Physics Q93 – AP-EAMCET 2026

For a planet having uniform density, the Gravitational field inside the planet varies with the distance from the centre as

Clue 1 of 3: Gravitational Field Inside a Sphere

For a uniformly dense spherical planet, the gravitational field at an internal point depends only on the mass enclosed within the sphere passing through that point, and not on the mass outside it.

🧠 Full Solution Path

Step 1: Identify the relevant physical principle✦ Active

The problem asks for the variation of the gravitational field *inside* a uniformly dense planet. For points inside a spherically symmetric mass distribution, the gravitational field is determined only by the mass enclosed within a sphere of radius equal to the distance from the center to the point. This is a direct application of Gauss's Law for Gravitation.

💡 Teacher's Secret Hint

Remember that outside mass does not contribute to the gravitational field inside a spherical shell. This concept is crucial for problems involving fields inside planets.

Step 2: Calculate the mass enclosed within a given radius○ Expand

Let the planet have a uniform density $ρ$ . Consider a point inside the planet at a distance $r$ from its center ( $r < R$ , where $R$ is the planet's total radius). The mass enclosed within a sphere of radius $r$ is:

M_{e n c l o s e d} = ρ \times V_{r} = ρ \times (\frac{4}{3} π r^{3})

The gravitational field at this point is due only to this enclosed mass.

💡 Teacher's Secret Hint

Ensure you use the volume of the sphere up to radius $r$ , not the total volume of the planet, as only the mass inside the Gaussian surface contributes.

Step 3: Apply the formula for gravitational field○ Expand

The gravitational field $E_{g}$ at a distance $r$ from the center due to the enclosed mass $M_{e n c l o s e d}$ is given by Newton's law of gravitation (or Gauss's law for gravity) as:

E_{g} = \frac{G M_{e n c l o s e d}}{r^{2}}

Now, substitute the expression for $M_{e n c l o s e d}$ from Step 2 into this equation:

E_{g} = \frac{G (\frac{4}{3} π r^{3} ρ)}{r^{2}}

💡 Teacher's Secret Hint

Remember to use the distance $r$ as the denominator for the field, as it's the distance from the center to the point where the field is being calculated.

Step 4: Simplify the expression to find the variation with

r

○ Expand

Simplify the equation for $E_{g}$ by canceling out the $r^{2}$ term in the denominator:

E_{g} = G \frac{4}{3} π ρ \frac{r^{3}}{r^{2}} = G \frac{4}{3} π ρ r

Since $G$ (gravitational constant), $\frac{4}{3} π$ (mathematical constant), and $ρ$ (uniform density of the planet) are all constants, the gravitational field $E_{g}$ is directly proportional to $r$ .

E_{g} \propto r

This means the gravitational field inside a uniformly dense planet varies linearly with the distance from its center.

💡 Teacher's Secret Hint

Always look for constant terms in your final expression to determine the functional dependence of the variable in question. In this case, the proportionality with $r$ is clear.

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Physics Question 93 – AP-EAMCET 2026

Clue 1 of 3: Gravitational Field Inside a Sphere

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