A massive black hole of mass m and radius R is spinning with angular velocity omega . power P radiat

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1.Units, Dimensions and Measurement

hard

A massive black hole of mass $m$ and radius $R$ is spinning with angular velocity $\omega$. The power $P$ radiated by it as gravitational waves is given by $P=G c^{-5} m^{x} R^{y} \omega^{z}$, where $c$ and $G$ are speed of light in free space and the universal gravitational constant, respectively. Then,

$x=-1, y=2, z=4$

$x=1, y=1, z=4$

$x=-1, y=4, z=4$

$x=2, y=4, z=6$

(KVPY-2017)

Solution

$(d)$ Given, $P=G c^{-5} m^{x} R^{y} \omega^{z} \quad \ldots (i)$

Here, dimensions of various physical quantities are

Angular speed, $\omega=\left[ T ^{-1}\right]$

Power, $P=\left[ ML ^{2} T ^{-3}\right]$

Mass, $m=[ M ]$

Radius, $R=[ L ]$

Speed, $c=\left[ LT ^{-1}\right]$

Universal gravitational constant,

$G=\left[ M ^{-1} L ^{3} T ^{-2}\right]$

Substituting dimensions in Eq. $(i)$, we have

${\left[ ML ^{2} T ^{-3}\right]=} {\left[ M ^{-1} L ^{3} T ^{-2}\right]\left[ L ^{-5} T ^{5}\right][ M ]^{x} }$

${\left[ L ^{y}\left[ T ^{-z}\right]\right.}$

Equating dimensions of same quantity, we get

$1=-1+x \Rightarrow x=2$

$2=3-5+y \Rightarrow y=4$

$-3=-2+5-z \Rightarrow z=6$

Standard 11

Physics

Einstein’s mass-energy relation emerging out of his famous theory of relativity relates mass $(m)$ to energy $(E)$ as $E = mc^2$, where $c$ is speed of light in vacuum. At the nuclear level, the magnitudes of energy are very small. The energy at nuclear level is usually measured in $MeV$, where $1\,MeV = 1.6\times 10^{-13}\,J$ ; the masses are measured i unified atomicm mass unit (u) where, $1\,u = 1.67 \times 10^{-27}\, kg$

$(a)$ Show that the energy equivalent of $1\,u$ is $ 931.5\, MeV$.

$(b)$ A student writes the relation as $1\,u = 931.5\, MeV$. The teacher points out that the relation is dimensionally incorrect. Write the correct relation.

medium

If the velocity of light $c$, universal gravitational constant $G$ and planck's constant $h$ are chosen as fundamental quantities. The dimensions of mass in the new system is

hard

(JEE MAIN-2023)

A force is represented by $\mathrm{F}=a \mathrm{x}^2+\mathrm{bt}^{1 / 2}$. Where $\mathrm{x}=$ distance and $\mathrm{t}=$ time. The dimensions of $\mathrm{b}^2 / \mathrm{a}$ are :

hard

(JEE MAIN-2024)

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easy

If $P$ represents radiation pressure, $c$ represents speed of light and $Q$ represents radiation energy striking a unit area per second, then non-zero integers $x,\,y$ and $z$ such that ${P^x}{Q^y}{c^z}$ is dimensionless, are

hard

(AIPMT-1992)

Solution

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