Hi, I have an upcoming project where I need to use the following topics to descr

Question

Hi, I have an upcoming project where I need to use the following topics to describe a 3-D space. I have chosen to describe a mountain range with valleys and plains, as well as trees and roads. I need to describe movement in this space and how the space would be depicted using calculus. Here are the topics:

Differential Equations

Parametric Equations

Three-Dimensional Space, Vectors

Partial Derivatives

Second Order Differential Equations

Here are a few specific things I need to include.

Design roads in your imaginary space and describe your designs using the concepts from this week

Explanation / Answer

The unit for the horizontal deflections used in this paper is I millisecond = 0.001

seconds of arc. A deviation of I millisecond corresponds to a horizontal

component, acting in the azimuth of the deviation, of 4.85 x I O - ~ the normal

of

gravitational force.

The vertical component of the variation of gravity was determined by Frost

Gravimeter No. 54. The instrument and its mode of operation have already

been described in a previous paper (2). The unit for the vertical variations

milligals. This unit, I pgal, corresponds

used in this paper is I microgal = 0-001

of

to 1-02x I O - ~ the normal vertical gravitational acceleration.

All three instruments were located in the I.C.I. Salt Mine at Winsford

(Cheshire), Latitude : 53" 12' fi.; Longitude : 2" 30' W. The depth of the

observation point was 490 feet = 143 m below ground level, corresponding to

381 feet= 116 m below sea level. The nearest actual sea level at Liverpool Bay

was already below the horizontal plane laid through the observation point due

to the curvature of the Earth's surface. The horizontal distance between the

gravimeter and the horizontal pendulums was about IOO yards.

3. Influences acting on the instruments.-There exist several types of influence

which have to be taken into account when evaluating the results. They are:-

( a ) Tidal influences

These are of a three-fold nature:

(i) The gravitational influences of Sun and Moon.

(ii) The influence of the elastic deformation of the whole Earth due to

the gravitational influences under (i).

(iii) The influence of the maritime tides.

This last influence is itself complex as it consists of:

(a) Gravitational attraction of the tidal waters.

(/3) Elastic deformation of the Earth's crust due to tidal loading.

(7) Change of shape of the gravitational field of the Earth due to the

deformation by tidal load.

(b) Meteorological influences

These are also complex:

(i) Annual cycles.

(ii) Shorter periodic variations (diurnal and semidiurnal).

(iii) Irregular variations.

The variations due to the meteorological influences seem to be mainly

produced by the Merent loading of the Earth's crust with changing barometric

pressure. There may also be included a certain amount of variation due to

wind, precipitation, melting and displacement of oceanic water on a large scale.

Owing to the great depth of the observation point the influence of local

meteorological changes is entirely eliminated.

However, there exists an influence of barometric pressure on the gravimeter.

It has been determined, as already reported in paper (2), and the readings have

therefore to be corrected for this influence.

(c) Influences of temperature

The temperature in the mine was constant within 01 deg. C over the period

'

of observations. As will be shown in a paper to be published later, there exists

no influence of the temperature variation in the mine on the indications of the

542

horizontal pendulums. The gravimeter was controlled by two thermostats, so

that there exists no influence of external changes of temperature on the gravimeter.

But the observations show that the working of the thermostats themselves gives

residual effects, and these will be discussed in Section 7.

( d ) Influences of elastic residuals in the instruments

The horizontal pendulums have been in operation at this station since

1950 March. The elastic hysteresis effects at the period of the observations

described here (1951 April) are entirely negligible. The gravimeter has been

in operation over several years and its hysteresis effects are also very small during

the period of observation.

4. Results : Horizontal components.-The readings of the records were taken

at hourly intervals, and the mean of the readings of three observers was taken.

The mean deviation of a reading is of the order of 0.05 millimetres corresponding

to about I millisecond in Sp and 0.5 milliseconds in Wp. T o eliminate the

accidental deviations, twice the mean of two-hourly Observations was taken.

The difference between the original and the smoother curves in most cases does

not exceed the uncertainty due to the limitations in the measurements of the

records. However, it has to be borne in mind that this sort of averaging

diminishes systematically the ordinates of semidiurnal periodicities by 7-2 per cent

and those of diurnal ones by 1-7 per cent. This has to be taken into account when

evaluating the measurements.

The immediate observation of the records shows a progressive uniform

drift. This is not due to elastic hysteresis but is part of an annual tilting

influence (3). For the period 1951 April 17 to 24, this drift can be taken as

linear with sufficient accuracy, as the annual curve shows. The amount of the

correction due to this drift was: South pendulum (south taken as positive):

- 1.7662 milliseconds per hour ; West pendulum (west taken as positive) :

+0.6298 milliseconds per hour. At the period of observation therefore there

occurred a steady tilt approximately towards the south-east by an amount of

1-87 milliseconds per hour in the azimuth S. 39O-6E.

Having removed the linear seasonal drift and the chance oscillations by the

smoothing process, mentioned above, we obtain Fig. I. We shall regard this

curve as the fundamental curve for further investigations.

Vertical component.-The results of the gravimetric observations have been

already partly treated in paper (2). The aim was then to determine the

gravimetric factor for Winsford, which was shown to be G=1.2ok0.04. In

this paper the load tilt and the meteorological influences will also be discussed.

The treatment of the data is therefore somewhat Merent from that used in the

former paper, where an adapted curve of the experimental and theoretical values

had been given, the values having been reduced to a common zero line. We

start this paper with the original readings. Fig. 2 shows the gravity curve after

pressure correction, and we shall regard this curve as the fundamental curve

for our further investigations.

The mean accuracy of one reading is f 4pgals. As I pgal= 1-02 x ~ o - ~ g ,

the mean accuracy in this scale is the same as that of the horizontal pendulum

readings as given above.

The accuracy of the gravity curve was increased by using half-hourly

intervals, so that the number of observations is double the number used for the

tilt curves.

R. Tomaschek

Variations of the total vector of gravity at WinsfM.d (Cheshire)

The general drift is very small and certainly less than the amplitudes of the

residual influences; it is included in Fig. 2. As this curve shows, there is a

slight tendency to rise with increasing time, It can be accounted for by assuming

_1

0bserv.d values o f t i l t in the azimuth WPand Np

Constant seaso~l r i f t removed

d

-

FIG. 1.

Observed qravity values. pressure corrected

APRIL 1951

-

FIG.2.

a linear drift of + 4.8 pgals per day. Its origin does not lie in elastic hysteresis

of the instrument, which seems to cause a slight decrease of the zero point wt

ih

time, as observations with the unclamped instrument over more than half a year

544

showed. Continued daily readings of the meter at the same location (Kirklington

Hall, Nottinghamshire), with clamping of the instrument between the

readings, showed variations of the overall drift from zero over a period of about

three months to a gradient of +5pgal/day for the following period of three

months. Therefore, long-term variations of gravity of this order of magnitude

and presumably of geophysical origin seem to occur. But as small systematic

changes of the temperature of the thermostat could also cause a drift, only

parallel measurements of two very reliable gravimeters could give a positive

indication of real gravity changes over such long periods.

5. Elimination of the infuences (a) of the gravitational tide of the Sun and

the Moon, and (b) of the elastic yielding of the Earth

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