DP IB Physics: SL

D. Fields

D.1 Gravitational fields

DP IB Physics: SL

D. Fields

D.1 Gravitational fields

Understandings
Students should understand:
a) Kepler’s three laws of orbital motion
b) Newton’s universal law of gravitation as given by [math]F = G \frac{m_1 m_2}{r^2}[/math] for bodies treated as point masses
c) Conditions under which extended bodies can be treated as point masses
d) That gravitational field strength g at a point is the force per unit mass experienced by a small point mass at that point as given by

[math]g = \frac{F}{m} = \frac{G M}{r^2}[/math]
e) Gravitational field lines.

 

Additional higher level: 7 hours

Understandings

Students should understand:
a) That the gravitational potential energy Ep of a system is the work done to assemble the system from infinite separation of the components of the system
b) The gravitational potential energy for a two-body system as given by

[math]E_p = -G \frac{m_1 m_2}{r}[/math]

Where r is the separation between the centre of mass of the two bodies
c) That the gravitational potential Vg at a point is the work done per unit mass in bringing a mass from infinity to that point as given by

[math]V_g = -\frac{G M}{r}[/math]
d) The gravitational field strength g as the gravitational potential gradient as given by

[math]g = -\frac{\Delta V_g}{\Delta r}[/math]
e) The work done in moving a mass m in a gravitational field as given by

[math]W = m∆V_g[/math]
f) Equipotential surfaces for gravitational fields
g) The relationship between equipotential surfaces and gravitational field lines
h) The escape speed [math]v_{\text{esc}}[/math] at any point in a gravitational field as given by

[math]v_{\text{esc}} = \sqrt{\frac{2GM}{r}}[/math]
i) The orbital speed [math]v_{\text{orbital}}[/math] of a body orbiting a large mass as given by

[math]v_{\text{orbital}} = \sqrt{\frac{GM}{r}}[/math]
j) The qualitative effect of a small viscous drag force due to the atmosphere on the height and speed of an orbiting body.

  • a) Kepler’s Three Laws of Orbital Motion

  • These laws describe how planets (and other celestial bodies) move around a central body like the Sun.
  • 1. Kepler’s First Law – The Law of Ellipses
  • Every planet moves in an elliptical orbit with the Sun at one of the two foci.
  • – An ellipse is an oval-shaped curve.
  • – The focus (plural: foci) is one of two fixed points used to draw the ellipse.
  • – For planets, the Sun is not at the center of the orbit—it’s at one focus.
  • This law tells us: Orbits are not perfect circles (though some are nearly circular).
  • Figure 1 Kepler’s 1st Law
  • 2. Kepler’s Second Law – The Law of Equal Areas
  • A line joining a planet and the Sun sweeps out equal areas in equal times.
  • Meaning:
  • – When a planet is closer to the Sun (perihelion), it moves faster.
  • – When it is farther from the Sun (aphelion), it moves slower.
  • This law expresses the conservation of angular momentum.
  • Figure 2 Kepler’s Second Law
  • 3. Kepler’s Third Law – The Law of Periods
  • The square of the orbital period T is proportional to the cube of the average distance r from the Sun.
  • [math]T^2 ∝ r^3[/math]
  •             Or
  • [math]\frac{T^2}{r^3} = \text{constant}[/math]
  • For circular or nearly circular orbits:
  • – T: Time taken for one full orbit (orbital period)
  • – r: Average orbital radius or semi-major axis
  • This law helps compare the orbits of different planets or satellites.
  • Figure 3 Kepler’s third law

  • b) Newton’s Law of Universal Gravitation

  • [math]F = G \frac{m_1 m_2}{r^2}[/math]
  • Where:
  • – F: Gravitational force between two objects
  • – G: Gravitational constant ([math]6.674 × 10^{-11}Nm^2kg^2[/math])
  • – ​[math]m_1, m_2[/math]: Masses of the two bodies
  • – r: Distance between their centers of mass
  • The force is always attractive
  • It acts along the line joining the centers of mass
  • It’s mutual—both objects feel the same magnitude of force
  • Inversely proportional to the square of the distance—if r doubles, F becomes one-fourth
  • Figure 4 Newton’s law of gravitational
  • [math]F_1 = F_2 = \frac{G m_1 m_2}{r^2}[/math]

