Point and spherical masses (5.4.1)
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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.
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.
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.
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.