(D) body becomes weightless at the equator when the centrifugal force equals the gravitational force,which is expressed as $R \omega^2 = g$,or $\omega

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(D) body becomes weightless at the equator when the centrifugal force equals the gravitational force,which is expressed as $R \omega^2 = g$,or $\omega^2 = \frac{g}{R}$.The rotational kinetic energy of the earth is given by $K = \frac{1}{2} I \omega^2$.For a solid sphere (assuming the earth is a solid sphere),the moment of inertia is $I = \frac{2}{5} MR^2$.Substituting the values of $I$ and $\omega^2$ into the kinetic energy formula:$K = \frac{1}{2} \times (\frac{2}{5} MR^2) \times (\frac{g}{R})$.Simplifying the expression:$K = \frac{1}{2} \times \frac{2}{5} \times M \times R \times g = \frac{1}{5} MgR$.

Class 11 Physics Gravitation Mix Examples-Gravitation

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$A$ body situated on the surface of the earth becomes weightless at the equator when the rotational kinetic energy of the earth reaches a critical value '$K$'. The value of '$K$' is given by [$g$ = gravitational acceleration on earth's surface,$M$ = mass of the earth,and $R$ = radius of the earth].

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A
$\frac{1}{2} MgR$
B
$\frac{1}{3} MgR$
C
$\frac{1}{4} MgR$
D
$\frac{1}{5} MgR$

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$A$ body situated on the surface of the earth becomes weightless at the equator when the rotational kinetic energy of the earth reaches a critical value '$K$'. The value of '$K$' is given by [$g$ = gravitational acceleration on earth's surface,$M$ = mass of the earth,and $R$ = radius of the earth].

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The mass density of a spherical body is given by $\rho(r) = \frac{k}{r}$ for $r \leq R$ and $\rho(r) = 0$ for $r > R$,where $r$ is the distance from the centre. The correct graph that describes qualitatively the acceleration,$a$,of a test particle as a function of $r$ is

$A$ satellite is orbiting the Earth at a height $h$ above the surface (where $R$ is the radius of the Earth and $h \ll R$). What is the minimum increase in the orbital velocity required for the satellite to escape the Earth's gravitational field? (Ignore the effect of the atmosphere.)

Three particles of mass $1 \, kg$ each are placed at $(0, 0)$,$(0, 0.2 \, m)$,and $(0.2 \, m, 0)$. What is the net gravitational force on the particle placed at the origin?

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