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Gravity • Newton • http://csep10.phys.utk.edu/astr161/lect/history/newtongrav.html • http://www.britannica.com/bcom/eb/article/9/0,5716,109169+2+106265,00.html • http://www.nelsonitp.com/physics/guide/pages/gravity/g1.html

Gravity • The moon is performing circular motion round the earth. • The centripetal force comes from the gravity. v Fc moon earth

Gravity • Newton found that the gravity on the moon is the same force making an apple fall. W Ground

F F m1 m2 r Newton’s Law of Gravitation • Objects attract each other with gravitational force. • In the diagram, m1 and m2 are the masses of the objects and r is the distance between them.

F F m1 m2 r Newton’s Law of Gravitation • Every particle of matter attracts every other particle with a force whose magnitude is G is a universal constant G = 6.67 10-11 m3kg-1s-2 Note that the law applies to particles only.

Shell Theorem • Extends the formula to spherical objects like a ball, the earth, the sun and all planets.

m2 m1 F F r Theorem 1a. Outside a uniform spherical shell. • The shell attracts the external particle as if all the shell’s mass were concentrated at its centre. O

Theorem 1b. Outside a uniform sphere. • The sphere attracts the external particle as if all the sphere’s mass were concentrated at its centre. m2 m1 F F O r

m1 F F O m2 r earth Example 2 Outside a uniform sphere. • The earth is almost a uniform sphere.

m1 m2 The two forces on m2 cancel. Theorem 2a. Inside a uniform spherical shell. • The net gravitational force is zero on an object inside a uniform shell.

m2 m1 F r Theorem 2b. Inside a uniform sphere. where m1 is the mass of the core with r the distance from the centre to the mass m2

Gravitational Field • A gravitational field is a region in which any mass will experience a gravitational force. • A uniform gravitational field is a field in which the gravitational force in independent of the position. • http://saturn.vcu.edu/~rgowdy/mod/g33/s.htm

F test mass m Field strength, g • The gravitational field strength, g, is the gravitational force per unit mass on a test mass. F is the gravitational force m is the mass of the test mass g is a vector, in the same direction of F. SI unit of g is Nkg-1.

F test mass m Field strength, g • The gravitational field strength, g, is the gravitational force per unit mass on a test mass. F is the gravitational force m is the mass of the test mass SI unit of g is Nkg-1.

field strength at X X M r Field strength, g, outside an isolated sphere of mass M • The gravitational field strength, g, outside an isolated sphere of mass M is O Prove it by placing a test mass m at a point X with distance r from the centre of the isolated sphere M.

Example 4 • The field strength of the earth at the position of the moon.

Field strength, g • Unit of g is Nkg-1. • g is also a measure of the acceleration of the test mass. • g is also the acceleration due to gravity, unit is ms-2.

Field strength, g. Unit Nkg-1. A measure of the strength of the gravitational field. Acceleration due to gravity, g. Unit ms-2. A description of the motion of a test mass in free fall. Field strength, g

Field lines • We can represent the field strength by drawing field lines. • The field lines for a planet are radially inward. planet Radial field

Field lines • We can represent the field strength by drawing field lines. • The field lines for a uniform field are parallel. Uniform field earth’s surface

Field lines • The density of the field lines indicates the relative field strength. g1= 10 Nkg-1 g2= 5 Nkg-1

direction of the force test mass Field lines • The arrow and the tangent to the field lines indicates the direction of the force acting on the test mass.

Example 5 • The gravity on the earth’s surface, go.

Apparent Weight • Use a spring-balance to measure the weight of a body. • Depending on the case, the measured weight R (the apparent weight) is not equal to the gravitational force mgo. R mgo

Apparent Weight • The reading on the spring-balance is affected by the following factors: • The density of the earth crust is not uniform. • The earth is not a perfect sphere. • The earth is rotating.

Apparent Weight • The density of the earth crust is not uniform. • Places have different density underneath. Thus the gravitational force is not uniform.

Apparent Weight 2. The earth is not a perfect sphere. Points at the poles are closer to the centre than points on the equators. rpole < requator gpole > gequator N-pole Equator S-pole

X Y Apparent Weight 3. The earth is rotating. Except at the pole, all points on earth are performing circular motion with the same angular velocity .However the radii of the circles may be different.

Apparent Weight m 3. The earth is rotating. Consider a mass m is at point X with latitude . The radius of the circle is r = Re.cos . X Y r Re O

Apparent Weight Fc m 3. The earth is rotating. The net force on the mass m must be equal to the centripetal force. X Y r Re O Note that Fnet points to Y.

Apparent Weight R R Fc 3. The earth is rotating. The net force on the mass m must be equal to the centripetal force. So the apparent weight (normal reaction) R does not cancel the gravitational force mgo. X Y r m mgo O

Apparent Weight R R Fc 3. The earth is rotating. The apparent weight R is not equal to the gravitational force mgo in magnitude. X Y r m mgo O

Apparent weight R on the equator mgo R The apparent field strength on the equator is

Apparent weight R at the poles R mgo The apparent field strength at the poles is

Apparent weight at latitude R Fc X Y r m mgo O Note that the apparent weight R is not exactly along the line through the centre of the earth.

Me r g m O h Re Variation of g with height and depth • Outside the earth at height h. h = height of the mass m from the earth’s surface

Me m r g O h Re Variation of g with height and depth • Outside the earth at height h. where go is the field strength on the earth’s surface.

Me m r g O h Re Variation of g with height and depth • Outside the earth at height h. where go is the field strength on the earth’s surface.

Me m r g O h Re Variation of g with height and depth • Outside the earth at height h close to the earth’s surface. h<<Re. where go is the field strength on the earth’s surface.

Me r O d Re Variation of g with height and depth • Below the earth’s surface. Only the core with colour gives the gravitational force. g r = Re-d

Me r O d Re Variation of g with height and depth • Below the earth’s surface. Find the mass Mr of g r = Re-d

Me r O d Re Variation of g with height and depth • Below the earth’s surface. g r = Re-d

Me r O d Re Variation of g with height and depth • Below the earth’s surface. g g r r = Re-d