Let us consider a system of units in which mass and angular momentum are dimensionless. If length ha

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1.Units, Dimensions and Measurement

easy

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of $L$, which of the following statement ($s$) is/are correct ?

$(1)$ The dimension of force is $L ^{-3}$

$(2)$ The dimension of energy is $L ^{-2}$

$(3)$ The dimension of power is $L ^{-5}$

$(4)$ The dimension of linear momentum is $L ^{-1}$

$1,2,4$

$1,2,3$

$1,2$

$1,3$

(IIT-2019)

Solution

Mass $= M ^0 L ^0 T ^0$

$MVr = M ^0 L ^0 T ^0$

$M ^0 \frac{ L ^1}{ T ^1} \cdot L ^1= M ^0 L ^0 T ^0$

$L ^2= T ^1$ $. . . . . . .(1)$

Force $= M ^1 L ^1 T ^{-2} \quad \text { (in SI) }$

$= M ^0 L ^1 L ^{-4}$(In new system from equation $(1)) $

$= L ^{-3}$

Energy $= M ^1 L ^2 T ^{-2} \quad \text { (In SI) }$

$= M ^0 L ^2 L ^{-4}$ (In new system from equation $ (1)) $

$= L ^{-2}$

Power $=\frac{\text { Energy }}{\text { Time }}$

$= M ^1 L ^2 T ^{-5} \quad \text { (in SI) }$

$=M^0 L^2 L^{-6} $ (In new system from equation $(1)) $

$= L ^4$

Linear momentum $= M ^1 L ^1 T ^{-1} \text { (in SI) }$

$=M^0 L^1 L^{-2} $ (In new system from equation $(1)) $

$= L ^{-1}$

Ans. $(1,2,4)$

Standard 11

Physics

Turpentine oil is flowing through a tube of length $l$ and radius $r$. The pressure difference between the two ends of the tube is $P .$ The viscosity of oil is given by $\eta=\frac{P\left(r^{2}-x^{2}\right)}{4 v l}$ where $v$ is the velocity of oil at a distance $x$ from the axis of the tube. The dimensions of $\eta$ are

medium

(AIPMT-1993)

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hard

(JEE MAIN-2024)

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easy

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medium

Stokes' law states that the viscous drag force $F$ experienced by a sphere of radius $a$, moving with a speed $v$ through a fluid with coefficient of viscosity $\eta$, is given by $F=6 \pi \eta a v$.If this fluid is flowing through a cylindrical pipe of radius $r$, length $l$ and a pressure difference of $p$ across its two ends, then the volume of water $V$ which flows through the pipe in time $t$ can be written as

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where, $k$ is a dimensionless constant. Correct value of $a, b$ and $c$ are

normal

(KVPY-2015)

Solution

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