#### Gravitational Motion, Physics tutorial

Law of Universal Gravitation:

Sir Isaac Newton deduced law of universal gravitation in 1686 from speculations concerning fall of the apple toward earth. His suggestion, principia (mathematical principles of natural knowledge) was that gravitational attraction of sun for planets is a source of centripetal force that maintains orbital motion of the planets round the sun. Newton also asserts that this was like attraction of earth for apple. According to Newton also, there is gravitational force between all objects in universe.

This Newton's law of universal gravitation, may be stated thus:

Every particle of matter in the universe attracts other particles with a force which is directly proportional to the product of their masses and inversely proportional to the square of their distances apart.

Gravitational attraction, F between two bodies of masses M1 and M2 that are a distance r apart is given by

F∝ m1m2/r2

that is

F= Gm1m2/r2

Where G is the constant known as universal gravitational constant. It is supposed to have same value everywhere for all matter.

Newton thought that force was directly proportional to mass of each particle as force in the falling body is proportional to mass (F = ma = mg = m x constant, thus F ∝ m), i.e., mass of the attracted body. From point of view of third law, Newton also argued that the falling body exerts the equal and opposite force which is proportional to mass of earth. Then it was concluded that gravitational force between the bodies should also be proportional to mass of the attracting body.

Newton law of gravitation refers to force between two particles. It can also be illustrated that force of attraction exerted on or by the homogeneous sphere is the same as if mass of sphere were concentrated at its centre. Gravitational force exerted on the body by the homogeneous sphere is the same as if whole mass of sphere were concentrated in the point at its centre. Therefore if earth were a homogeneous sphere of mass ME, the force exerted by it on the small body of mass m1 at the distance r from its centre, would be

Fg = GmME/r2

The force of same magnitude would be exerted on earth by the body.

Magnitude of gravitational constant G can be found experimentally by estimating force of gravitational attraction between two bodies of known masses at a known separation. For bodies of moderate sizes, force is very small, but it can be estimated with the instrument invented by Rev. John Michell and first used for this by Sir Henry.

Two small spheres of mass m are mounted at ends of the horizontal portion of the T, and the small mirror M, fastened to vertical portion, reflects the beam of light onto the scale. To utilize balance, two large spheres of mass m are brought up to positions. Forces of gravitational attraction between large and small spheres result in the couple that twists system through the small angle, thus moving reflected light bean along the scale. By using very fine fibre, the deflection of mirror may be made adequately large so that gravitational force can be estimated fairly correctly. Gravitational constant, estimated in this way, is found to be

G = 6.67x10-11Nm2kg-2

Kepler's Laws of Planetary Motion:

These laws define:

• During equal time intervals, the radius vector from the sun to the planet sweeps out equivalent areas
• If T is the time that it takes for the planet to make one complete revolution round sun, and if R is half the major axis of the ellipse ( R reduces to radius of the planet's orbit if that orbit is circular), then

T2/R3 = C

Where C is the constant whose value is same for all planets.

If the time taken by the planet to travel from P1 to Q1 is equivalent to time taken to travel from P2 to Q2, areas covered are equal i.e. shaded region.

Mass and Weight:

The weight of the body can be stated more generally as resultant gravitational force applied on body by all other bodies in universe. Earth's attractive force on the object on surface is much greater than all other gravitational forces on object so we neglect all the other gravitational forces. Weight of object for practical purposes then results exclusively from earth's gravitational attraction on it. Likewise if object is on the surface of moon or of another planet, its weight will result exclusively from gravitational attraction of moon or planet on it. Therefore, assuming earth to be homogeneous sphere of radius R and mass of ME, weight w of the small object of mass M in its surface would be

W = Fg = GmmE/R2

The weight of the given body or object differs by a few tenths of percent from location to location on earth's surface. It is partially as there could be local deposits of ore, oil or other substances, with varying densities or partially as earth is not ideal sphere but flattened at its poles. It is identified that distance from the poles to centre of the earth is shorter than that from equator to the earth's centre, so, acceleration because of gravity differs at these locations. Also weight of the given body reduces inversely with square of the distance from the earth's centre. For instance, at the radial distance of two earth radii, weight of the given object has reduced to one quarter of value at the earth's surface.

Rotation of earth about its axis is also part of what causes evident weight of the body to vary somewhat in magnitude and direction from earth's gravitational force of attraction. For practical reasons ignore this minor difference and suppose that earth is the inertial reference system. Then, when the body is permitted to fall freely, force accelerating it is its weight, w and acceleration generated by this force is that because of gravity, g. General relation

F = ma

Therefore becomes, for special case of freely falling body,

w = mg

Now,

w = mg = GmmE/R2

It follows that,

g = GmE/R2

This illustrates that acceleration because of gravity is same for all bodies or objects (as m cancelled out ). It is also extremely nearly constant (as G and are constants and R differs only somewhat from point to point on earth) Weight of the body is force and its unit is Newton, N in mks system. So Equation provides relation between mass and weight of the body in any consistent set of units. For instance, weight of object of mass 1kg at the point where g=9.80ms-2 is

w = mg = 1kg x9.80ms-2

= 9.80N

At another place where g = 9.78ms-2

Its weight is w = 9.78N

Thus, we see that weight varies from one point to another. Mass does not.

Mass of the Earth:

Applying Newton's law of gravitation we have that

w = mg = GmmE/R2

This provides the mass of earth as

mE = R2g/G

Where R is earth's radius. As all quantities on R.H.S of the Equation are known, we can compute mass of earth.

Hence,

For R = 6370km, G = 6.37x106 m and g = 9.80 ms-2

ME = 5.98x1024kg

The volume VE of earth is

VE=4/3ΠR3 = 1.09 x 1021m3

Thus the average density of the earth is

ρE = ME/VE = 5500kgm-3

or

5.5gcm-3

Density of water is 1g cm-3 = 1.000kgm-3).  Density of most rock near earth's surface, like granites and gneisses, is about 3g cm-3 = 3000kg m-3. The interior of earth have higher density than surface.

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