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Gravitational Fields (DP IB Physics)

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    What is Newton's law of gravitation?

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  • What is 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.

  • What is the gravitational force between two identical masses separated by a distance, r?

    The gravitational force between the two identical masses is: F space equals space fraction numerator G M squared over denominator r squared end fraction

    Where:

    • G = gravitational constant (6.67 × 10-11 N m2 kg–2)

    • M = magnitude of each mass, measured in kilograms (kg)

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

  • True or False?

    The value of, G, (gravitational constant) is the same in all of space.

    True.

    The value of, G, (gravitational constant) is the same in all of space.

  • True or False?

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

    False.

    The gravitational force between two masses is always attractive.

  • What is the relationship between gravitational force and distance?

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

    This means that when the distance between two point masses doubles, the gravitational force between them falls by 1 fourth.

  • What assumptions are made 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

  • Draw arrows to show the gravitational force between the Earth and the Moon.

    Diagram showing Earth and Moon. The Earth is labeled on the left, larger and with shaded continents. The Moon is on the right, smaller and shown with craters.

    The gravitational force between the Earth and the Moon:

    • is attractive (arrows point towards each other)

    • has the same magnitude on each object (arrows are the same size)

    • treats the objects as point masses (arrows begin from the centres)

    Diagram showing Earth and the Moon with green arrows representing force (F) directed towards each other, illustrating gravitational attraction.

  • Define 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.

  • What are the two equivalent units of gravitational field strength?

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

  • What is the 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.

    This is because gravitational field strength is a vector quantity.

  • 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.

  • Which two quantities does the strength of a gravitational field at the surface of a planet depend on?

    The strength of a gravitational field at the surface of a planet depends on:

    • the radius of the planet

    • the mass of the planet

  • What is the gravitational field strength due to a point mass of magnitude, M, at a distance, r?

    The gravitational field strength due to a point mass is: g space equals space fraction numerator G M over denominator r squared end fraction

    Where:

    • G = gravitational constant (6.67 × 10-11 N m2 kg–2)

    • M = magnitude of the point mass, measured in kilograms (kg)

    • r = distance from the mass to the point, measured in metres (m)

  • 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.

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

  • How is the resultant gravitational field due to multiple masses determined?

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

    • using simple addition (if the point lies on a line joining the masses)

    • using Pythagoras (if the point makes a right-angled triangle with the masses)

  • Sketch the variation of gravitational field strength with distance from the centre of Earth.

    Graph with gravitational field strength on the y-axis and distance from centre of Earth on the x-axis.

    The variation of gravitational field strength with distance from the centre of Earth is:

    Graph showing gravitational field strength (N/kg) on the y-axis and distance from the centre of Earth (m) on the x-axis. The line begins at the origin and is linear with a very steep upward gradient, peaks at R distance, and then curves steeply downward tending toward zero g at 5R

  • What do gravitational field lines represent?

    Gravitational field lines represent

    • the strength of the gravitational field

    • the direction of the gravitational field

  • What is a uniform gravitational field?

    A uniform gravitational field is one where the field strength is the same at all points.

  • True or False?

    Radial gravitational field lines always point towards the centre of mass of a body.

    True.

    Radial gravitational field lines always point towards the centre of mass of a body.

  • True or False?

    Radial fields are considered uniform fields.

    False.

    Radial fields are considered non-uniform fields.

  • Draw the gravitational field lines around a planet.

    A planet

    The gravitational field lines around a planet are radial:

    The gravitational field lines around a planet point radially inwards.

  • Draw the gravitational field lines close to the Earth's surface.

    A horizontal line labelled "Earth's surface"

    The gravitational field lines close to the Earth's surface are uniform:

    The gravitational field lines are uniform near Earth's surface

  • What is the difference between radial and uniform gravitational fields?

    The difference between radial and uniform gravitational fields is:

    • 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

  • Draw the gravitational field lines between larger mass P and smaller mass Q. Indicate the neutral point with an X.

    Two circles on a white background. The larger circle on the left is labeled "P," and the smaller circle on the right is labeled "Q."

    The gravitational field lines between P and Q are:

    Gravitational field lines between masses P and Q, with arrows pointing towards them. A neutral point is marked between P and Q.

  • What is the relationship between gravitational field strength and line density?

    The density of field lines represents the strength of a gravitational field

    • the closer together the field lines, the stronger the field

    • the further apart the field lines, the weaker the field

  • State Kepler's first law.

    Kepler's first law states that the orbit of a planet is an ellipse, with the Sun at one of the two foci.

  • Define gravitational potential at a point.

    The gravitational potential at a point is the work done per unit mass in bringing a small test mass from infinity to a defined point.

