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A planet of radius $R$ and mass $M$ has a gravitational field strength of $g$ at its surface - AQA - A-Level Physics - Question 9 - 2022 - Paper 2

Question 9

A planet of radius $R$ and mass $M$ has a gravitational field strength of $g$ at its surface. Which row describes a planet with a gravitational field strength of $4... show full transcript

Worked Solution & Example Answer:A planet of radius $R$ and mass $M$ has a gravitational field strength of $g$ at its surface - AQA - A-Level Physics - Question 9 - 2022 - Paper 2

Step 1

Determine the gravitational field strength formula

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Answer

The gravitational field strength at the surface of a planet is given by the formula: $g = \frac{GM}{R^2}$ where $G$ is the universal gravitational constant. For a planet with gravitational field strength $4g$ , we have: $4g = \frac{GM'}{R'^2}$ where $M'$ is the mass of the new planet and $R'$ is its radius.

Step 2

Set up the relationship between the two planets

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Answer

Equating the two expressions for gravitational field strength: $4 \left( \frac{GM}{R^2} \right) = \frac{GM'}{R'^2}$ Dividing both sides by $G$ and rearranging gives: $4M = \frac{M'R^2}{R'^2}$ This can be rewritten to find a relationship between the mass and radius of the new planet: $M' = 4M \frac{R'^2}{R^2}$

Step 3

Evaluate the options

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Answer

We can evaluate the given options:

Row A: Radius $2R$ , Mass $2M$ :
- $M' = 4M \frac{(2R)^2}{R^2} = 4M \cdot 4 = 16M$ , (incorrect)
Row B: Radius $\frac{R}{\sqrt{2}}$ , Mass $\frac{M}{2}$ :
- $M' = 4M \frac{(\frac{R}{\sqrt{2}})^2}{R^2} = 4M \cdot \frac{1}{2} = 2M$ , (incorrect)
Row C: Radius $R$ , Mass $\frac{M}{\sqrt{2}}$ :
- $M' = 4M \frac{R^2}{R^2} = 4M$ , (incorrect)
Row D: Radius $\frac{R}{\sqrt{2}}$ , Mass $2M$ :
- $M' = 4M \frac{(\frac{R}{\sqrt{2}})^2}{R^2} = 4M \cdot \frac{1}{2} = 2M$ , (correct)

From the evaluation, Row D provides a condition in which the gravitational field strength is $4g$ .

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A planet of radius $R$ and mass $M$ has a gravitational field strength of $g$ at its surface - AQA - A-Level Physics - Question 9 - 2022 - Paper 2

Worked Solution & Example Answer:A planet of radius $R$ and mass $M$ has a gravitational field strength of $g$ at its surface - AQA - A-Level Physics - Question 9 - 2022 - Paper 2

Determine the gravitational field strength formula

Set up the relationship between the two planets

Evaluate the options

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