IB Physics SL D.1 Gravitational fields Questions

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Practise D.1 by relating orbital motion to Kepler’s laws, applying point-mass gravity, calculating field strength and interpreting gravitational-field diagrams.

55 questions in this syllabus scope, with question parts counted together.

Exam points & syllabus

Exam points

Distinguish Kepler’s laws: the star is at an ellipse focus, equal times sweep equal areas, and T² scales with the cube of semi-major axis.
Use equal-area reasoning to explain changing orbital speed and kinetic energy: motion is slower farther from the star.
Compare orbital periods and radii around a common central body with the 3/2 or inverse 2/3 power, maintaining ratio direction.
Connect circular orbital observations to central mass by identifying gravitation as the centripetal force and combining it with v=2πr/T.
Justify point-mass treatment geometrically: a uniform spherical planet acts externally as a central mass, and satellite size is negligible relative to distance.
State the attractive inverse-square force law for point masses, and identify its predictive role without claiming it explains gravity’s origin.
Distinguish field from force and source from test mass: g is force per unit test mass, so test mass cancels in GM/r² and a=g for gravity-only motion.
Use the full centre separation in inverse-square calculations, adding radius to altitude and retaining units when converting kilometre or day data.
Read graph axes before recognising an inverse-square law: F versus reciprocal separation squared is linear through the origin.
Compare surface fields through M/R², using density to infer how mass changes when radius changes rather than assuming equal masses.
Infer a source mass or size from field data by rearranging GM/r²; distinguish a squared-size ratio from the square root needed to solve for radius.
For a field change calculate both values and subtract: moving from d to 3d leaves one ninth of g, so the decrease is eight ninths.
Combine collinear fields with directions: balance opposing magnitudes for cancellation and compare distances before choosing a between-body resultant.
Compare gravity-only acceleration by position: different starting radii give different accelerations, while arrival at the same surface gives the same value.
Judge whether a distant attraction matters by evaluating its field at the true separation and comparing it with the relevant local acceleration or force.
For a cross-field comparison, compute the proton’s gravitational field and divide by its electric field, keeping the resulting C kg⁻¹ unit.
Draw radial field lines with inward arrows, and use the tangent to a field line for gravity-only acceleration rather than the tangent to the trajectory.

Covered syllabus points

D.1.1—Kepler’s three laws of orbital motion D.1.2—Universal gravitation D.1.3—Point-mass approximation D.1.4—Gravitational field strength D.1.5—Gravitational field lines

[Maximum number: 3]

A satellite powered by solar cells directed towards the Sun is in a polar orbit about the Earth.

The satellite is orbiting the Earth at a distance of 6600 km from the centre of the Earth.

Determine the orbital period for the satellite.

Mass of Earth $=6.0 \times 10^{24} \mathrm{~kg}$

$\begin{aligned} & \frac{m v^{2}}{r}=G \frac{M m}{r^{2}} & \text { leading to } T^{2}=\frac{4 \pi^{2} r^{3}}{G M} & T=5320 \text { «s» } \end{aligned}$

Alternative 2
$\begin{aligned} & \text { « } V=\sqrt{\frac{G m_{E}}{r}} \text { » }=\sqrt{\frac{6.67 \times 10^{-11} \times 6.0 \times 10^{24}}{6600 \times 10^{3}}} \text { OR } 7800 \text { « } \mathrm{ms}^{-1} \text { » } & \text { distance }=2 \pi r=2 \pi \times 6600 \times 10^{3} \text { « } \mathrm{m} \text { » or } 4.15 \times 10^{7} \text { « } \mathrm{m} \text { » } & \text { « } T=\frac{d}{v}=\frac{4.15 \times 10^{7}}{7800} \text { » }=5300 \text { « } \mathrm{S} \text { » } \end{aligned}$

Accept use of $\omega$ nistead of v

[Maximum number: 3]

B2. This question is in two parts. Part 1 is about simple harmonic motion and the superposition of waves. Part 2 is about gravitational fields.
Part 1 Simple harmonic motion and the superposition of waves
An object of mass m is placed on a frictionless surface and attached to a light horizontal spring. The other end of the spring is fixed.

The equilibrium position is at B . The direction B to C is taken to be positive. The object is released from position A and executes simple harmonic motion between positions A and C .

