Understanding Weight and Gravity with Height

This question asks us to evaluate two statements related to the weight of a body and the acceleration due to gravity at different heights on Earth.

Analyzing Statement I: Weight on a Hilltop vs. Earth's Surface

Statement I says: "A body weighs less on a hill top than on earth's surface even though its mass remains unchanged."

• Mass: Mass is an intrinsic property of a body, representing the amount of matter it contains. It remains constant regardless of location or external conditions.
• Weight: Weight is the force exerted on a body due to gravity. It is calculated using the formula: $W = m \times g$, where $W$ is weight, $m$ is mass, and $g$ is the acceleration due to gravity at that location.

Since mass ($m$) remains unchanged, any change in weight ($W$) must be due to a change in the acceleration due to gravity ($g$). If the body weighs less on a hilltop compared to the Earth's surface, it implies that the acceleration due to gravity ($g$) is lower on the hilltop.

Based on physics principles, acceleration due to gravity does vary with location, specifically with distance from the center of the Earth. Hilltops are at a greater distance from the Earth's center than the Earth's surface at sea level.

Therefore, Statement I is true, assuming that gravity is indeed weaker at higher altitudes like hilltops.

Analyzing Statement II: Acceleration Due to Gravity and Height

Statement II says: "The acceleration due to gravity of the earth decreases with height."

The acceleration due to gravity ($g$) at a distance $r$ from the center of the Earth (with mass $M$) is given by the formula: $g = \frac{GM}{r^2}$, where $G$ is the gravitational constant.

When a body is on the Earth's surface, its distance from the center is approximately the Earth's radius, $R$. So, gravity at the surface is $g_0 \approx \frac{GM}{R^2}$.

When a body is at a height $h$ above the Earth's surface (like on a hilltop), its distance from the center is $r = R + h$. The acceleration due to gravity at height $h$ is $g(h) = \frac{GM}{(R+h)^2}$.

We can express $g(h)$ in terms of $g_0$:

$\qquad g(h) = \frac{GM}{(R+h)^2} = \frac{GM}{R^2 \left(1 + \frac{h}{R}\right)^2} = g_0 \left(1 + \frac{h}{R}\right)^{-2}$

Alternatively, using a binomial approximation for small $h/R$, $g(h) \approx g_0 \left(1 - \frac{2h}{R}\right)$.

From both the exact formula $g(h) = \frac{GM}{(R+h)^2}$ and the approximation $g(h) \approx g_0 \left(1 - \frac{2h}{R}\right)$, it is clear that as height $h$ increases (since $h$ is positive), the value of $g(h)$ decreases.

Therefore, Statement II is true.

Relating Statement I and Statement II: Is it an Explanation?

Statement I claims a body weighs less on a hilltop (higher location) because mass is unchanged. Statement II claims gravity decreases with height.

Weight $W = m \times g$.

Since mass ($m$) is constant, a decrease in weight ($W$) implies a decrease in acceleration due to gravity ($g$).

Statement II states that gravity ($g$) *does* decrease with height. A hilltop is at a greater height than the Earth's surface (usually taken at sea level).

Thus, the lower gravity on the hilltop, as explained by Statement II, directly causes the lower weight mentioned in Statement I (since $W=m \times g$ and $m$ is constant).

Therefore, Statement II is not only true but also correctly explains why Statement I is true.

Evaluating the Options

• Option 1: Both statements are individually true and Statement II is the correct explanation of Statement I. This aligns with our analysis.
• Option 2: Both statements are individually true but Statement II is not the correct explanation of Statement I. Our analysis shows Statement II *is* the correct explanation.
• Option 3: Statement I is true but Statement II is false. Our analysis shows Statement II is true.
• Option 4: Statement I is false but Statement II is true. Our analysis shows Statement I is true.

Based on the detailed analysis, both statements are true, and Statement II provides the correct explanation for Statement I.

Conclusion

Both statements are true, and Statement II accurately explains the reason behind Statement I. The decrease in acceleration due to gravity with increasing height (Statement II) directly causes a body to weigh less on a hilltop compared to the Earth's surface (Statement I), assuming mass remains constant.

Revision Table: Key Physics Concepts

Additional Information: Factors Affecting Gravity

While height is a significant factor, the acceleration due to gravity ($g$) on Earth's surface is not perfectly uniform and is also affected by other factors:

• Latitude: The Earth is not a perfect sphere; it bulges slightly at the equator. This means points at the poles are slightly closer to the center than points at the equator. Also, the rotation of the Earth causes a centrifugal force that slightly counteracts gravity, most significantly at the equator and least at the poles. Both effects cause gravity to be slightly stronger at the poles than at the equator.
• Local Geology: Variations in the density of the Earth's crust beneath a location can cause slight local variations in gravity.
• Depth: As one goes below the Earth's surface, gravity initially increases slightly below the surface due to increasing mass below, but then decreases as depth increases further, eventually becoming zero at the center.

The question focuses specifically on the effect of height, which is a primary reason for the difference in weight between the Earth's surface and a hilltop.