Physics Gravitation

(a) Decreases - Acceleration due to gravity at depth h is given by $g_h$ = (1 – $\frac{2h}{R_e}$ )g, where $R_e=Radiusoftheearth,$ g = acceleration due to gravity on the surface of the Earth. From this equation, it is clear that acceleration due to gravity decreases with increase in height

(b) Decreases – Acceleration due to gravity at depth d is given by $g_d$ = (1- $\frac{d}{R_e}$ )g. So the acceleration due to gravity decreases with increase in depth.

(c) Mass of the body – Acceleration due to gravity of body mass m is given by the relation g = $\frac{GM}{R^2}$ , where G = Universal gravitation constant, M = mass of the Earth and R = radius of the Earth. Hence, it can be inferred that acceleration due to gravity is independent of the mass of the body.

(d) More – Gravitational potential energy of two points $r_1$ and $r_2$ distance away from the centre of the Earth is respectively given by:

V( $r_1$ ) = $-$ - $\frac{GmM}{r_1}$ and V( $r_2$ ) = $-$ - $\frac{GmM}{r_2}$

Difference in potential energy, V = V( $r_2$ ) $-$ V( $r_1$ ) = $-GmM(\frac{1}{r_2}$ $-\frac{1}{r_1}$ )

Hence this formula is more accurate than mg( $r_2$ - $r_1$ )

It is due to a concept taken from Newton's Law of Gravitation. It says that each mass element of the shell puts an attractive force on a particle inside but due to the fact that the shape of the shell is symmetrical, these forces cancel out in every direction.
To prove it mathematically, one can use Gauss's Law for Gravitation. It states that inside a uniformly distributed spherical shell, the net gravitational field is zero. It differs from electrostatics, which states that a conducting shell blocks or stops the external electric fields. The shielding concept is not part of gravitation where the external bodies exert a gravitational force on an object inside the shell. It is the fundamental property of astrophysics and helps in understanding why the gravitational field inside a star or planet depends only on the mass enclosed within a given radius.

Gravitational Potential Energy (U) is the energy an object has due to its position in a gravitational field. It is defined as:

Where M represents the mass of the larger body, m is the mass of the smaller body, r is the distance between them, and G stands for the gravitational constant.
G is written with a negative sign to show that the gravitational potential energy is always lower than zero and the values even decrease as the objects move apart from each other. It implies that to separate two masses, work should be done against the gravity. At infinity, U = 0, implying that no external force is required to keep them apart. When two objects are closer, potential energy is more negative which implies a stronger gravitational attraction. The concept is very important in fields such as orbital mechanics and astrophysics.