Problem 701 - JEE Physics

Question

The radii of two planets are respectively \(\mathrm{R}_{1}\) and \(\mathrm{R}_{2}\) and their densities are respectively \(\rho_{1}\) and \(\rho_{2}\) the ratio of the accelerations due to gravity at their surface is (A) \(g_{1}: g_{2}=\left(\rho_{1} / R_{1}^{2}\right) \cdot\left(\rho_{2} / R_{2}^{2}\right)\) (B) \(\mathrm{g}_{1}: \mathrm{g}_{2}=\mathrm{R}_{1} \mathrm{R}_{2}: \rho_{1} \rho_{2}\) (C) \(g_{1}: g_{2}=R_{1} \rho_{2} \cdot R_{2} p_{1}\) (D) \(g_{1}: g_{2}=R_{1} \rho_{1}: R_{2} \rho_{2}\)

Step-by-Step Solution

Answer

The short answer is: The ratio of the accelerations due to gravity at the surface of the two planets is \(g_1 : g_2 = R_1 \rho_1 : R_2 \rho_2\). The correct option is (D).

Step 1: Recall the formula for gravitational acceleration on a planet's surface

The gravitational acceleration at the surface of a planet can be expressed as \(g = \frac{GM}{R^2}\), where G is the gravitational constant, M is the mass of the planet, and R is its radius. In this problem, we will need to find the mass of each planet in terms of their density and radius, and then plug those expressions into the formula for gravitational acceleration.

Step 2: Find the mass of each planet using density and volume

The mass of a planet can be found using its density and volume. We know that the density is mass divided by volume, or \(\rho = \frac{M}{V}\). We can solve for mass by multiplying both sides by the volume: \(M = \rho V\). The volume of a sphere is given by \(\frac{4}{3}\pi R^3\), where R is the radius. Thus, the mass of each planet can be expressed as: - Planet 1: \(M_1 = \rho_1 \frac{4}{3}\pi R_1^3\) - Planet 2: \(M_2 = \rho_2 \frac{4}{3}\pi R_2^3\)

Step 3: Find the gravitational acceleration at the surface of each planet

Now, we will substitute the mass expressions we found in Step 2 into the formula for gravitational acceleration: \(g = \frac{GM}{R^2}\) - Planet 1: \(g_1 = \frac{G(\rho_1 \frac{4}{3}\pi R_1^3)}{R_1^2} = \frac{4}{3}G\pi \rho_1 R_1\) - Planet 2: \(g_2 = \frac{G(\rho_2 \frac{4}{3}\pi R_2^3)}{R_2^2} = \frac{4}{3}G\pi \rho_2 R_2\)

Step 4: Find the ratio of the gravitational accelerations

To find the ratio of the gravitational accelerations, we can divide \(g_1\) by \(g_2\): \[ \frac{g_1}{g_2} = \frac{\frac{4}{3}G\pi \rho_1 R_1}{\frac{4}{3}G\pi \rho_2 R_2} = \frac{\rho_1 R_1}{\rho_2 R_2} \] Therefore, the ratio of the accelerations due to gravity at the surface of the two planets is \(g_1 : g_2 = R_1 \rho_1 : R_2 \rho_2\). We can compare this result with the given options and conclude that the correct answer is: (D) \(g_1 : g_2 = R_1 \rho_1 : R_2 \rho_2\)

Key Concepts

Surface Gravity

Surface gravity refers to the gravitational acceleration experienced at the surface of a planet. It's what keeps everything in place on a planet’s surface by pulling objects towards the center of the planet. Understanding surface gravity involves recognizing it depends on two main factors:

• The mass of the planet
• The radius of the planet

The formula to calculate the gravitational acceleration or surface gravity on a planet is:\[ g = \frac{GM}{R^2} \]where:- \( g \) is the gravitational acceleration- \( G \) is the universal gravitational constant- \( M \) is the mass of the planet- \( R \) is the radius of the planetThe mass \( M \) of the planet is further determined using the planet’s density and volume, specifically:\[ M = \rho \times \frac{4}{3} \pi R^3 \]Once the mass is found, it is possible to compute the gravitational acceleration accurately and understand how strong or weak gravity would feel standing on the planetary surface.

Planetary Radius

The planetary radius is a measure of the size of a planet from its center to its surface. It plays a crucial role in determining the gravitational force experienced on the surface. A larger radius typically means a lower surface gravity if the mass remains constant because the gravitational pull weakens with distance.
In mathematical terms, the relationship is seen in how surface gravity \( g \) decreases with an increase in radius \( R \), as indicated by the \( R^2 \) term in the denominator of the formula:\[ g = \frac{GM}{R^2} \]Consequently, two planets may have the same mass but will have different surface gravities if their radii are different. For students trying to understand or compare surface gravity between planets, considering both radius and density becomes important. By analyzing how the radius comes into play in different scenarios, one can predict how likely or unlikely a similar weight will feel on two distinct planets.

Planetary Density

The concept of planetary density is integral to determining the mass and subsequently the surface gravity of a planet. Density is defined as mass per unit volume, and when dealing with planetary bodies, this density can vary greatly.
The formula to express density \( \rho \) is:\[ \rho = \frac{M}{V} \]where \( M \) represents mass and \( V \) represents volume. In the context of planets, it is the planet's density that, in combination with its volume determined by its radius \( R \), gives us the mass. The volume of a sphere, which planets closely resemble, is \( \frac{4}{3} \pi R^3 \), allowing us to express mass as:\[ M = \rho \times \frac{4}{3}\pi R^3 \]Higher density indicates a larger amount of mass within a given radius, contributing to a stronger gravitational pull at the planet's surface. Grasping this helps in understanding why planets can have varying surface gravitational forces even if they have similar sizes.