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**The law of gravitation By the end of this topic you should be able to:**

appreciate that there is an attractive force between any two point masses that is directed along the line joining the two mases. state the definition of gravitational field strength.

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**Newton’s law of gravitation**

Newton’s second law implies that, whenever a mass moves with acceleration, a force must be acting on it. An object falling freely under gravity is accelerating at 9.8 ms-2 and thus experiences a net force in the direction of the acceleration. This force is the weight of the mass. Similarly, a planet that revolves around the sun also experiences acceleration and thus a force is acting on it.

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**Newton’s law of gravitation**

Newton proposed that the attractive force of gravitation between two point masses is given by the formula where M1 and M2 are the masses of the attractive bodies, r the separation between them and G a new constant of physics called Newton’s constant of universal gravitation. It has the value G = 6.67 x Nm2kg-2. The direction of the force is along the line joining the two masses.

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**Newton’s law of gravitation**

Newton proved that for bodies which are spherical and of uniform density one can assume that the entire mass of the body is concentrated at the centre – as if the body is a point mass.

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**Newton’s law of gravitation**

The weight of a mass (mg) is actually the force of gravitational attraction between the earth of mass Me and the mass of the body in question. The mass of the earth is assumed to be concentrated at its centre and thus the distance that goes in Newton’s formula is the radius of the earth, Re.

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Questions Find the force between the sun and the earth (the average distance is 1.5 x 1011m, the mass of the earth is 5.98 x 1024 kg and that of the sun 1.99 x 1030 kg). If the distance between two bodies is doubled, what happens to the gravitational force between them? Find the acceleration due to gravity on a planet 10 times as massive as the earth and with radius 20 times as large. Find the acceleration due to gravity at a height of 300 km from the surface of the earth.

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**Gravitational field strength**

Physicists wondered how a mass ‘knows’ about the presence of another mass nearby that will atract it. By the 19th century, the idea of a ‘field’ was developed. A mass M is said to create a gravitational field in the space around it. The gravitational field strength at a certain point is the force per unit mass experienced by a small point mass m at that point. The units of gravitational field strength are Nkg-1.

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**Gravitational field strength**

The gravitational field strength is a vector quantity whose direction is given by the direction of the force a point mass would experience if placed at the point of interest. The gravitational field strength around a single point M is radial, which means that it is the same for all points equidistant from the mass and is directed towards the mass.

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Questions Two stars have the same density but star A has double the radius of star B. Determine the ratio of the gravitational field strength at the surface of each star. Show that the gravitational field strength at the surface of a planet of density ρ has a magnitud given by g = 4GπρR/3.

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