(B) The given equation is $P = P_0 \exp(-\alpha t^2)$.In any exponential function of the form $e^x$,the exponent $x$ must be dimensionless.Therefore,t

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(B) The given equation is $P = P_0 \exp(-\alpha t^2)$.In any exponential function of the form $e^x$,the exponent $x$ must be dimensionless.Therefore,the term $\alpha t^2$ must be dimensionless.This implies that the dimensions of $\alpha$ multiplied by the dimensions of $t^2$ must equal $1$ (dimensionless).$[\alpha] [t^2] = [M^0 L^0 T^0] = 1$.Since $[t] = [T]$,we have $[\alpha] [T^2] = 1$.Thus,$[\alpha] = [T^{-2}]$.

Class 11 Physics Units and Measurements Dimensional Analysis, Uses and Limitations

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The time dependence of a physical quantity $P$ is given by $P = P_0 \exp(-\alpha t^2)$,where $\alpha$ is a constant and $t$ is time. The constant $\alpha$:

AIPMT 1993 All AIPMT

A
is dimensionless
B
has dimensions $T^{-2}$
C
has dimensions of $P$
D
has dimensions $T^2$

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Dimensional Analysis, Uses and Limitations

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AIPMT 1993 Paper All AIPMT years →

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The time dependence of a physical quantity $P$ is given by $P = P_0 \exp(-\alpha t^2)$,where $\alpha$ is a constant and $t$ is time. The constant $\alpha$:

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Assertion $(A)$: Energy per unit volume and angular momentum can be added dimensionally.Reason $(R)$: Physical quantities having same dimensions can be added or subtracted.

$A$ physical quantity $x$ depends on quantities $y$ and $z$ as follows: $x = Ay + B \tan(Cz)$,where $A, B,$ and $C$ are constants. Which of the following do not have the same dimensions?

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Assertion $(A)$: Energy per unit volume and angular momentum can be added dimensionally.
Reason $(R)$: Physical quantities having same dimensions can be added or subtracted.