Dependency relations

Physical quantities have dependency relations. For school level, we need to understand dependency relations between the quantities under the early classical mechanics point of view. We will study about some physical quantities.

Example (2): Gravitational field strength (or just gravitational field) at a point at a time in space:

Gravitational field at a point P in space at the time t is the gravitational force that would act on (a particle with unit mass) if it is located at P at that time t. If the gravitational field is static (it does not change with time), then we may drop the time t from discussion.  When there is no particle at P,  the gravitational field at P is still defined the same way.  

If (a particle A with mass m > 0) located at P experiences gravitational force F, then the field at P is taken to be F/m. Note that the gravitational field felt by A when it is located at P is the combined gravitational field of all bodies in universe except A. Suppose, a moment later, the particle A moves to stay at another point Q.  There is no particle at P. Now, the gravitational field at P is the combined field of all bodies including the particle A at Q.

We can also talk about gravitational field of (a specific gravitating body B) at a specific point P at  a specific time t. According to Newton's law of gravitational force, gravitational field (strength) of (a spherically symmetric body B) of mass M at a point P (which is outside of the body B)  has the magnitude given by:

In this case, g-value depends on the mass of the gravitating body and the distance r.  If B is earth, then r becomes distance between the point P and mass-centre of earth. If we choose the point P to be located near the sea level surface, then r = R, the mean radius of earth.  Assuming earth is spherically symmetric, g-value of earth measured at a point P depends on earth mass M and distance between P and earth's mass-centre.

If P moves up a few kilometers above the earth's surface, the distance r changes by that few km.  Since earth's radius is about 6370 km, that few km change will not significantly change the g-value of earth. Therefore, we usually assume earth's g-value at P be a fixed number even when the location of P changes by a few km above earth's surface.

Suppose we throw a stone straight up from ground with initial speed of u. Neglecting the air resistance, we have, for u > 0 and g > 0,

It turn out that if we do not have the g-value of earth at its surface, we can perform the experiment of throwing a stone straight upward and measure the maximum height H and time taken T to reach there. Then , we can calculate the g-value by either  g = u2/2H or g = u/T. It looks like g-value depend on the pair u and H or the pair u and T.  It is not true. The g-value at point P still depend on earth mass M and the location of P. These two formulas give g-value indirectly. It is just that u2/2H and u/T happen to be the same as g-value of earth at that place under early classical mechanics.

Maximum height H depends on u and g. 

And, time T needed to reach the maximum height depends on u and g as well.

Example 1: applications of a formula Example 2: dependency relations