Gravitational force is a universal force that acts between any two bodies in the universe. In 1687, Sir Isaac Newton formulated the Universal Law of Gravitation after observing that an apple falls vertically to the ground and that the Moon revolves around the Earth due to mutual attraction.
See how \(g\) increases linearly inside the Earth and decreases with the inverse square of distance outside the Earth.
\(g\) is directly proportional to \(r\). At the centre of the Earth, \(r = 0\), so \(g = 0\).
\(g\) is inversely proportional to \(r^2\). If \(g(R)\) is the surface value, then \(g(2R) = \frac{1}{4}g(R)\) and \(g(3R) = \frac{1}{9}g(R)\).
The gravitational force between two bodies is directly proportional to the product of the masses of both bodies and inversely proportional to the square of the distance between the centres of the two bodies.
\(F = \dfrac{G m_1 m_2}{r^2}\)
\(F\) = gravitational force (N), \(m_1\) and \(m_2\) = masses (kg), \(r\) = distance between centres (m), \(G = 6.67 \times 10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}\).
On the Earth's surface, \(g = \dfrac{GM}{R^2} = 9.81\ \mathrm{m\,s^{-2}}\). At height \(h\), \(g = \dfrac{GM}{(R+h)^2}\).
For a satellite or planet in circular orbit, gravitational force provides the centripetal force: \(F = \dfrac{mv^2}{r}\), with centripetal acceleration \(a = \dfrac{v^2}{r}\).
Determining the Mass of the Earth or the Sun
Equate gravitational force and centripetal force, then substitute \(v = \dfrac{2\pi r}{T}\).
Surface gravitational acceleration differs because each body has a different mass and radius. Approximate values are the Moon \(1.62\ \mathrm{m\,s^{-2}}\), Mars \(3.71\ \mathrm{m\,s^{-2}}\), Earth \(9.81\ \mathrm{m\,s^{-2}}\), and the Sun \(274\ \mathrm{m\,s^{-2}}\). An astronaut's mass remains the same, but the astronaut's weight, \(W=mg\), changes.
The value of \(g\) is needed to plan launches, landings, orbits, fuel, and mission payloads. Gravity also helps retain an atmosphere and influences the movement of fluids and living organisms. Long exposure to low gravity can reduce bone and muscle density, redistribute body fluids, and affect lung development and function, blood circulation, and blood pressure. Knowing \(g\) is therefore important for life-support systems and for human development and health during space exploration.
Answer in your mind, then press “Check Answer”.
Two identical spheres of mass \(5.0\ \mathrm{kg}\) each are separated by a distance of \(0.5\ \mathrm{m}\) between their centres. Calculate the gravitational force between them. (\(G = 6.67 \times 10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}\))
What is the formula for the centripetal acceleration, \(a\), of a satellite orbiting Earth at linear speed \(v\) and orbital radius \(r\)?
What two physical measurements of an orbiting satellite are needed to determine the mass of the Earth?
A satellite of mass \(1.00\times10^3\ \mathrm{kg}\) is \(4.00\times10^5\ \mathrm{m}\) above Earth. Calculate the gravitational force on it if \(M=5.97\times10^{24}\ \mathrm{kg}\), \(R=6.37\times10^6\ \mathrm{m}\), and \(G=6.67\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}\).
The Moon orbits Earth with a mean radius of \(3.84\times10^8\ \mathrm{m}\) and a period of \(27.3\) days. Determine the mass of Earth.
There is no gravity acting on satellites orbiting the Earth, which is why astronauts float.
Satellites and astronauts are within Earth's gravitational field. They experience free fall towards the Earth, and this gravitational force provides the centripetal force required to keep them in orbit.
Gravitational force only acts between planets and stars, not between ordinary people or small objects.
Gravitational force is universal and exists between any two masses. However, because \(G\) is extremely small, the force between everyday small masses is negligible and cannot be felt.
Gravitational acceleration increases continuously as you go deeper towards the Earth's centre.
Below the Earth's surface (\(r < R\)), gravitational acceleration decreases linearly with distance towards the centre and becomes zero at the centre of the Earth.
Objective: Identify gravitational force, gravitational acceleration, centripetal force and mass determination.
1 What happens to the gravitational force between two bodies if the distance between their centres is doubled while their masses remain unchanged?
2 What provides the centripetal force that keeps a man-made satellite in orbit around the Earth?
3 For a position above the surface of the Earth (\(r \ge R\)), the value of gravitational acceleration is inversely proportional to the of the distance from the centre of the Earth.
4 The formula for centripetal acceleration of a body moving at linear speed \(v\) in a circle of radius \(r\) is a = .
Drag each quantity to the correct formula or value. If using a phone, tap the answer first, then tap the matching box.
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