Practice Questions

Propose a method to 'weigh' a star that has a planet orbiting it, without landing on either body.

Define escape speed and recall the formula for the escape speed from the surface of the Earth in terms of its mass $M_E$ and radius $R_E$.

Analyze Kepler's second law (law of areas) to determine where a planet's speed is maximum and minimum in its elliptical orbit around the Sun.

Define the universal gravitational constant, G, and recall its SI unit and value.

Propose a reason why the Moon has no significant atmosphere, while Earth does, based on the concept of escape speed.

State Newton's universal law of gravitation in words.

Examine the formula for escape speed, $v_e = \sqrt{2gR_E}$. Does the escape speed of an object depend on its mass? Justify your answer.

Evaluate whether the gravitational potential energy of a system of two masses can be positive. Justify your answer based on the conventional choice of zero potential energy at infinity.

List the three laws of planetary motion as stated by Johannes Kepler.

Name the scientist who first determined the value of G experimentally and identify the key principle of the apparatus he used.

Recall the formula for the orbital period $T$ of a planet in a circular orbit of radius $R$ around the Sun of mass $M_s$.

Summarize the findings regarding the gravitational force exerted by a uniform hollow spherical shell on a point mass for two cases: (a) when the point mass is located outside the shell, and (b) when it is located inside the shell.

Describe how the acceleration due to gravity, $g$, varies with depth $d$ below the Earth's surface, assuming uniform density. Recall the formula for $g$ at a depth $d$.

Identify the two physical quantities that are conserved for a planet revolving around the Sun in an elliptical orbit.

Evaluate the statement: 'If the Earth stopped rotating, the value of acceleration due to gravity, $g$, would be slightly greater at the equator.' Justify your answer.

Recall the value of acceleration due to gravity $g$ on the surface of the Earth and calculate its value at a height of $64$ km. Use the approximate formula and assume the radius of Earth $R_E = 6400$ km.

Summarize the difference between the geocentric and heliocentric models of the planetary system. Name one key proponent for each model.

Two point masses, $m_1 = 4$ kg and $m_2 = 9$ kg, are separated by a distance of $10$ m. Calculate the position of a third mass $m = 1$ kg on the line joining them where the net gravitational force on it is zero.

Calculate the escape speed from the surface of the Moon. (Given: Mass of Moon $M_M = 7.35 \times 10^{22}$ kg, Radius of Moon $R_M = 1.74 \times 10^6$ m, $G = 6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2$).

Critique the statement: 'An astronaut in an orbiting satellite is weightless because they are far from Earth's gravity.' Justify your reasoning using the concept of free fall.

Critique Newton's shell theorem, which states that a uniform spherical shell of matter attracts a particle outside it as if all the shell's mass were concentrated at its center. Why is this a non-trivial result, and what mathematical principle allows for this simplification?

Explain why the gravitational potential energy of a two-particle system is conventionally considered to be negative.

If the Earth were to shrink to half its present radius without any change in its mass, calculate the new value of acceleration due to gravity on its surface. (Assume $g = 9.8 \text{ m/s}^2$ on the current surface).

Analyze the total energy of a satellite in a circular orbit. Why is the total energy always negative for a bound orbit?

A satellite of mass $m = 500$ kg is in a circular orbit at a height $h = R_E$ above the Earth's surface. Calculate its (a) kinetic energy, (b) potential energy, and (c) total energy in this orbit. (Given: Mass of Earth $M_E = 6 \times 10^{24}$ kg, Radius of Earth $R_E = 6.4 \times 10^6$ m, $G = 6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2$).

Analyze the statement: 'An astronaut in an orbiting space station experiences weightlessness.' Is this because the gravitational force of the Earth is zero at that altitude? Explain your reasoning.

A body weighs $90$ N on the surface of the Earth. Solve for the gravitational force on it at a height equal to twice the radius of the Earth from the surface.

Propose why gravitational shielding is considered impossible, unlike electrical shielding inside a hollow conductor.