  • c) Conditions When Extended Bodies Can Be Treated as Point Masses

  • Sometimes, large objects like planets or stars are approximated as point masses in calculations.
  • 1. Spherically symmetric mass distributions
  • – If the object is spherical and mass is evenly distributed (like a uniform solid sphere), its entire mass acts as if concentrated at its center.
  • – This is a result of the Shell Theorem in gravity.
  • 2.Distances are much larger than object sizes
  • – When the distance r between the objects is much greater than their sizes, they can be modeled as points.
  • ⇒ Example:
  • Earth orbiting the Sun.
  • Even though both are large spheres, they can be treated as point masses located at their centers, because:
  • – Their distance apart is very large compared to their radii.
  • – They are both roughly spherically symmetric.
  • ⇒ Real-World Applications
  • – Calculating satellite orbits (Earth–satellite system)
  • – Predicting planetary motion in the solar system
  • – Estimating mass of stars or galaxies using orbital motion
  • – Determining gravitational fields in space missions

  • d) Gravitational Field Strength (g)

  • The gravitational field strength g at a point in space is the gravitational force per unit mass experienced by a small test mass placed at that point.
  • [math]g = \frac{F}{m}[/math]
  • Where:
  • – F is the gravitational force acting on the mass
  • – g is the intensity of the gravitational field at that location
  • From Newton’s Universal Law of Gravitation
  • Figure 5 Gravitational field strength
  • We know:
  • [math]F = G \frac{M m}{r^2}[/math]
  • Where:
  • – M: Mass of the object creating the gravitational field (e.g., a planet or star)
  • – m: Test mass
  • – r: Distance from the center of mass of object M
  • – G: Gravitational constant ([math]6.674 × 10^{-11}Nm^2/kg^2[/math])
  • Substitute into the definition of g:
  • [math]g = \frac{F}{m} \\
    g = \frac{GM}{r^2}[/math]
  • So, gravitational field strength only depends on the source mass and distance from it.
  • – It does not depend on the test mass
  • – It is a vector quantity (points towards the mass generating the field).
  • – As distance r increases, field strength decreases by the inverse square
  • – On Earth’s surface, [math]g ≈ 9.8m/s^2[/math]

  • e)  Gravitational Field Lines

  • Gravitational fields are visualized using field lines, similar to electric and magnetic fields.
  • Field lines show the direction of the gravitational force on a small test mass.
  • The closer the lines are to each other, the stronger the field.
  • They always point toward the mass generating the field (because gravity is always attractive).
  • They never cross, because at any one point, the direction of gravitational force is unique.
  • Figure 6 Gravitational field lines
  • ⇒  Field Lines Around Different Masses
  • 1. Isolated Point Mass (e.g., a planet or star)
  • – Field lines are radial (spread out in straight lines from the center).
  • – Lines converge toward the mass.
  • – The field is spherically symmetric.
  • 2. Uniform Field (approximation near Earth’s surface)
  • – Close to Earth’s surface, field lines are shown as parallel and equally spaced.
  • – This suggests a constant gravitational field strength ([math]g ≈ 9.8m/s^2[/math]).
  • ⇒ Real-Life Applications
  • – Satellites and orbits (calculating escape velocity, orbital height)
  • – Predicting how objects fall on different planets (e.g., Moon’s gravity is ~1/6 of Earth’s)
  • – Building gravity maps of Earth for geology or mining
  • – Space travel and planning interplanetary missions

  • Additional higher level: 7 hours

  • a)   Gravitational Potential Energy [math]E_p[/math]

  • The gravitational potential energy of a system is the work required to bring two masses from infinite separation to a distance r

  • b)Gravitational potential energy for a two-body

  • This is especially important when dealing with space, orbits, and astronomy, where gravity acts over long distances.
  • [math]E_p = -\frac{G m_1 m_2}{r}[/math]
  • Where:
  • – [math]E_P[/math]: Gravitational potential energy of the two-body system
  • – G: Gravitational constant ([math]6.674 × 10^{-11}Nm^2/kg^2[/math])
  • – ​[math]m_1, m_2[/math]: Masses of the two bodies
  • – r: Distance between their centres of mass
  • Negative sign:
  • – Gravitational potential energy is zero at infinite separation.
  • – As you bring two masses closer, they attract each other, and work is done by the field, not against it.
  • – So, the potential energy becomes negative—it indicates that energy is released (not required) to bring them closer.
  • ⇒ Important Insights:
  • – ​[math]E_p[/math] becomes more negative as masses get closer.
  • – A system with negative total energy (kinetic + potential) is gravitationally bound.
  • – This formula is valid for point masses or spherically symmetric bodies.

