gravity physics

Newton's law of gravitation

Newton's law of gravitation states that the gravitational force between two point masses is directly proportional to the product of the masses and inversely proportional to the square of their separation.

Gravitational force between two identical masses

The gravitational force between the two identical masses is: F = G M² / r²

Gravitational constant (G)

G = gravitational constant (6.67 × 10⁻¹¹ N m² kg⁻²)

Separation of the masses (r)

r = separation of the masses, measured in metres (m)

True or False: The value of G is the same in all of space.

True.

True or False: The gravitational force between two masses can be attractive or repulsive.

False. The gravitational force between two masses is always attractive.

Relationship between gravitational force and distance

The relationship between gravitational force and distance is an inverse square law.

Effect of doubling distance on gravitational force

When the distance between two point masses doubles, the gravitational force between them falls by 1 fourth.

Assumptions about planets in Newton's law of gravitation

The assumptions made about planets in Newton's law of gravitation are: they are perfectly spherical, they are point masses (all the mass acts at their centres), their separations are much greater than their radii.

Gravitational force between Earth and Moon

The gravitational force between the Earth and the Moon is attractive, has the same magnitude on each object, and treats the objects as point masses.

Gravitational field strength at a point

The gravitational field strength at a point is the force per unit mass experienced by a test mass at that point.

Units of gravitational field strength

The two equivalent units of gravitational field strength are N kg⁻¹ and m s⁻².

Significance of a test mass

Test masses are used to define the strength of a field at a point and the direction a mass will move in the field.

True or False: An object's mass changes depending on the strength of the gravitational field.

False. An object's mass remains the same at all points in space, but the force it experiences changes depending on the strength of the gravitational field.

Factors affecting gravitational field strength at a planet's surface

The strength of a gravitational field at the surface of a planet depends on the radius of the planet and the mass of the planet.

Gravitational field strength due to a point mass

The gravitational field strength due to a point mass is: g = G M / r²

True or False: The variation of gravitational field strength around a planet and a point mass are identical.

False. The variation of gravitational field strength around the outside of a planet and a point mass are identical.

Gravitational field strength inside a planet

Inside the planet, the gravitational field strength decreases linearly from a maximum value (at the surface) to zero at the centre.

Resultant gravitational field due to multiple masses

The resultant gravitational field due to multiple masses is determined by vector addition.

Methods for determining resultant gravitational field

Using simple addition (if the point lies on a line joining the masses) or using Pythagoras (if the point makes a right-angled triangle with the masses).

Gravitational field lines

Represent the strength and direction of the gravitational field.

Uniform gravitational field

A field where the field strength is the same at all points.

Radial gravitational field lines

Always point towards the centre of mass of a body.

Radial fields

Considered non-uniform fields.

Gravitational field lines around a planet

Point radially inwards.

Gravitational field lines close to Earth's surface

Are uniform.

Difference between radial and uniform gravitational fields

In a uniform gravitational field, field strength is the same at all points; in a radial gravitational field, field strength varies with distance from the centre.

Gravitational field lines between larger mass P and smaller mass Q

Indicate a neutral point marked between P and Q.

Relationship between gravitational field strength and line density

The density of field lines represents the strength of a gravitational field; closer lines indicate a stronger field, while further apart lines indicate a weaker field.

Kepler's first law

States that the orbit of a planet is an ellipse, with the Sun at one of the two foci.

Gravitational potential at a point

The work done per unit mass in bringing a small test mass from infinity to a defined point.

Kepler's second law

States that a line segment joining the Sun to a planet sweeps out equal areas in equal time intervals.

Gravitational potential

Is a scalar quantity.

Kepler's third law

States that for planets or satellites in a circular orbit about the same central body, the square of the time period is proportional to the cube of the orbital radius.

Unit of gravitational potential

J kg-1.

Derivation of Kepler's third law

Derived by equating the centripetal force and the gravitational force on an orbiting mass.

Gravitational potential is always positive

False; gravitational potential is always negative.

Gravitational potential due to a point mass

V = -G M / r

Mass (M)

Mass producing the gravitational field, measured in kilograms (kg)

Distance (r)

Distance from the centre of the mass to the point mass, measured in metres (m)

Fastest point in a comet's orbit

Comets travel fastest when they are at the closest point to the Sun in their elliptical orbit.

Gravitational potential increases with distance

True.

Slowest point in a comet's orbit

Comets travel slowest when they are at the furthest point from the Sun in their elliptical orbit.

Gravitational potential as r approaches infinity

As r approaches infinity, gravitational potential approaches zero.

Graph of log T against log r for planets

A graph of log T against log r for the planets in the Solar System is a straight line.

Combined gravitational potential due to multiple point masses

False. The combined gravitational potential at a point due to multiple point masses is determined by adding together the potential due to each mass.

Gravitational potential energy of a system

The gravitational potential energy of a system is the work done when bringing all the masses in a system to their positions from infinity.

Gravitational potential energy of a point mass

The gravitational potential energy of a point mass is the work done in bringing the point mass from infinity to a point.

Gravitational potential energy near the Earth's surface

E_p = m g Δh

Gravitational field strength (g)

9.8 N kg-1

Gravitational potential energy on the surface of a planet

True. The gravitational potential energy on the surface of a planet is taken to be zero.