Part 2
Gravitational fields

State Newton's universal law of gravitation.

[Maximum number: 1]

Ion-thrust engines can power spacecraft. In this type of engine, ions are created in a chamber and expelled from the spacecraft. The spacecraft is in outer space when the propulsion system is turned on. The spacecraft starts from rest.

The mass of ions ejected each second is $6.6 \times 10^{-6} \mathrm{~kg}$ and the speed of each ion is $5.2 \times 10^{4} \mathrm{~m} \mathrm{~s}^{-1}$ . The initial total mass of the spacecraft and its fuel is 740 kg . Assume that the ions travel away from the spacecraft parallel to its direction of motion.

On arrival at the planet, the spacecraft goes into orbit as it comes into the gravitational field of the planet.

Newton's law of gravitation applies to point masses. Suggest why the law can be applied to a satellite orbiting a spherical planet of uniform density.

Exam points

Define gravitational field strength as force per unit mass on a small point or test mass at the specified position; force alone is insufficient.
Derive g=GM/r² by substituting the gravitational force into g=F/m and cancelling the test mass, keeping source mass M distinct.
Separate force from acceleration: at the same position doubling a satellite’s mass doubles its gravitational force but leaves g and acceleration unchanged.
Calculate g from source mass and centre-to-point distance, converting kilometre data and using N kg⁻¹ or the equivalent m s⁻².
Add planetary radius to altitude before using the inverse square; a satellite one radius above the surface is at 2R and accelerates at gsurface/4.
Compare surface fields with mass divided by radius squared; diameter ratios are also radius ratios, so square the size factor.
For equal-density spherical planets, combine M proportional to R³ with g proportional to M/R² to show surface g proportional to R.
Rearrange field and size ratios to infer a source mass or radius; preserve the squared radius and take a square root only when solving for radius.
For a change of field, subtract the new value from the old: at three times the centre distance g becomes g₀/9, so the decrease is 8g₀/9.
Use a=g for gravity-only free fall or orbital motion; field at the position sets acceleration, rather than the small object’s mass.
For objects released at different heights, compare their different initial radial fields and the same surface field immediately before impact.
At a collinear zero-field point, equate the opposing GM/r² magnitudes using each source-to-point distance before inferring a mass ratio.
Check field directions before adding: outside two attracting masses the fields reinforce, while between them the net direction depends on position.
Judge a distant body’s influence by calculating its field at the actual location and comparing it with the relevant local acceleration or force.
When comparing gravitational and electric fields, calculate GM/r² first and divide by the electric field; retain the resulting unit C kg⁻¹.
If gravity is the only force, acceleration follows the gravitational field vector tangent to its field line, rather than the tangent to the object’s path.

[Maximum number: 4]

The equilibrium position is at B . The direction B to C is taken to be positive. The object is released from position A and executes simple harmonic motion between positions A and C .

(a)

Part 2
Gravitational fields

[ 4 ]

(i)

Deduce that the gravitational field strength g at the surface of a spherical planet of uniform density is given by

g=\frac{G M}{R^{2}}

where M is the mass of the planet, R is its radius and G is the gravitational constant. You can assume that spherical objects of uniform density act as point masses.

[ 2 ]

(ii)

The gravitational field strength at the surface of Mars $g_{\mathrm{M}}$ is related to the gravitational field strength at the surface of the Earth $g_{\mathrm{E}}$ by

g_{\mathrm{M}}=0.38 \times g_{\mathrm{E}}

The radius of Mars $R_{\mathrm{M}}$ is related to the radius of the Earth $R_{\mathrm{E}}$ by

R_{\mathrm{M}}=0.53 \times R_{\mathrm{E}}

Determine the mass of Mars $M_{\mathrm{M}}$ in terms of the mass of the Earth $M_{\mathrm{E}}$ .

[ 2 ]

[Maximum number: 1]

The equilibrium position is at B . The direction B to C is taken to be positive. The object is released from position A and executes simple harmonic motion between positions A and C .

Part 2
Gravitational fields

On the diagram below, draw lines to represent the gravitational field around the planet Mars.
Mars

Question 1

Question 1

Question 1

Question 1

Question (a)

Question (i)

Question (ii)

Question 1