Time (T),velocity (C),and angular momentum (h | vedclass

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Question

(A) Let mass be related to the fundamental quantities as $M \propto T^x C^y h^z$.The dimensional formula for mass is $[M^1 L^0 T^0]$.The dimensional f

Vedclass · Accepted Answer

$[M] = [T^{-1} C^{-2} h]$

Vedclass · Answer

(A) Let mass be related to the fundamental quantities as $M \propto T^x C^y h^z$.The dimensional formula for mass is $[M^1 L^0 T^0]$.The dimensional formulas for the given quantities are: $[T] = [T]$,$[C] = [L T^{-1}]$,and $[h] = [M L^2 T^{-1}]$.Substituting these into the proportionality equation:$[M^1 L^0 T^0] = [T]^x [L T^{-1}]^y [M L^2 T^{-1}]^z$$[M^1 L^0 T^0] = [M^z] [L^{y+2z}] [T^{x-y-z}]$Comparing the powers of $M, L,$ and $T$ on both sides:For $M$: $z = 1$For $L$: $y + 2z = 0 \implies y + 2(1) = 0 \implies y = -2$For $T$: $x - y - z = 0 \implies x - (-2) - 1 = 0 \implies x + 1 = 0 \implies x = -1$Thus,the dimensions of mass are $[M] = [T^{-1} C^{-2} h^1]$.

List-$I$	List-$II$
$(A)$ Angular momentum	$(I)$ $[ML^2T^{-1}]$
$(B)$ Torque	$(II)$ $[ML^2T^{-2}]$
$(C)$ Stress	$(III)$ $[ML^{-1}T^{-2}]$
$(D)$ Pressure gradient	$(IV)$ $[ML^{-2}T^{-2}]$

Time $(T)$,velocity $(C)$,and angular momentum $(h)$ are chosen as fundamental quantities instead of mass,length,and time. In terms of these,the dimensions of mass would be

Similar Questions

Match List-$I$ with List-$II$.

List-$I$ List-$II$

$(A)$ Angular momentum $(I)$ $[ML^2T^{-1}]$

$(B)$ Torque $(II)$ $[ML^2T^{-2}]$

$(C)$ Stress $(III)$ $[ML^{-1}T^{-2}]$

$(D)$ Pressure gradient $(IV)$ $[ML^{-2}T^{-2}]$

Choose the correct answer from the options given below:

Dimensions of resistance in an electrical circuit in terms of dimension of mass $M$,of length $L$,of time $T$,and of current $I$ would be:

The dimension of $R$ (Resistance) is:

Given below are two statements :
$Statement$ $(I)$ : Planck's constant and angular momentum have same dimensions.
$Statement$ $(II)$ : Linear momentum and moment of force have same dimensions.
In the light of the above statements,choose the correct answer from the options given below :

The dimensions of permittivity ${\varepsilon _0}$ are

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