  • State Kepler's second law.

    Kepler's second law states that a line segment joining the Sun to a planet sweeps out equal areas in equal time intervals.

  • True or False?

    Gravitational potential is a vector quantity.

    False.

    Gravitational potential is a scalar quantity.

  • State Kepler's third law.

    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.

    • T squared space proportional to space R cubed or R cubed over T squared space equals space constant

  • What is the unit of gravitational potential?

    The unit of gravitational potential is J kg-1.

  • How is Kepler's third law derived?

    Kepler's third law is derived by equating the centripetal force and the gravitational force on an orbiting mass and substituting the expression for the speed of an object in circular motion:

    • F space equals space fraction numerator m v squared over denominator r end fraction space equals space fraction numerator G M m over denominator r squared end fraction

    • v squared space equals space fraction numerator G M over denominator r end fraction space equals space open parentheses fraction numerator 2 straight pi r over denominator T end fraction close parentheses squared

    • T squared space equals space fraction numerator 4 straight pi r cubed over denominator G M end fraction

  • True or False?

    Gravitational potential is always positive.

    False.

    Gravitational potential is always negative.

  • On the diagram, label the Sun, a planet and a comet.

    Three black dots each enclosed in separate dashed oval shapes representing the orbits of a planet and a comet around the Sun.

    The Sun is at the centre of the near-circular orbit of a planet and one of the two foci of the elliptical orbit of a comet.

    Three black dots each enclosed in separate dashed oval shapes representing the orbits of a planet and a comet around the Sun.

  • What is the gravitational potential due to a point mass of magnitude, M, at a distance, r?

    The gravitational potential due to a point mass is: V space equals space minus fraction numerator G M over denominator r end fraction

    Where:

    • G = gravitational constant (6.67 × 10-11 N m2 kg–2)

    • M = mass producing the gravitational field, measured in kilograms (kg)

    • r = distance from the centre of the mass to the point mass, measured in metres (m)

  • At what point in a comet's orbit does it travel fastest?

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

  • True or False?

    Gravitational potential increases with distance from a planet.

    True.

    Gravitational potential increases (becomes less negative) with distance from a planet.

  • At what point in a comet's orbit does it travel slowest?

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

  • What happens to gravitational potential as r approaches infinity?

    As r approaches infinity, gravitational potential approaches zero.

  • True or False?

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

    True.

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

  • True or False?

    The combined gravitational potential at a point due to multiple point masses is determined by vector addition.

    False.

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

  • Sketch the graph of log space T against log space r and label the position of Earth.

    Logarithmic graph with the log of the orbital period on the y-axis ranging from 0.1 to 1000 and the log of the orbital radius on the x-axis ranging from 0.1 to 100.

    The graph of log space T against log space r is a straight line. Earth is at (1, 1) as the orbital period is in years and the orbital radius is in astronomical units (AU)

    Logarithmic graph with the log of the orbital period on the y-axis ranging from 0.1 to 1000 and the log of the orbital radius on the x-axis ranging from 0.1 to 100. The green straight line shows the relationship with Earth positioned at (1, 1)

  • Define 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.

  • Define 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.

  • What is the gravitational potential energy near the Earth's surface?

    The gravitational potential energy near the Earth's surface is: E subscript p space equals space m g increment h

    Where:

    • m = mass, measured in kilograms (kg)

    • g = gravitational field strength (9.8 N kg–1)

    • increment h = change in height above surface, measured in metres (m)

  • True or False?

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

    True.

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

  • True or False?

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

    True.

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

  • What is the work done in moving a mass, m, in a gravitational field?

    The work done in moving a mass, m, in a gravitational field is: W space equals space m increment V

    Where:

    • W = work done in moving mass, measured in joules (J)

    • m = magnitude of mass moving in the field, measured in kilograms (kg)

    • increment V = potential difference between two points, measured in joules per kilogram (J kg−1)

  • Define gravitational potential difference.

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

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

  • What is the potential difference due to a point mass M when a mass is moved from a distance r subscript 2 to r subscript 1?

    The potential difference due to a point mass is: increment V space equals space G M open parentheses 1 over r subscript 1 space minus space 1 over r subscript 2 close parentheses

    Where:

    • G = gravitational constant (6.67 × 10-11 N m2 kg–2)

    • M = magnitude of mass producing the potential, measured in kilograms (kg)

    • r subscript 2 = initial distance from charge M, measured in metres (m)

    • r subscript 1 = final distance from charge M, measured in metres (m)

  • State the equation for the gravitational potential energy of two point masses.