Work required to move a mass against a gravitational field

True. An input of work is required to move a mass against a gravitational field.

Work done in moving a mass in a gravitational field

W = m ΔV

Gravitational potential difference

Gravitational potential difference is the difference in gravitational potential between two points.

Work done when a mass is moved between two points

It is equal to the work done when a mass of 1 kg is moved between the points.

Potential difference due to a point mass

increment V = G M (1/r1 - 1/r2)

Magnitude of mass (M)

Measured in kilograms (kg)

Initial distance (r2)

Measured in metres (m)

Final distance (r1)

Measured in metres (m)

Gravitational potential energy of two point masses

Ep = - (G M m / r)

Mass moving within the field (m)

Measured in kilograms (kg)

Distance between the centres of the masses (r)

Measured in metres (m)

Gravitational potential energy as a satellite moves away from a planet

Increases

Gravitational potential energy as a satellite moves towards a planet

Decreases

Area under a force-distance graph

Represents the change in gravitational potential energy or the work done in moving the mass from one point to another.

Change in gravitational potential energy when moving away from a larger mass

increment Ep = G M m (1/r1 - 1/r2)

Potential gradient of a gravitational field

Rate of change of gravitational potential with respect to displacement in the direction of the field.

Negative sign in the potential gradient equation

Indicates that the direction of the field strength g opposes the direction of increasing potential.

Gradient of a V-r graph

Represents the gravitational field strength at that point.

Area under a g-r graph

Represents the potential difference between the two points.

V-r graph relation

Follows a negative 1/r relation.

Curve of a g-r graph compared to V-r graph

Steeper than its corresponding V-r graph.

Relation of g-r graphs

Follow a 1/r² relation.

Graph of gravitational potential with distance

Gravitational potential V on the y-axis and distance r on the x-axis; potential is negative at all values but approaches zero with distance.

Gravitational equipotential surface

Equipotential surfaces (or lines) connect points of equal gravitational potential.

Equipotential lines and gravitational field lines

Equipotential lines are always perpendicular to gravitational field lines.

Work done on equipotential surface

No work is done when a mass moves along an equipotential surface.

Key features of equipotential lines in a radial gravitational field

The key features of the equipotential lines in a radial gravitational field are concentric circles that become further apart with distance.

Key features of equipotential lines in a uniform gravitational field

The key features of the equipotential lines in a uniform gravitational field are horizontal straight lines that are parallel and equally spaced.

Escape speed

Escape speed is the minimum speed that will allow an object to escape a gravitational field with no further energy input.

Escape speed and mass of escaping object

Escape speed is the same for all masses in the same gravitational field.

Equation for escape speed

The equation for escape speed is: v_{esc} = sqrt(2GM/r) where G = gravitational constant (6.67 × 10^-11 N m² kg⁻²), M = mass of the object to be escaped from, measured in kilograms (kg), and r = distance from the centre of mass, measured in metres (m).

Derivation of escape speed equation

The equation for escape speed is derived by equating the kinetic energy and the gravitational potential energy of a mass: 1/2 mv_{esc}² = GMm/r.

Escape speed and planet's surface

Escape speed is the speed needed to escape a planet's gravitational field altogether.

Relationship between escape speed and mass of a planet

The relationship between escape speed and the mass of a planet is: v_{esc} ∝ sqrt(M). Therefore, the greater the mass of a planet, the greater the escape speed.

Orbital Speed

The minimum speed required for an object to maintain a circular orbit.

Escape Speed Equation

v subscript e s c end subscript space proportional to space square root of 1 over r end root

Orbital Speed Equation

The equation for orbital speed is: v subscript o r b i t a l end subscript space equals space square root of fraction numerator G M over denominator r end fraction end root

Kinetic Energy of an Orbiting Satellite

E subscript k space equals space 1 half m open parentheses v subscript o r b i t a l end subscript close parentheses squared space equals space 1 half m open parentheses fraction numerator G M over denominator r end fraction close parentheses

Total Energy of an Orbiting Satellite

E subscript T space equals space E subscript k space plus space E subscript p

Potential Energy of an Orbiting Satellite

E subscript T space equals space minus 1 half fraction numerator G M m over denominator r end fraction

Effect of Drag on Satellite's Orbit

The effect of drag on a satellite's orbit over time is a decrease in height and an increase in orbital speed.

Factors for Launching a Satellite

The increase in gravitational potential energy, required kinetic energy for orbital speed, overcoming frictional forces, and additional energy for thermal energy dissipation.

Effect of Lower Orbit on Orbital Speed

As a satellite's orbit becomes lower due to drag, its orbital speed increases.

Effect of Lower Orbit on Total Energy

As a satellite's orbit becomes lower due to drag, its total energy decreases.

True or False: All satellites at the same orbital radius have the same orbital speed.

True.

True or False: As a satellite's orbital radius decreases, its kinetic energy increases.

True.

True or False: As a satellite's orbital radius increases, its potential energy decreases.

False.

True or False: Satellites in low orbits are not affected by air resistance.

False.

True or False: As a satellite's orbital radius increases, its potential energy increases.

True.