Relation Between Height and Depth for the Same Change in g:
The acceleration due to gravity decreases both when moving above the Earth’s surface (height) and below it (depth). At a height h above the surface, gravity decreases slightly based on the distance from the Earth’s center. Similarly, at a depth d below the surface, gravity decreases because only the mass within the radius (R – d) contributes to gravity.

For the same decrease in gravity at height (h) and depth d, the relation between them can be found by comparing the two effects. It turns out that the change in gravity at a height h above the surface is the same as that at a depth d below the surface if the depth is twice the height. In other words, d = 2h.

This relationship holds true only when the height h is much smaller than the Earth’s radius, as approximations are used in this derivation. Thus, for small heights and depths, gravity behaves symmetrically with this relation.

Effect of Altitude on g:
Consider the Earth as a sphere with mass M, radius R , and center O . The acceleration due to gravity at a point on the surface of the Earth depends on its mass and radius. At a height h above the Earth’s surface, the gravity decreases due to the increased distance from the Earth’s center.

The gravity at the Earth’s surface is proportional to (1/R²), while at a height h, it is proportional to (1/(R + h)²). By comparing the two, the ratio of gravity at the height h to gravity at the surface can be expressed as ( (R/(R + h))²).

Using the binomial theorem for approximation when h is much smaller than R, the expression simplifies to show that the change in gravity is proportional to (1 – 2h/R). This means that as h increases, gravity decreases.

For very small heights compared to the Earth’s radius, higher-order terms in h/R can be ignored. This gives an approximate linear relationship showing a decrease in gravity with height.

Key Points:
– Gravity decreases with altitude because the distance from the Earth’s center increases.
– At greater heights, such as on mountains, the value of g is less than at lower elevations or plains.
– For significant heights relative to the Earth’s radius, the more accurate proportional relationship ((R/(R + h))²) should be used. For small heights, the approximate relationship (1 – 2h/R) suffices for practical calculations.

Introduction:
The conflict between communities X and Y demands a strategic approach to promote harmony and shared governance.

Strategies:
1. Dialogue and Mediation: Engage in open discussions to comprehend grievances and negotiate amicable solutions.
2. Power-sharing Mechanisms: Institute systems allowing both communities equitable representation in decision-making processes.
3. Conflict Resolution Mechanisms: Establish legal frameworks or local committees to address disputes impartially.
4. Socio-Economic Development: Implement initiatives fostering economic growth and social harmony.
5. Educational Initiatives: Promote programs cultivating tolerance, cultural understanding, and community integration.
6. Impartial Assessment: Consider an unbiased assessment or referendum to gauge preferences for self-administration.

Conclusion:
A harmonious resolution requires a blend of dialogue, inclusivity, equitable resource distribution, and a commitment to fostering mutual respect and cooperation.

Mr. Y opts for a loan from a bank primarily due to two reasons:
i) Cheap and affordable since the rate of interest is lower: Banks usually offer loans at a lower interest rate compared to other informal sources of credit, making it a cost-effective option for borrowing funds. This affordability reduces the financial burden on the borrower.

ii) Free from falling into a debt trap: Bank loans often come with structured repayment plans and regulated interest rates, reducing the risk of falling into a debt trap. Structured repayments ensure that the borrower can manage the loan without facing overwhelming financial stress, thereby avoiding a debt trap.