Gravitational fields (Topic 12)
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An object with mass generates a gravitational field around it. Objects with mass are attracted to each other: an object in a gravitational field is attracted to the source of that field.
The strength of a gravitational field depends on the mass of the object and the distance from the object.
The gravitational field strength decreases as the distance from the mass increases. It follows an inverse-square relationship:
This relationship holds true for all gravitational fields, regardless of the mass generating it. For example, doubling the distance from a mass decreases the gravitational field strength by a factor of four.
Gravitational field strength, (at a given distance) is directly proportional to the mass of the object.
Larger masses produce stronger gravitational fields at the same distance. For example, a person standing on the Earth’s surface experiences a gravitational force of towards the centre of the Earth.
On the other hand, the gravitational force between two electrons is a factor of less than the Coulomb force of repulsion between them, so can be ignored.
In a gravitational field, the field lines:
- represent the direction and strength of a gravitational field,
- indicate the force on a mass at each point in a gravitational field,
- always point inward towards the mass, as gravity is always an attractive force, and
- never cross each other.
If the gravitational field lines are closer together, the field is stronger at that point. When the gravitational field lines are spaced further apart, the field is weaker. Gravitational field-line diagrams show that the field strength decreases with distance from the centre of mass.
A spherical mass produces a radial, symmetrical gravitational field, equivalent to a point mass at its centre. Non-spherical masses only approximate this pattern at large distances; close to the object, the field lines follow the actual shape of the mass distribution.
In a uniform gravitational field, the field lines are parallel and equidistant, indicating a constant gravitational field strength over the region.
The gravitational field close to the surface of a planet is approximately uniform. The gravitational field strength remains approximately constant over relatively small distances from the surface.
Although the planet’s gravitational field is radial, the distance between the test mass (the object experiencing gravity) and the centre of the planet does not change much relatively when you are near the surface of a planet.
Gravitational field lines always point inward towards the mass, and never away. This is because gravity is an attractive force. On the other hand, electric field lines can point inward or outward.
Gravitational field strength, , is the force per unit mass at a point in a gravitational field. It is equal to the force exerted on a mass at a point in a gravitational field.
The formula for gravitational field strength is:
Where:
- is the gravitational force acting on an object in the gravitational field in newtons (), and
- is the mass of the object in kilograms ().
The unit for gravitational field strength is newtons per kilogram . This is equivalent to metres per second squared which is the SI unit for acceleration.
The equivalence in units of gravitational field strength and acceleration arises from the following two equations:
While both are expressions of Newton’s second law, they serve distinct purposes:
- : applies to any object experiencing a force, regardless of the cause.
- : a specific application describing an object under the influence of a gravitational field.
The higher the mass and the smaller the radius of a celestial body, the greater the gravitational field strength at its surface.
Different celestial bodies have different values of , which refers to the gravitational field strength at the body’s surface. The gravitational field strength on a planet determines the force acting on an object or person due to gravity. The stronger the gravitational field strength, the heavier the object will feel, and the more force is needed to lift it.
Question walkthrough
Astronaut's Weight on Mars vs Earth
Converts an astronaut's weight on Earth into mass, then uses Mars's gravitational field strength to find their weight on Mars.
Newton’s law of gravitation states that the gravitational force between two point masses, separated by a distance, is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Newton’s law of gravitation can be expressed mathematically as:
Where:
- is the gravitational force in newtons .
- and are the masses of the two point bodies in .
- is the separation distance in metres .
- is the gravitational constant, equal to
The negative sign indicates that gravitational force is attractive – gravity pulls masses toward each other.
The gravitational force acting on objects on the surface of the Earth and other planets is known as weight.
The weight of an object near a planet’s surface is given by:
Where:
- is the object’s mass, and
- is the gravitational field strength on the Earth’s surface, which is .
Different celestial bodies have different gravitational field strengths at their surface, depending on their mass and radius.
Since weight is equivalent to the gravitational force, it has the same unit, the newton, .
The gravitational force between two masses decreases as the distance between them increases, following an inverse-square relationship:
The inverse-square relationship holds true regardless of the specific masses involved. For example, doubling the distance between any two masses decreases the gravitational force by a factor of four.
The graph above shows the inverse-square relationship between the gravitational force, , and the distance from the centre of a spherical object, .
It is important to note that the curve is initially very steep and then becomes shallower. The gravitational force exerted by a mass (like a planet or star) diminishes rapidly as an object moves further away. However, this rate of decrease lessens significantly at greater distances.
The point mass approximation can be used for spherically symmetrical objects, where the mass distribution is uniform.
