The energy E of a system is a function of tim | vedclass

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(C) The energy of the system is given by the equation $E(t) = \alpha t - \beta t^3$.According to the principle of homogeneity of dimensions,the dimens

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$[ML^2 T^{-3}]$ and $[ML^2 T^{-5}]$

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(C) The energy of the system is given by the equation $E(t) = \alpha t - \beta t^3$.According to the principle of homogeneity of dimensions,the dimensions of each term in an equation must be the same as the dimensions of the physical quantity on the other side of the equation.Since $E$ represents energy,its dimensional formula is $[ML^2 T^{-2}]$.For the first term,the dimension of $\alpha t$ must be equal to the dimension of energy:$[\alpha][T] = [ML^2 T^{-2}]$$[\alpha] = [ML^2 T^{-2}] / [T] = [ML^2 T^{-3}]$For the second term,the dimension of $\beta t^3$ must be equal to the dimension of energy:$[\beta][T^3] = [ML^2 T^{-2}]$$[\beta] = [ML^2 T^{-2}] / [T^3] = [ML^2 T^{-5}]$Thus,the dimensions of $\alpha$ and $\beta$ are $[ML^2 T^{-3}]$ and $[ML^2 T^{-5}]$ respectively.

The energy $E$ of a system is a function of time $t$ and is given by $E(t) = \alpha t - \beta t^3$,where $\alpha$ and $\beta$ are constants. The dimensions of $\alpha$ and $\beta$ are

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