Gravitation

Distance of the Earth from the Sun, $r_e$ = 1.50 × 10⁸ km = 1.5 × 10¹¹ m

Time period of the Earth = $T_e$ , Time period of the Saturn $T_s$ = ${29.5T}_e$

Distance of Saturn from the Sun = $r_s$

From Kepler’s 3rd law of planetary motion, we have T =( ${\frac{4{\pi }^2{r}^3}{GM})}^{1/2}$

For Saturn and Sun, we can write, $\frac{r_s^3}{r_e^3}$ 

Orbital radius of the Earth around the Sun, r = 1.5 × 10⁸ km = 1.5 × 10¹¹ m

Time taken by the Earth to complete 1 revolution around the Sun, T = 1 year = 365.25 days = 365.25 $\times$ 24 $\times 24\times 60\times 60s$

Universal gravitational constant, G = 6.67 $\times {10}^{-11}$ N $m^2/{kg}^2$

Thus, mass of the Sun can be calculated as,

M

If the upper half of the spherical shell is cut out, then the net gravitational force acting on a particle at an arbitrary point P will be in the downward direction. Since gravitational intensity at a point is defined as the gravitational force per unit mass at that point, it will also act in the do

Gravitational potential (V) is constant at all points in a spherical shell. Hence the gravitational gradient ( $\frac{dV}{dr})$ is zero everywhere inside the spherical shell. The gravitational potential gradient is equal to the negative of gravitational intensity. Hence intensity is also zero at all points ins

(a) Legs hold the entire mass of a body in standing position due to gravitational pull. In space, an astronaut feels weightlessness because of the absence of gravity. Therefore swollen feet of an astronaut do not affect him/her in space

(b) A swollen face is caused generally because of appare

Angular momentum and total energy at all points of the orbit of a comet moving in a highly elliptical orbit around the Sun are constant. Is linear speed, angular speed, kinetic and potential energy varies from point to point in the orbit.

(a) No

(b) No

(c) Yes

(d) No

(e) No

(f) Yes

(a) Escape velocity of a body from the Earth is given by the relation:

$v_{esc}$ = $\sqrt{2gR}$ …. (1)

Where g = acceleration due to gravity and R = radius of the Earth

So escape velocity is independent of the mass of the body.

(b) It does not depend on the location from where it is projected.

(c) Does not

(a) Total mechanical energy of a satellite is the sum of its kinetic energy (always positive) and potential energy (can be either positive or negative). At infinity, the gravitational potential energy of the satellite is zero. As the Earth-Satellite system is a bound system, the total energy of the

Mass of our galaxy Milky Way, M = 2.5 $\times {10}^{11}$ solar mass

Solar mass = Mass of the Sun = 2.0 $\times {10}^{30}$ kg

Mass of our galaxy = 2.5 $\times {10}^{11}\times$ 2.0 $\times {10}^{30}$ = 5 $\times {10}^{41}$ kg

Diameter of the Milky Way, d = 10⁵ ly

Radius of the Milky Way = 5 $\times {10}^4$ ly

1 ly = 9.46 $\times {10}^{15}$ m

r = 5 $\times {10}^4\times$ 9.46&n

The rotation period of the satellite Io, $T_{ju}$ = 1.769 days = 1.769 x 24 x 60 x 60s

Radius of the Orbit is given by the relation, $R_{ju}$ = 4.22 $\times {10}^8$ m

Mass is given by the relation: $M_j$ = $\frac{4{\pi }^2{R}_{ju}^3}{G{T}_{ju}^2}$ ……(i)

Where $M_j$ = mass of Jupiter and G = Universal gravitational constant

O