IB Physics SL D 1 Gravitational Fields Questions

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

Exam points

Identify elliptical orbits and equal-area motion to apply Kepler’s first and second laws.
Apply T²∝R³ to compare orbital periods, infer orbital radii or estimate a star’s mass.
Use F=Gm₁m₂/r² to compare gravitational forces as masses or separation change.
Justify a point-mass model for a satellite orbiting a much larger, uniform spherical planet.
Calculate g=F/m=GM/r² and compare field strength as planetary mass or distance changes.
Combine gravitational fields to locate a zero-field point or determine the net field direction.
Draw a planet’s radial gravitational field lines with arrows pointing towards its centre.

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: 8]

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 .

(a)

Part 2
Gravitational fields

[ 8 ]

(i)

State Newton's universal law of gravitation.

[ 3 ]

(ii)

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 ]

(iii)

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 ]

(iv)

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

[ 1 ]

[Maximum number: 3]

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.

(a)

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

[ 3 ]

(i)

Outline what is meant by the gravitational field strength at a point.

[ 2 ]

(ii)

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.

[ 1 ]

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Covered syllabus points

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Question 2

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