In Newton’s law of gravitation, the point mass approximation states that the separation distance between two masses is equal to the distance between their centres rather than the distance between their surfaces.
Use the point mass approximation when calculating gravitational field strength or gravitational force between masses.
When multiple objects are present, the total gravitational force acting on any single object is determined by summing the vectors of the individual gravitational forces exerted by all the other objects.
- One-dimensional cases: If objects are aligned along a straight line, the magnitudes of the gravitational forces can be added by considering their directions.
- Two-dimensional cases: For objects positioned at angles (e.g. triangular configurations), individual forces can be resolved into components using Pythagoras’ theorem or trigonometric functions (sine and cosine). Then, sum the components along each axis to find the net force.
The diagram above shows a triangular arrangement of masses. The total gravitational force on mass is equal to the vector addition of the gravitational forces and due to each of the masses and .
Question walkthrough
Resultant Gravitational Force on a Collinear Mass
Calculate the resultant gravitational force on a mass from two other masses aligned along the same straight line.
Newton’s shell theorem states that a spherical shell of mass exerts the same gravitational pull on external objects as if all its mass were concentrated at its centre.
Calculations related to gravitational fields of spherical objects can be simplified by treating their mass as concentrated at a single point at their centre – this is the point mass approximation.
This approximation holds for spherically symmetric objects where the mass distribution is uniform.
The point mass approximation is commonly used for planets, stars, and other celestial bodies to calculate gravitational effects on nearby objects.
For example, the Earth can be approximated to a point mass of at the centre.
Without this approximation, finding the gravitational force acting on an object due to the Earth would require summing up the effects from each point on the Earth, which would be extremely difficult.
Remember to use the point mass approximation when calculating gravitational field strength or gravitational force between masses.
The point mass approximation applies only at distances greater than the radius of the spherical mass.
Inside the spherical mass, the gravitational field strength is influenced by the mass distribution.
Inside a spherical mass, the gravitational field strength at a point depends on the amount of mass inside within the radius equal to the distance at that point.
As you move away from the centre, more mass is enclosed within that radius, so the gravitational field strength actually increases.
This increase turns out to be directly proportional to the distance from the centre, which explains why the beginning of the graph above is a straight-line.
At distances greater than the radius of the mass, the inverse-square relationship is observed.
Gravitational field strength, is the force per unit mass at a point in a gravitational field. It equals the force exerted on a mass at a point in a gravitational field.
Gravitational field strength is expressed mathematically as:
Where
- is the gravitational force in newtons (N), and
- is the object’s mass in the gravitational field in kilograms ().
The units for gravitational field strength are (or ), the same as for acceleration.
Combining the formulas for gravitational field strength and Newton’s law of gravitation leads to an expression for gravitational field strength .
Substituting Newton’s law for gravitation:
into the equation for gravitational field strength:
gives:
Simplifying the expression gives the gravitational field strength, , at a distance from the centre of an object of mass as:
The negative sign indicates that the gravitational field strength at a distance is in the opposite direction to the displacement from the centre of mass , showing that the gravitational force is attractive, since is force per unit mass.
Note that the gravitational field strength does not depend on the object’s mass in the gravitational field.
The magnitude of gravitational field strength , at a given and constant distance, is directly proportional to the mass of the object creating the gravitational field.
Moreover, the magnitude of the gravitational field strength decreases as the distance from the mass increases, following an inverse-square relationship.
The gravitational potential at a point is the work done per unit mass to bring a test mass from an infinite distance to a chosen point inside a body’s gravitational field.
At an infinite distance away from an object, the gravitational potential is defined as zero:
A mass at this point feels no force due to the object’s gravitational field.
Gravitational potential is always negative because:
- Gravitational forces are attractive. Energy is required to move a mass away from another mass. Therefore, the work done to move a mass from infinity towards another mass is negative.
- At any point within a gravitational field, the potential is lower than at infinity (where ).
Gravitational potential can be thought of as a ‘gravitational well’. Moving a mass to infinity is like climbing out of the well – energy is needed to reach the ‘zero’ level. A larger mass has a stronger gravitational field, corresponding to a larger ‘gravitational well’.
Gravitational potential energy is the energy a mass has due to its position in a gravitational field. It can be found from by multiplying by the object’s mass :
Gravitational potential at a point is the gravitational potential energy per unit mass.
Gravitational field strength is given by:
Where:
- is Newton’s gravitational constant ,
- is the mass of the body producing the gravitational field (kg),
- is the distance (m) from the centre of the mass to the point in the field, and
- is measured in
Comparing this to the equation for gravitational potential shows that
Where:
- is the energy per unit mass at a point in a gravitational field.
