H2 Physics Gravitation Summary

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. mm m m Fg : is a vector, formula give the magnitude of Fg ∝ 1 2 2 ⇒ Fg = G 1 2 2 the force only. r r : always attractive, acts along the joining -11 2 -2 G: 6.67 x 10 Nm kg the 2 centres of mass Unit of Fg : N : to find resultant Fg due to multiple masses, use vector addition. Fg / N 4 mg Inside Earth (r < RE): F = Gmπρr ⇒ F ∝ r g

Outside Earth (r>RE): r/m

RE

g=

Fg = mg

M r

2

g : vector -1 : SI unit: N kg : to find resultant g at a point, use vector addition. Inside Earth (r < RE): g = 4 Gπρr ⇒ g ∝ r

9.81

3

Outside Earth (r>RE):

GRAVITATION

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. dU → negative of gradient of U-r Fg = − curve gives the gravitational dr force F .

mg − N = mRE ω 2

mg

mg ' = mg − mRω 2 N (= mg’)

U φ= m

g ' = g − Rω 2

g =−

o

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

2GM E r

o

ET = KE + GPE

GM E M Em v2 = ma → G 2 = m → v= r r r

M m 4π 3  2π  G E2 = mrω 2 = mr   → T 2 = R → T 2 ∝ R3 r GM E T  2

or

Gravitional. field lines E/J

Orbiting Satellite

Fnet

Total Mechanical Energy of Satellite

ET = KE + GPE =

dφ dr

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. φ : scalar -1 U M : SI unit : J kg φ = = −G : to find resultant φ, use algebraic addition. m r

Motion and G-field o Escape Velocity

M m 1 2 mve ≥ G E → ve ≥ r 2

1 GM E m GM E m GM E m )=− + (− 2 r r 2r

KE RE

r/m ET GPE

ME 1 ⇒g∝ 2 2 r r

Fnet = ma

g

o

g =G

r/m RE 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:

Gravitational Potential Energy o Work done by an external force in bringing the mass from infinity to that point in a gravitational field. U : scalar mm : SI unit : J U = −G 1 2 : to find resultant U, use algebraic addition. r

o

m-1

⇒ g =G

g / N kg

M Em 1 ⇒ Fg ∝ 2 2 r r

dU F =− g dr

o

Fg

g

3

Fg = 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.

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.