A space agency plans to launch a satellite into a geostationary orbit. Evaluate the challenges in achieving this specific orbit compared to a much lower Earth orbit (e.g., $400$ km altitude). Your evaluation should consider orbital speed, energy requirements, and trajectory precision.

Justify Kepler's second law (law of areas) as a direct consequence of the conservation of angular momentum. Prove that the rate of change of area swept by the radius vector is constant.

A projectile is launched from Earth's surface with an initial speed $v_i$ equal to half its escape speed. Formulate an equation, in terms of $G$, $M_E$, and $R_E$, to calculate the maximum height $h$ it will reach above the surface. Justify the steps in setting up the energy conservation equation.

Design a mission to move a satellite of mass $m=500$ kg from a circular orbit of radius $r_1 = 2R_E$ to a circular orbit of radius $r_2 = 3R_E$. Formulate the steps to calculate the total energy required for this transfer and calculate its value. Justify why energy must be added. (Use $M_E = 6 \times 10^{24}$ kg, $R_E = 6.4 \times 10^6$ m, $G = 6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2$)

Compare and contrast the change in acceleration due to gravity when moving from the Earth's surface upwards by a distance $h$ and downwards by the same distance $h$. Which change is greater for a small value of $h$? Demonstrate this using the relevant approximate formulas.

Solve for the gravitational potential energy of a system of three equal masses, each of mass $m = 2$ kg, placed at the vertices of an equilateral triangle with side length $a = 1$ m. (Given: $G = 6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2$).

A satellite is orbiting the Earth in a circular orbit at an altitude of $600$ km above the surface. Calculate its orbital speed. (Given: Mass of Earth $M_E = 6 \times 10^{24}$ kg, Radius of Earth $R_E = 6400$ km, $G = 6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2$).

Explain the relationship between the total energy, kinetic energy, and potential energy of a satellite in a circular orbit around the Earth. Recall the formulas for each.

Demonstrate that Kepler's second law of planetary motion (the law of areas) is a direct consequence of the conservation of angular momentum for a central force like gravitation. Use the expression for the area swept by the radius vector $\Delta \mathbf{A} = \frac{1}{2}(\mathbf{r} \times \mathbf{v} \Delta t)$ to derive the relationship between the rate of change of area and angular momentum.

Compare the value of acceleration due to gravity at a height $h$ and a depth $d$ below the Earth's surface, where $h=d$. Then, solve for the depth $d$ at which the acceleration due to gravity is exactly half of its value on the surface. (Radius of Earth $R_E = 6400$ km)

Formulate an expression for the total energy of a binary star system where two stars of equal mass $M$ orbit their common center of mass in a circular path of radius $R$ from the center.

Design an experiment to determine the mass of an unknown planet by observing one of its moons. Justify the formulas and measurements required, assuming the moon has a nearly circular orbit.

Apply Kepler's third law of periods to solve the following. An artificial satellite has an orbital period of $T_1 = 90$ minutes. If another satellite has a period $T_2 = 8 T_1$, calculate the ratio of their orbital radii $(r_2/r_1)$. Also, calculate the orbital radius $r_1$ of the first satellite. (Given: $M_E = 6 \times 10^{24}$ kg, $G = 6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2$).

Formulate a proof to show that the escape speed ($v_e$) from a planet's surface is $\sqrt{2}$ times the orbital speed ($v_o$) of a satellite in a circular orbit just above the surface. Justify each step and explain the physical significance.

Explain how the formula for acceleration due to gravity at a height $h$, $g(h) = \frac{GM_E}{(R_E+h)^2}$, is simplified for cases where the height $h$ is much smaller than the Earth's radius $R_E$.

Explain why an astronaut in an orbiting space station feels weightless.

Create a hypothetical scenario where the gravitational force follows an inverse cube law ($F \propto 1/r^3$) instead of an inverse square law. Predict and justify how this would alter Kepler's laws of planetary motion, specifically the law of periods and the stability of orbits.