u₁ = n₂ u₂ . For acceleration, the dimensional formula is [LT⁻²] . Given: n₁ = 10 , u₁ = m/s ² n₂ = 129600 L₂ = 1 km = 1000 m Equating the two: n₁ [L₁ T₁⁻²] = n₂ [L₂ T₂⁻²] 10 ( 1 m 1 s ² ) = 129600 ( 1000 m T₂² ) Rearranging to solve for T₂² : T₂² = 129600 1000 10 s ² T₂² = 12960000 s ² Taking the square root: T₂ = 3600 s Since 1 hour = 3600 s , the new unit of time is 1 hour . Ans