  • c)    Gravitational Potential [math]V_g[/math]

  • The gravitational potential [math]V_g[/math]​ at a point in space is the work done per unit mass to bring a small test mass from infinity to that point.
  • [math]V_g = -\frac{GM}{r}[/math]
  • Where:
  • – [math]V_g[/math]: Gravitational potential at a point due to mass
  • – M: Mass creating the gravitational field
  • – r: Distance from the center of the mass to the point
  • Negative Sign:
  • – By definition, potential is zero at infinity.
  • – To bring a mass closer, gravity does the work, so the potential decreases (goes negative).
  • – This means the object is trapped in the gravitational well of M.
  • Relationship between ​[math]V_g[/math] and [math]E_p[/math]:
  • [math]E_p = mV_g[/math]
  • This makes sense because:
  • Multiply potential per unit mass [math]V_g[/math]​ by mass mmm, and you get total potential energy.

  • d)   Gravitational Field Strength as a Potential Gradient

  • Gravitational field strength g can be thought of in two equivalent ways:
  • – As the force per unit mass:
  • [math]g = \frac{F}{m}[/math]
  • – As the rate of change of gravitational potential with distance:
  • [math]g = -\frac{\Delta V_g}{\Delta r}[/math]
  • Or
  • [math]g = -\frac{dV_g}{dr}[/math]
  • Where:
  • – g: Gravitational field strength
  • – ​[math]V_g[/math]: Gravitational potential
  • – r: Distance from the mass creating the field
  • The negative sign indicates that the gravitational potential decreases as you move closer to the mass (gravity pulls inward).
  • Figure 7 Gravitational field strength and gravitational potential
  • Interpretation:
  • A steep potential gradient (large change in [math]V_g[/math]​ over small distance) = strong gravitational field
  • Flat potential gradient = weak gravitational field.
  • ⇒  Example:
  • Near Earth’s surface, potential changes linearly, and g≈8m/s2.
  • But near a massive object like a neutron star, potential changes rapidly—so the gradient (and g) is huge.

  • e)    Work Done in a Gravitational Field

  • When a mass is moved in a gravitational field, work is done against or by the field, depending on the direction.
  • [math]W = m∆V_g[/math]
  • Where:
  • – W: Work done
  • – m: Mass being moved
  • – [math]∆V_g[/math]: Change in gravitational potential between two points
  • ⇒ Positive or Negative:
  • – If object moves away from the mass, [math]ΔV_g > 0[/math]:
    → Work is done against gravity (positive work)
  • – If it moves toward the mass, [math]ΔV_g < 0[/math]:
    → Gravity does the work (negative work by external force)

  • f)     Equipotential Surfaces

  • Equipotential surfaces are surfaces in a gravitational field where the gravitational potential is constant.
  • Properties:
  • – No work is required to move an object along an equipotential surface.
  • – Gravitational field lines are always perpendicular to equipotential surfaces.
  • – The closer the equipotential surfaces are, the stronger the gravitational field (steeper gradient).
  • ⇒ Examples of Equipotential Surfaces:
Situation Shape of Equi-potentials
Around a point mass (e.g., planet) Concentric spheres around the mass
Near Earth’s surface Parallel horizontal planes (approximate)
  • ⇒ Analogy:
  • Imagine a topographic map where elevation lines are equi-potentials.
  • Gravitational field lines are like the direction water would flow—steepest slope, always perpendicular to the lines.
  • Figure 8 Equi-potential Surface
  • ⇒  Real-World Applications
  • Satellites: Placed on equipotential orbits with predictable energy/work calculations.
  • Black holes: Have tightly packed equi-potentials—extreme gravitational gradients.
  • Geophysics: Mapping Earth’s gravity for oil exploration uses equipotential data.
  • Engineering: Ensures that movement within a system doesn’t require unnecessary energy input.

