The famous physicist Isaac Newton discovered that all objects with mass attract one another. We now know that this is due to gravitational forces. These forces are always attractive, meaning that they will never act to repel one object from another.
The gravitational forces exerted by the mass of an object forms a gravitational field around it. A field here refers to a region of space where objects will have certain forces exerted on them. In other words, any object placed within a gravitational field will generate a force of attraction on the object whose mass created the field.
There are two main types of fields. The first type is radial fields. These fields have field lines arranged around, and pointing towards, an object’s centre of mass. The closer you get to the centre, the stronger the force. The second type is uniform fields. In this case, the magnitude and direction of the gravitational field strength are identical throughout. This is often the case at the surface of the Earth.
At any point in a gravitational field, the gravitational field strength (g) is described as the force per unit mass at that point. This concept can be summarised by the following formula:
- Formula: Gravitational field strength = Force (N) / Mass (kg)
- Simplified formula: g = F / m
- SI Unit: Newton per kilogram (Nkg-1)
Newton’s Law of Universal Gravitation
Newton’s Law of Universal Gravitation states that the gravitational force between two objects with mass is directly proportional to the masses of each object, and inversely proportional to the square of the distance between each object. This is summarised in the following equation:
- Formula: Gravitational force = (Universal gravitational constant (Nm2kg-2) x Mass of object 1 (kg) x Mass of object 2 (kg)) / (Separation of centres of mass (m))2
- Simplified formula: F = Gm1m2 / r2
- SI Unit: Newton (N)
The universal gravitational constant is approximately 6.7 x 10-11 Nm2kg-2. The constant is exceptionally small, especially compared to the constant of the equation for electrostatic force, which is 1020 times larger. This demonstrates that the force of gravity is actually very weak, except in the case of objects with massive amounts of mass.
Gravitational field strength on the Earth’s surface
Upon substituting the equation for gravitational force (F = Gm1m2 / r2) into that of gravitational field strength (g = F / m), you also derive an equation for gravitational field strength related to the Earth’s mass. This can be used to deduce g at a particular point due to a spherical mass, like the Sun or the Earth.
- Formula: Gravitational field strength = (Universal gravitational constant (Nm2kg-2) x Mass of the Earth (kg)) / (Separation of Earth’s centre and the point (m))2
- Simplified formula: g = GM / r2
- SI Unit: Newton per kilogram (Nkg-1)
The mass of the Earth is approximately 6 × 1024 kg, while the distance between the centre of the Earth and its surface is about 6.4 × 106 m. It is important to note that this equation only applies to points beyond the surface of the Earth. If you were to tunnel far into the Earth, the mass ‘above’ you would pull you away from the centre of the Earth, invalidating the equation.
On the surface of the Earth, the gravitational field strength is approximately constant at g = 9.81 Nkg-1, which is often approximated to 10 Nkg-1 for calculations. It is also equal to the acceleration of free fall, i.e. 9.81 ms-2.
While gravitational force is weak as a proportion of mass, it can become significant when dealing with massive objects, like the Earth. It is crucial to understand the fundamental concept of gravitational fields before you move on to more complex topics, such as gravitational potential and the effect of gravitational force on centripetal acceleration.
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