## Earth weight and mass

The Earth is an oblate spheroid, with slightly larger diameter in the equatorial plane than in a meridional (pole-to-pole) plane. The distance from the center to the poles

 Quantity Unit Abbreviation mass kilogram kg length meter m time second s force newton N pressure pascal Pa = Nm-2 = 0.01 : energy joule J temperature degree Celsius ◦C temperature degree Kelvin K speed ms 1 density kgm-3 specific heat J kg-1 K-1

Table 1.3 Greek prefixes applied to SI units

Prefix Numerical meaning Example Abbreviation

Table 1.3 Greek prefixes applied to SI units

Prefix Numerical meaning Example Abbreviation

 nano 10-9 nanometer nm micro 10-6 micrometer ^m milli 10-3 millimeter mm centi 10-2 centimeter cm hecto 102 hectopascal hPa kilo 103 kilogram kg mega 106 megawatt MW giga 109 gigawatt GW tera 1012 terawatt tW

Table 1.4 Selected conversions to SI units

Quantity

Conversion energy pressure distance temperature

4.186 J = 1 cal 1 kWh = 3.6x 106 J 1 atm = 760 mm Hg

Table 1.5 Some relationships between SI units

Quantity Equivalent

1N lkgms-2

1J lkgm2 s-2

### 1 Pa 1Nm-2

is 6356.91 km and the radius in the equatorial plane is 6378.39 km. About two thirds of the Earth's atmosphere lies below 10 km above the surface, hence the atmosphere and the oceans (depth averaging 4-5 km) only form a very thin skin of about 1/60 the radius of the sphere.

The weight of a mass is the force applied to that mass by the force of gravity. It may be expressed as the mass in kilograms times the acceleration due to gravity, g = 9.81ms-2:

Weight is expressed in newtons, abbreviated N; N = kgms 2. The acceleration due to gravity varies slightly with altitude above sea level gz = gQ(1 - 3.14 x 10-7z) z in meters. (1.3)

There is also a slight variation (< 0.3%) with latitude due to the ellipsoidal shape of the Earth (due to both centrifugal force and the equatorial bulge). In most meteorological applications these variations are negligible. However, in calculations of satellite orbits such variations are extremely important.

Example 1.1 The density of water in old fashioned units (cgs) is

1 gram

To express this in SI units, we can multiply by

103 gram

We obtain:

103 cm3 liter

This gives us an intuitive measure of the kilogram. Now we can multiply by

The final result is

Physics refresher: vertical motion of a particle The acceleration due to gravity is g = 9.81m s-2. A particle falling from a height z0 with no initial velocity, has a velocity -gt after time t. After the same time interval it will have fallen 1 gt2 meters. These are both obtained by simple integration:

vz (t) = I -g dt =-g dt =-gt, since g = constant (1.4)

Jo Jo and z(t) - z(0) =[ vz (t) dt =[ -gt dt = -gt2. (1.5)

### Jo Jo 2

More vertical motion mechanics The minimum work necessary to lift a particle a vertical distance z against the force of gravity is force x distance = Mgz (Mg is the weight or vertical force necessary to lift the mass without accelerating it). This work done in lifting the particle is equal to the change in its potential energy Mgz. If the particle is released, work will be done by the gravitational force applied to the particle. The kinetic energy of the particle during its fall is 1 Mv2. The conservation of mechanical energy says the sum of these two forms of energy is conserved: E = PE + KE = constant, or more explicitly

where the subscript 0 denotes the initial time and the subscript t denotes evaluation at a later time.

The conservation law is derived by first writing Newton's Second Law:

Now multiply through by v dt and integrate with respect to t. The left-hand side becomes f t dv 1 2 1 2

/0 di 2 1 2 On the other side of Newton's equation we have f -vMg dt = f' -Mg dz = -Mg(zt - zo). (1.9)

JO Jz.o

Equating these expressions gives our answer (1.6).

0 0