Gravitational Force and Field We already know that; 1.Masses attract each other

Gravitational Force and Field We will know that; 2. Mass/energy is conserved (E = mc 2 )

Gravitational Force and Field The force between masses was formulated (discovered?) by Isaac Newton in 1687

Gravitational Force and Field Newton found that the force between two masses is proportional to the product of the two masses F α m 1 x m 2 and inversely proportional to the square of the distance (r) between the masses F α 1/r 2

Newton’s law of universal gravitation It follows that F α m 1 m 2 r 2 or F = Gm 1 m 2 r 2

Newton’s law of universal gravitation F = Gm 1 m 2 r 2 The constant G is known as “Big G” and is equal to x Nm 2 kg -2

Newton’s law of universal gravitation F = Gm 1 m 2 r 2 For large objects like the earth, r is the distance to the centre of mass

Calculations using Newton’s law What is the force of attraction between 2 people? 2 m 63kg ?70kg ?

Calculations using Newton’s law F = Gm 1 m 2 = x x 63 x 70 = 7.3 x N r m 63kg ?70kg ?

Force of gravity due to earth on a person? F = Gm 1 m 2 = x x 63 x 6 x = 615 N (= mg) r 2 (6400 x 10 3 ) 2 63kg ? R = 6400 km, m = 6 x kg Jan’s weight

Force of gravity due to earth on person 2? F = Gm 1 m 2 = x x 63 x 6 x = 615 N (= mg) r 2 (6400 x 10 3 ) 2 In other words, for any planet; g = Gm p r p 2

Gravitational field An area or region where a mass feels a gravitational force is called a gravitational field. The gravitational field strength at any point in space is defined as the force per unit mass (on a small test mass) at that point. g = F/m (in N.kg -1 )

Gravitational field around a point mass If we have two masses m 1 and m 2 distance r apart F = Gm 1 m 2 /r 2 Looking at the force on m 1 due to m 2, F = gm 1 F = Gm 1 m 2 /r 2 = gm 1 g (field due to m 2 ) = Gm 2 /r 2 m1m1 m2m2

Gravitational field around a point mass If we have two masses m 1 and m 2 distance r apart F = Gm 1 m 2 /r 2 Looking at the force on m 1 due to m 2, F = gm 1 F = Gm 1 m 2 /r 2 = gm 1 g (field due to m 2 ) = Gm 2 /r 2 m1m1 m2m2

Gravitational field Gravitational field is a vector, and any calculations regarding fields (especially involving adding the fields from more than one mass) must use vector addition. m1m1 m2m2 Field here due to both masses?

Gravitational field Gravitational field is a vector, and any calculations regarding fields (especially involving adding the fields from more than one mass) must use vector addition. m1m1 m2m2 Field due to m 1 Field here due to both masses?

Gravitational field Gravitational field is a vector, and any calculations regarding fields (especially involving adding the fields from more than one mass) must use vector addition. m1m1 m2m2 Field due to m 1 Field due to m 2

Gravitational field Gravitational field is a vector, and any calculations regarding fields (especially involving adding the fields from more than one mass) must use vector addition. m1m1 m2m2 Field due to m 1 Field due to m 2 Resultant Field

Gravitational field patterns A gravitational field can be represented by lines and arrows on a diagram, in a similar ways to magnetic field lines.

Gravitational field patterns A gravitational field can be represented by lines and arrows on a diagram, in a similar ways to magnetic field lines. This is an example of a radial field The closer the lines are together, the stronger the force felt. Note, gravity is ALWAYS attractive

Field around a uniform spherical mass

Field close to the earth’s surface Uniform

Why is Ep Negative? Gravitational potential energy is the work done against gravity to bring a mass to a given point in space. The force of gravity approaches zero at very large distances, therefore it makes sense to choose the zero of gravitational potential energy as zero at infinity. The gravitational potential energy near a planet is then negative as gravity does positive work as the mass is brought closer.

This negative potential means that the mass is in a bound state – it is trapped unless enough energy can be provided to allow it to escape. This may also be shown mathematically by using the formula Work = F.R Integrating the gravitational force law w.r.t R shows that the gravitational potential energy is negative.