Motion, Forces and EnergyGravitation: That Sinking Feeling

Newton’s Law of Gravitation (1686):

Every particle of matter in the Universe attracts every other particle with a force that is directly proportional to the product of the masses and inverselyproportional to the square of the distance between them.

221

r

mmGFg

G is the gravitational constant = 6.67x10-11 Nm2kg-2.r

Fg

Fg

Two particles separated by a distance r exert attractivegravitational forces of equal magnitude on each other.

laser

m2

m2

m1

m1

Fg

Fg

scale

mirror

GHenry Cavendish first measured thevalue of G using a torsion balance (1798).

Modern versions use lasers andmirrors – the reflected laser beamis displaced from its original positionas two large spheres m2 are broughtclose to the smaller spheres, m1.

For m1=m2=1 kg, and a separation of 1 cm, the forcebetween m1 and m2 is 6.67x10-7 N. The accelerationof each mass will be 6.67x10-7 ms-2.

Weight

2

2

2

)(hR

GMhg

R

GMg

R

mGMmg

E

E

E

E

E

E

We can develop a fundamental definition of g. Because the force acting on a mass near the Earth’ssurface is mg, we can say:

For an object of mass m locateda distance h above the Earth’ssurface, we can write:

Free-fall Accelerations g(h)

Altitude, h (km) g (ms-2)

1000 7.33 2000 5.68 5000 3.0810000 1.4920000 0.5750000 0.13

Acceleration due to gravity on other planets, gp.

2

2

p

pp

p

pp

R

GMg

R

mGMmg

Planet Mass (kg) Mean radius (m) gEq (ms-2)

Mercury 3.24x1023 2.34x106 3.95Venus 4.86x1024 6.10x10 8.72Earth 5.97x1024 6.37x10 9.78Mars 6.40x1023 3.32x106 3.84Jupiter 1.89x1027 69.8x106 23.16Saturn 5.67x1026 58.2x106 8.77Uranus 8.67x1025 23.8x106 9.46Neptune 1.05x1026 22.4x106 13.66Pluto 6.60x1023 2.90x106 5.23Moon 7.34x1022 1.74x106 1.62

The Concept of Gravitational Field

rr

GMg

magnituder

GM

m

Fg g

ˆ

)(

2

2

A mass creates a gravitational field around it. We can use a test mass asa detector of gravitational field by taking it to various points and measuringthe gravitational force that acts on it and defining the field g as:

Here is the unit vector along the line joining M and m.r̂

We can express the vectorial natureof the field as:

r̂M

g m

Gravitational force between a particle and a bar

x

Lh

dx

x

y

The (red) segment of the bar of length dx has mass dM. The mass perunit length is dM/dx or M/L.

Analysis

Lhh

GmM

xL

GmMF

dxxL

MGmF

x

dxGmF

Lhhg

Lh

h

g

Lh

h

g

1

12

2

As L tends to zero, the force varies as 1/h2 as expected for two point masses.If h>>L, the force also varies as 1/h2; in other words, when two objects are separated by huge distances, they behave as point masses even though theymay both be extended objects.