    The gravitational potential energy of two point masses is: E subscript p space equals space minus fraction numerator G M m over denominator r end fraction

    Where:

    • G = gravitational constant (6.67 × 10-11 N m2 kg–2)

    • M = larger mass producing the field, measured in kilograms (kg)

    • m = mass moving within the field of M, measured in kilograms (kg)

    • r = distance between the centres of the masses, measured in metres (m)

  • What happens to the gravitational potential energy as a satellite moves away from a planet?

    The gravitational potential energy increases as a satellite moves away from a planet.

    When a small mass moves away from a large mass, the gravitational potential energy always increases.

  • What happens to the gravitational potential energy as a satellite moves towards a planet?

    The gravitational potential energy decreases as a satellite moves towards a planet.

    When a small mass moves towards a large mass, the gravitational potential energy always decreases.

  • What does the area under a force-distance graph represent?

    The 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.

  • When a small mass m moves away from a larger mass M, what is the change in gravitational potential energy?

    The change in gravitational potential energy (or work done) is: increment E subscript p space equals space G M m open parentheses 1 over r subscript 1 space minus space 1 over r subscript 2 close parentheses

    Where:

    • G = gravitational constant (6.67 × 10-11 N m2 kg–2)

    • M = larger mass producing the field, measured in metres (kg)

    • m = mass moving within the field of M, measured in kilograms (kg)

    • r subscript 1 = initial separation between masses, measured in metres (m)

    • r subscript 2 = final separation between masses, measured in metres (m)

  • Define the potential gradient of a gravitational field.

    Gravitational gradient is the rate of change of gravitational potential with respect to displacement in the direction of the field.

  • What does the negative sign indicate in the potential gradient equation?

    In the potential gradient equation g space equals space minus fraction numerator increment V over denominator increment r end fraction the negative sign indicates that the direction of the field strength g opposes the direction of increasing potential.

  • What does the gradient of a V minus r graph represent?

    The gradient of a potential-distance open parentheses V minus r close parentheses graph represents the gravitational field strength at that point.

  • What does the area under a g minus r graph represent?

    The area under a field-distance open parentheses g minus r close parentheses graph represents the potential difference between the two points.

  • True or False?

    The V minus r graph follows a 1 over r relation.

    False.

    The potential-distance open parentheses V minus r close parentheses graph follows a negative 1 over r relation.

    All values of gravitational potential are negative.

  • True or False?

    The curve of a g minus r graph is steeper than its corresponding V minus r graph.

    True.

    The curve of a g minus r graph is steeper than its corresponding V minus r graph.

    This is because g minus r graphs follow a 1 over r squared relation whereas V minus r graphs follow a negative 1 over r relation.

  • Sketch the relationship between the gravitational potential with distance above the surface of a planet.

    Graph with gravitational potential V on the y-axis and distance r on the x-axis

    The relationship between the gravitational potential with distance above the surface of a planet is:

    Graph with gravitational potential V on the y-axis and distance r on the x-axis. The green curve represents the potential which is negative at all values but approaches zero with distance.

  • Sketch the variation of potential with the distance between the surfaces of two planets A and B.

    Graph depicting gravitational potential against distance, with marks for the surface of planets A and B at zero potential.

    The variation of potential with the distance between the surfaces of two planets A and B is:

    Graph showing gravitational potential against distance, with the surfaces of planets A and B at zero potential.

  • What is a gravitational equipotential surface?

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

  • True or False?

    Equipotential lines are always perpendicular to gravitational field lines.

    True.

    Equipotential lines are always perpendicular to gravitational field lines.

  • True or False?

    Work is done when a mass moves along an equipotential surface.

    False.

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

  • What are the key features of the equipotential lines in a radial gravitational field?

    The key features of the equipotential lines in a radial gravitational field are:

    • concentric circles

    • become further apart with distance

  • What are the key features of the equipotential lines in a uniform gravitational field?

    The key features of the equipotential lines in a uniform gravitational field are:

    • horizontal straight lines

    • parallel

    • equally spaced

  • Draw the equipotential lines around the Earth.

    A planet such as the Earth

    The equipotential lines (dotted green lines) around the Earth are:

    A planet such as the Earth, with concentric dashed green circles radiating outward to indicate equipotential lines.

  • Draw the equipotential lines near the surface of the Earth.

    A horizontal black line representing the Earth's surface

    The equipotential lines (dotted green lines) near the Earth's surface are:

    A horizontal black line representing the Earth's surface with four equally spaced horizontal green dashed lines to indicate equipotential lines.

  • Draw the equipotential lines around two identical point masses.

    This is the equipotential surface for two identical point masses. The region of empty space in the centre indicates where the resultant field is zero.

    Equipotential lines between two identical masses where lines curve around each mass. At the midpoint between the masses is an empty space.