- is the force per unit mass acting on an object in a gravitational field.
It is important to note that and are directly proportional to each other, but they are distinctly different quantities. Moreover, both quantities are generally measured from the body’s centre of mass. However, for practical applications, we often express these quantities at different distances from the centre, including the surface.
The zero point for gravitational potential is at a distance of infinity. Hence, all distances closer than infinity have negative gravitational potential – this reflects that work is required to escape a gravitational field.
The gravitational potential at a distance from a mass is given by:
Where:
- is Newton’s gravitational constant ,
- is the mass of the body producing the gravitational field ,
- is the distance in from the centre of the mass to the point in the field.
is negative because gravitational forces are attractive, and has units of
The closer a point is to the mass the more negative the gravitational potential. As increases, becomes less negative, approaching zero at infinity.
Work is required to move a unit mass away from a planet or mass This work increases as the object moves further away.
Two points at different distances from a mass have different gravitational potentials because increases (becomes less negative) with distance.
The gravitational potential difference between two points is equal to:
Where:
- is the final gravitational potential,
- is the initial gravitational potential,
- is the change in gravitational potential.
All quantities are in
The change in potential between two distances (initial) and (final) from a mass is:
Where:
- is the gravitational constant
- is the mass of the object creating the field ()
- is the initial distance from the mass ()
- is the final distance from the mass ()
The change in potential is measured in
Question walkthrough
Work Done Moving a Satellite in a Gravitational Field
Calculate the work done moving a satellite between two distances from a planet's centre using the change in gravitational potential.
Gravitational fields are one of several types of fields that exert a force on objects.
Gravitational fields always attract objects, exerting a force toward the center of gravity. The force depends on the object’s mass and the distance from the source.
Other fields that give rise to a force include:
- Electric fields
Electric fields exert forces on charged particles, with the direction of the force determined by the charge’s polarity.
- Magnetic fields
Magnetic fields exert a force on moving charged particles, with the direction of the force being perpendicular to both the velocity of the particle and the direction of the magnetic field.
Gravitational fields share many similarities with electric fields:
Both fields obey the inverse square law – the force between two objects decreases with the square of the distance between them:
- The strength of both fields are represented by the force experienced by an object within the field divided by its respective property: mass in gravitational fields and charge in electric fields. In both cases, the field strength is a measure of the force per unit mass or unit charge:
- gravitational field:
- electric field:
- Both fields exert forces without physical contact, affecting objects at a distance.
- Both have a radial field for point masses and point charges.
- Both fields can be represented using field lines. Electric field lines point from positive to negative charges. Gravitational field lines point towards the mass creating the field.
Gravitational and electric fields are similar in many ways, but they have their differences:
- Electric fields arise from electric charges, whereas gravitational fields arise from mass.
- Electric fields can be either attractive or repulsive depending on the charges (like charges repel, opposite charges attract). Gravitational fields are always attractive – masses always attract each other.
- The direction of field lines in electric fields depends on the sign of the charge (field lines point away from positive charges and towards negative charges), whereas field lines in gravitational fields always point towards the source of the mass.
- Forces exerted by gravitational fields are weaker compared to those exerted by electric fields for everyday charged objects, but gravitational forces dominate on large scales, such as between planets.
The centripetal force acting on a planet is the gravitational force of the sun on the planet.
The direction of the centripetal force is towards the sun and perpendicular to the direction of motion of the planet.
It is important to note that for simplified models typically found at A level, the orbital path of most planets can be approximated as circular.
The centripetal force required to keep a planet in orbit is provided by the gravitational force between the planet and the Sun, such that:
Where:
- is the mass of the planet in
- is the orbital velocity of the planet in
- is the angular velocity of the planet in
- is the orbital radius (distance between the centre of masses of the planet and sun) in
- is the mass of the Sun, in
- is the gravitational constant
Using the relation between gravitational force and centripetal force the orbital velocity, , of a planet in a circular orbit can be derived:
The planet’s mass cancels out on both sides of the equation, meaning that the orbital velocity and centripetal force only depend on the Sun’s mass and the planet’s distance. This means that all planets, regardless of mass, travel at the same speed at a given orbital radius.
It is important to understand the role that force and acceleration play in circular motion.
Question walkthrough
Orbital Radius from Gravitational-Centripetal Force
Equate gravitational and centripetal force to calculate a planet's orbital radius from its speed and its star's mass.