IB Physics HL D.1 Gravitational fields Notes

Kepler’s laws are observational rules that Newton’s gravitation later explains. The first law fixes the orbit geometry. The second law describes changing speed around an ellipse. The third law links orbital period to orbit size, but only when the orbiting bodies share the same central mass.

State Kepler’s second and third laws and explain why a comet moves faster when it is closer to the Sun.

Newton’s law gives the magnitude of the attractive force between two masses. For spherical bodies, treat the mass as if concentrated at the centre when calculating the external field or force. The formula gives equal force magnitudes for both bodies; the acceleration can differ because acceleration depends on mass.

Two spherical masses attract each other gravitationally. State the formula for the force and explain what distance should be used for r.

The point-mass approximation is a modelling step before using inverse-square gravity. For planets and stars treated as spherical, external gravitational calculations use the centre as the location of the mass. This is why orbital radius is measured from the centre of the planet, not from the surface.

A satellite orbits 400 km above Earth’s surface. Explain why the orbital radius used in gravitational calculations is not 400 km.

A gravitational field describes what force a small test mass would experience at each point. Dividing force by the test mass gives g, so the field is a property of the source masses and position, not of the particular test mass. Around a spherical planet, g points radially inward and has magnitude GM/r^2 outside the planet.

Define gravitational field strength and derive the expression for the field strength outside a spherical mass M.

A field-line diagram is a vector map. The arrow direction tells the direction a small mass would be pulled. Around an isolated spherical mass, all field lines point inward toward the centre. Near Earth’s surface over small distances, the radial lines are almost parallel, so the field is often drawn as uniform.

Sketch the gravitational field lines around an isolated spherical planet and explain how the diagram shows field direction and strength.

Because gravity is attractive, two separated masses form a bound system with less energy than the same masses infinitely far apart. Setting E_p = 0 at infinity makes every finite separation negative. Work done by an external agent to move a mass away from a planet increases E_p toward zero.

Explain why the gravitational potential energy of a satellite-Earth system is negative when zero is set at infinity.

Force and potential energy describe the same interaction from different viewpoints. Force tells the instantaneous pull and has a direction. Potential energy tells how much energy is associated with the separation of the two-mass system. The formulas look similar, but the distance dependence and sign are different.

A student writes the gravitational potential energy of two masses as -Gm1m2/r^2. Identify the error and give the correct expression.

Potential strips away the test mass from potential energy. Instead of asking how much energy a particular mass has, V describes the field’s energy-per-kilogram value at a point. Around a single mass, V is negative because work must be supplied to move a test mass from that point to infinity.

Define gravitational potential and state the expression for potential at distance r from an isolated mass M.

A potential graph stores field information in its slope. Close to a mass, the potential changes rapidly with distance, so the field is strong. Far away, the graph flattens and the field is weak. The minus sign is a direction statement: an object accelerates in the direction of decreasing gravitational potential.

The gravitational potential V becomes less negative as distance r from a planet increases. Explain how the V-r graph can be used to find gravitational field strength.

Potential is energy per unit mass, so multiplying potential difference by mass gives the potential energy change. The sign depends on who does the work. If an external agent slowly lifts a mass away from a planet, it does positive work and gravitational potential energy increases. Gravity does negative work for that same displacement.

A 500 kg spacecraft is moved slowly between two points where the gravitational potential changes from -8.0 MJ kg^-1 to -6.0 MJ kg^-1. Calculate the external work done.

Equipotentials are contour lines for gravitational potential. Just as height contours connect equal altitude on a map, equipotentials connect equal potential in a field diagram. Moving along a contour does not change potential energy, but moving between contours does.

Explain why no work is done moving a satellite along a gravitational equipotential surface.

Equipotentials and field lines are two views of the same field. The equipotentials show scalar values of V, while field lines show the direction of force. A field line cannot have any component along an equipotential, because then the field would do work without a change in potential. Therefore the field line crosses the equipotential at right angles.

On a diagram of gravitational equipotential lines around a planet, add field lines and explain their direction.

At escape threshold, the object just reaches infinity with no kinetic energy left. Its initial kinetic energy must therefore equal the magnitude of its negative gravitational potential energy. This gives the escape-speed formula and explains why escape speed from a planet depends on planet mass and starting radius, not on the launched object’s mass.

Derive the escape speed from the surface of a planet of mass M and radius R, ignoring air resistance.

A circular orbit is continuous free fall around the central mass. Gravity provides the inward acceleration needed to keep the satellite moving in a circle. Equating gravitational force with centripetal force gives orbital speed. The force direction is inward while the velocity is tangential.

Derive the speed of a satellite in a circular orbit of radius r around a planet of mass M.

Drag acts opposite the satellite’s motion and removes mechanical energy. The satellite drops to a lower orbit with more negative gravitational potential energy. For a circular orbit around the same planet, smaller radius requires a larger orbital speed. The apparent paradox is resolved by remembering that total mechanical energy decreases even though kinetic energy can increase in the lower orbit.

Describe the qualitative effect of a small atmospheric drag force on the height and speed of a satellite in low Earth orbit.

Core D.1 is secure when the student chooses the correct gravitational representation. Kepler describes observed orbit patterns. Newton explains the attractive inverse-square force. Field strength describes force per kilogram and field lines show direction and relative strength. Circular orbit questions use centripetal force, not escape-energy arguments.

HL D.1 is an energy-field extension. The key move is separating scalar quantities from vector quantities. Potential and potential energy are scalar and negative near an attracting mass. Field strength is a vector found from the negative potential gradient. Work depends on potential difference. Escape speed uses total energy, while drag reduces total mechanical energy and drives orbital decay.