  • Draw the equipotential lines around the Earth and the Moon. The point P represents the point between them where the potential is zero.

    Gravitational field lines between the Earth and the Moon with point P between them to indicate the neutral point where potential is zero.

    The equipotential lines around the Earth and the Moon are perpendicular to the field lines.

    Gravitational field lines between the Earth and the Moon with point P between them to indicate the neutral point where potential is zero. The green lines represent equipotential lines.

  • Define escape speed.

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

  • True or False?

    Escape speed depends on the mass of the escaping object.

    False.

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

  • State the equation for escape speed.

    The equation for escape speed is: v subscript e s c end subscript space equals space square root of fraction numerator 2 G M over denominator r end fraction end root

    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)

    • r = distance from the centre of mass, measured in metres (m)

  • How is the equation for escape speed derived?

    The equation for escape speed is derived by equating the kinetic energy and the gravitational potential energy of a mass:

    • 1 half m v subscript e s c end subscript squared space equals space fraction numerator G M m over denominator r end fraction

    • v subscript e s c end subscript space equals space square root of fraction numerator 2 G M over denominator r end fraction end root

  • True or False?

    Escape speed is the speed needed to escape a planet's surface.

    False.

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

  • How does the mass of the planet affect escape speed?

    The relationship between escape speed and the mass of a planet is: v subscript e s c end subscript space proportional to space square root of M

    Therefore, the greater the mass of a planet, the greater the escape speed.

  • How does escape speed vary with distance from the centre of a planet?

    The relationship between escape speed and distance from the centre of a planet is: v subscript e s c end subscript space proportional to space square root of 1 over r end root

    Therefore, escape speed decreases with distance from the centre of a planet.

  • Why don't satellites being launched into an orbit around the Earth need to achieve escape speed?

    Satellites launched from Earth's surface do not need to achieve escape velocity to reach orbit because

    • the escape speed equation assumes the mass starts with zero kinetic energy, but the thrust from fuel provides a continuous source of energy

    • less energy is needed to achieve orbit than to escape from Earth's gravitational field

  • Define orbital speed.

    Orbital speed is the minimum speed required for an object to maintain a circular orbit.

  • True or False?

    All satellites at the same orbital radius have the same orbital speed, regardless of their mass.

    True.

    All satellites at the same orbital radius have the same orbital speed, regardless of their mass.

  • State the equation for orbital speed.

    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

    Where:

    • G = gravitational constant (6.67 × 10-11 N m2 kg–2)

    • M = mass of the larger mass being orbited, measured in kilograms (kg)

    • r = distance from the centre of mass, measured in metres (m)

  • How is the equation for orbital speed derived?

    The equation for orbital speed is derived by equating the centripetal force and the gravitational force between an orbiting mass and a larger mass:

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

    • 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

  • What is the kinetic energy of an orbiting satellite?

    The kinetic energy of an orbiting satellite is:

    • 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

    • E subscript k space equals space 1 half fraction numerator G M m over denominator r end fraction

  • What is the total energy of an orbiting satellite?

    The total energy of an orbiting satellite is equal to the sum of its kinetic energy and gravitational potential energy:

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

    • E subscript T space equals space 1 half fraction numerator G M m over denominator r end fraction space plus space open parentheses negative fraction numerator G M m over denominator r end fraction close parentheses

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

  • True or False?

    As a satellite's orbital radius decreases, its kinetic energy increases.

    True.

    As a satellite's orbital radius decreases, its kinetic energy increases.

  • True or False?

    As a satellite's orbital radius increases, its potential energy decreases.

    False.

    As a satellite's orbital radius increases, its potential energy increases.

  • True or False?

    Satellites in low orbits are not affected by air resistance.

    False.

    Satellites in low orbits (<600 km) can be affected by air resistance.

  • What is the effect of drag on a satellite's orbit over time?

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

  • What factors must be considered to provide enough energy to a satellite to put it into orbit?

    To provide enough energy to a satellite to put it into orbit, the factors that must be considered are:

    • the increase in gravitational potential energy between the Earth's surface and the height of the satellite's orbit

    • the required kinetic energy for the correct orbital speed of the satellite (to maintain a circular orbit)

    • the satellite must overcome frictional forces

    • additional energy must be allowed for thermal energy dissipation

  • What happens to the orbital speed of a satellite in a circular orbit as its orbit becomes lower due to drag?

    As a satellite's orbit becomes lower due to drag, its orbital speed increases. This is because some of its potential energy has been transferred to kinetic energy.

  • What happens to the total energy of a satellite in a circular orbit as its orbit becomes lower due to drag?

    As a satellite's orbit becomes lower due to drag, its total energy decreases. This is because some energy has been dissipated.

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