GRAVITATION

Newton’s Law of Gravitation o Every point mass attracts another point mass with a force that is proportional to

the product of the 2 masses and inversely proportional to the square of the distance between them.

221

221

rmmGF

rmmF gg =⇒∝

G: 6.67 x 10-11 Nm2 kg-2 Unit of Fg : N

Fg : is a vector, formula give the magnitude of the force only.

: always attractive, acts along the joining the 2 centres of mass

: to find resultant Fg due to multiple masses, use vector addition.

Inside Earth (r < RE): rFrGmF gg ∝⇒= πρ34

Outside Earth (r>RE): 221r

Fr

mMGF gE

g ∝⇒=

Gravitation Field o A region of space surrounding a body possessing mass, in which any other body

that has mass will experience a force of attraction. o Gravitational Field Strength g (at a point in a gravitational field) is defined as the

gravitational force per unit mass acting on a mass placed at that point.

2rMGg

mF

g g =⇒=

Factors affecting measurements of g o Earth’s density not being uniform o Earth is not a sphere but bulging at the equator o Rotation of Earth (apparent g = g’)

At the equator:

g : vector : SI unit: N kg-1 : to find resultant g at a point, use vector addition. mgFg =

RE

Fg / N

r / m

mg

RE

g / N kg-1

r / m

9.81 Inside Earth (r < RE): rgrGg ∝⇒= πρ34

Outside Earth (r>RE): 22

1r

gr

MGg E ∝⇒=

N (= mg’)

Gravitational Potential Energy o Work done by an external force in bringing the mass from infinity to that point in

a gravitational field.

rmm

GU 21−=

o U is always negative. Since the gravitational force is attractive, work must be

done by external force to bring it to infinity. As infinity is taken to be zero (reference), any other point in the g-field will have less GPE, hence negative. OR Since the gravitational force is attractive, positive work is done by the gravitational force to bring the mass from infinity to that point. Hence negative work is done by the external force, GPE is negative.

o

drdU

gF −=

U : scalar : SI unit : J : to find resultant U, use algebraic addition.

dr

dUFg −=

→ negative of gradient of U-r curve gives the gravitational force Fg.

Gravitational Potential o Work done per unit mass by an external force in bringing that body from infinity to

that point in a gravitational field.

rMG

mU

−==φ

o φ is always negative, as reference point, where φ = 0, is taken to be at infinity.

drdg φ

−=

φ : scalar : SI unit : J kg-1 : to find resultant φ, use algebraic addition.

mU

=φ

Motion and G-field o Escape Velocity

rmMGmv E

e ≥2

21

→ rGMv E

e2

≥

o Orbiting Satellite

maFnet = → rvm

rmMG E

2

2 = → rGMv E=

or 32322

22

42 RTRGM

TT

mrmrr

mMGE

E ∝→=→

==

ππω

o Total Mechanical Energy of Satellite

rmGM

rmGM

rmGMGPEKEE EEE

T 2)(

21

−=−+=+=

E / J

r / m RE

KE

GPE ET

ET = KE + GPE

Equipotential lines

Gravitional. field lines

mg

2

2'

2

' ω

ω

ω

RggmRmgmg

mRNmgmaF

E

net

−=

−=

=−

=

Motion and G-field (cont’d) o Geostationary Orbit: above a fixed point on Earth.

Conditions: Period = 24 hrs In plane of equator Moving from wst to east.