  • g)   Relationship Between Equipotential Surfaces and Gravitational Field Lines

  • ⇒ Equipotential Surfaces
  • These are imaginary surfaces in a gravitational field where the gravitational potential is the same at every point.
  • ⇒  Characteristics:
  • – No work is needed to move a mass along an equipotential surface.
  • – Since potential doesn’t change along the surface, gravitational force (and field) must act perpendicular to it.
  • ⇒  Gravitational Field Lines
  • These lines show the direction of gravitational force acting on a mass:
  • – Always point toward the mass generating the field (e.g., the Earth).
  • – The closer the lines, the stronger the field.
  • ⇒  Relationship:
  • Gravitational field lines are always perpendicular to equipotential surfaces.
  • – If field lines weren’t perpendicular, there would be a component of force along the surface.
  • – That would mean you could do work moving along the surface — but by definition, equi-potentials require zero work.
  • ⇒  Analogy:
  • Equipotential lines = paths at the same elevation (flat walking)
  • Field lines = direction of steepest descent (gravity pulling you down)
  • Walking along the path: no effort to go “up or down” (no work)
  • Walking across contours = climbing = doing work against gravity

  • h)   Escape Speed in a Gravitational Field

  • The escape speed is the minimum speed an object needs to break free from a planet or star’s gravitational field, without further propulsion.
  • [math]v_\text{esc} = \sqrt{\frac{2GM}{r}}[/math]
  • Where:
  • [math]v_\text{esc}[/math]​ : Escape speed
  • G: Gravitational constant ([math]6.674 × 10^{-11}Nm^2/kg^2[/math])
  • M: Mass of the planet/star
  • r: Distance from the center of the mass
  • Figure 9 Escape speed in a gravitational field
  • ⇒  Derivation from Energy Conservation:
  • To escape:
  • – Total energy = 0 at infinity
  • – Initial kinetic energy must cancel out gravitational potential energy
  • [math]\frac{1}{2} mv^2 = \frac{GMm}{r} \\
    v = \sqrt{\frac{2GM}{r}}[/math]
  • ⇒ Important Points:
  • Escape speed is independent of the mass of the escaping object.
  • It only depends on the mass of the body being escaped from and the distance from its center.
  • ⇒  Examples:
Body Approx. Escape Speed
Earth ≈11.2 km/s
Moon ≈2.4 km/s
Jupiter ≈60 km/s
Sun (from surface) ≈618 km/s

  • i)  Orbital Speed of a Body in Circular Orbit

  • The orbital speed ​ is the constant speed required for a body (like a satellite or planet) to maintain a stable circular orbit around a much larger mass (like a planet or star).
  • [math]v_\text{orbital} = \sqrt{\frac{GM}{r}}[/math]
  • Where:
  • – [math]v_\text{orbital}[/math]= orbital speed
  • – G = universal gravitational constant ([math]6.674 × 10^{-11}Nm^2/kg^2[/math])
  • – M = mass of the central body (e.g., Earth)
  • – r = radius of the orbit (distance from the center of the central body to the orbiting body)
  • ⇒  Derivation (from Newton’s laws):
  • For a circular orbit, the gravitational force provides the centripetal force:
  • [math]\frac{GMm}{r^2} = \frac{mv^2}{r} \\
    v^2 = \frac{GM}{r} \\
    v = \sqrt{\frac{GM}{r}}[/math]
  • This equation shows that orbital speed:
  • – Decreases with increasing altitude (higher r)
  • – Increases with a more massive central body (higher M)
  • ⇒  Example: Satellite around Earth
  • – [math]M_Earth ≈ 5.97 × 10^{24} kg[/math]
  • – Orbit at [math]r = R_{Earth} + h[/math], where h is height above Earth’s surface

  • j) Effect of Viscous Drag in Low Earth Orbit (LEO)

  • Even in Low Earth Orbit, there is some atmospheric drag (a small viscous force due to the thin atmosphere).
  • ⇒  Viscous Drag:
  • Drag force opposes motion
  • Caused by collisions between the orbiting body and atmospheric particles
  • Even tiny, it has significant long-term effects
  • Figure 10 Viscous drag
  • ⇒  Qualitative Effects on Orbiting Body:
  • 1. Decrease in Speed
  • – The drag removes energy from the satellite’s motion
  • – So orbital speed gradually decreases
  • 2. Reduction in Altitude
  • – As speed decreases, the satellite loses kinetic energy
  • – Its orbit decays → moves closer to Earth
  • – Since ​[math]v_\text{orbital} = \sqrt{\frac{GM}{r}}[/math], getting closer requires even higher speed to stay in orbit, but the satellite is slowing down → spirals inward
  • 3. Eventually Re-enters Atmosphere
  • – If not corrected, the satellite spirals downward and burns up due to atmospheric friction
  • ⇒Satellites in low orbits need:
  • – Periodic boosts (from onboard thrusters) to maintain altitude
  • – Or they are designed for short-term missions before reentry
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