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Chapter 5: Problem 20

The surface gravity at a measuring site is \(9.803243 \mathrm{~m} \mathrm{~s}^{-2}\). The site has a latitude \(43^{\circ} 32^{\prime} 16^{\prime \prime} \mathrm{N}\)

Short Answer

Expert verified
Gravity: 9.803243 m/s²; Latitude: 43.5378°.

Step by step solution

01

Convert Latitude to Decimal Degrees

The given latitude is written in degrees, minutes, and seconds (DMS). We need to convert it to decimal degrees. The formula is: \(\text{Degrees} + \frac{\text{Minutes}}{60} + \frac{\text{Seconds}}{3600}\).Applying this to the given latitude, we have:\[43^{\circ} 32' 16'' = 43 + \frac{32}{60} + \frac{16}{3600}\]\[= 43 + 0.5333 + 0.00444 = 43.5378^{\circ}\].
02

Relate Surface Gravity to Latitude

Knowing the surface gravity at a specific location, it might be investigated against theoretical models involving gravity differences due to Earth's rotation and shape (latitude-related). However, since no calculation for latitude effect on gravity or equatorial correction is requested beyond providing values, we interpret the gravity and latitude as separate.Measure the latitude as a conversion result and gravity as a known constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface Gravity
Surface gravity is the force that a planet exerts on objects at or near its surface. On Earth, this force is approximately 9.8 m/s², but it can vary slight based on location. This variation is due to both the Earth's shape and its rotation.
Earth's surface is not a perfect sphere; it's slightly flattened at the poles and bulging at the equator. This shape difference means that surface gravity is stronger at the poles than at the equator.
  • Factors affecting intensity: Shape of the Earth causes different distances to the center, reducing gravity at the equator.
  • Implications: Accurate measurements of surface gravity are crucial for tasks such as calibrating instruments in geophysics and engineering.
Although precision instruments measure surface gravity, understanding the factors contributing to its variance enhances comprehension of not just Earth but other celestial bodies as well.
Decimal Degrees
Decimal degrees provide a simple way to represent latitude and longitude, especially useful in computations. Unlike degrees, minutes, and seconds (DMS), decimal degrees express these measurements as a single value.
Converting DMS to decimal degrees involves dividing the minutes by 60 and the seconds by 3600, then adding these values to the degrees. For example, a latitude of 43° 32' 16'' converts to 43.5378° as follows:
  • Convert 32 minutes to decimal: 32/60 = 0.5333
  • Convert 16 seconds to decimal: 16/3600 = 0.00444
  • Add to degrees: 43 + 0.5333 + 0.00444 = 43.5378
This simplified format facilitates easier calculations in various fields, including mapping and geographic information systems (GIS).
Geophysics
Geophysics is the study of Earth's physical properties and the processes influencing them. This science covers a broad array of topics, from seismic activity to magnetic fields.
One key aspect of geophysics is understanding how surface gravity varies across different locations on Earth. This knowledge helps in:
  • Oil and mineral exploration: Variations can indicate deposits below the surface.
  • Earthquake research: Tremors affect surface gravity readings, aiding in studying fault lines.
  • Infrastructure development: Precise gravity measurements are used in designing stable structures.
The integration of geophysics with technologies like satellite imaging allows for improved predictive and analytical capabilities, which enhance our ability to respond to natural events.
Earth's Rotation
Earth's rotation is a fundamental factor influencing many geophysical phenomena. The planet rotates once approximately every 24 hours, a movement that affects surface gravity, among other things.
As the Earth spins, centrifugal force causes a reduction in gravity at the equator compared to the poles. This is why an object weighs slightly less at the equator than at the poles.
  • Key Effects: Centrifugal force stemming from rotation contributes to the equatorial bulging.
  • Global Implications: Rotation impacts weather patterns, ocean currents, and even time calculations.
Understanding Earth's rotation enhances our grasp of everyday phenomena and technical fields, like satellite deployment and global positioning systems (GPS).
By considering rotation, scientists can develop better models for weather prediction and climate studies.

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Most popular questions from this chapter

Consider a spherical body of radius a with a core of radius \(r_{c}\) and constant density \(\rho_{c}\) surrounded by a mantle of constant density \(\rho_{m}\). Show that the moment of inertia \(C\) and mass \(M\) are given $$ \begin{array}{l} \text { TABLE 5-1 Values of the Dimensionless Polar Moment of Inertia, } \mathrm{J}_{2} \text { , and the Polar Flattening for the Earth, } \\ \text { Moon, Mars, and Venus } \\ \begin{array}{lllll} \text { Earth } & \text { Moon } & \text { Mars } & \text { Venus } \\ \hline C / M a^{2} & 0.3307007 & 0.3935 & 0.366 & 0.33 \\ J_{2} \equiv \frac{1}{M a^{2}}\left(C-\frac{A+B}{2}\right) & 1.0826265 \times 10^{-3} & 2.037 \times 10^{-4} & 1.96045 \times 10^{-3} & 4.458 \times 10^{-6} \\\ f \equiv \frac{2}{(a+b)}\left(\frac{a+b}{2}-c\right) & 3.35281068 \times 10^{-3} & 1.247 \times 10^{-3} & 6.4763 \times 10^{-3} & \- \\ \hline \end{array} \end{array} $$ by $$ \begin{aligned} C &=\frac{8 \pi}{15}\left[\rho_{c} r_{c}^{5}+\rho_{m}\left(a^{5}-r_{c}^{5}\right)\right] \\ M &=\frac{4 \pi}{3}\left[\rho_{c} r_{c}^{3}+\rho_{m}\left(a^{3}-r_{c}^{3}\right)\right] \end{aligned} $$ Determine mean values for the densities of the Earth's mantle and core given \(C=8.04 \times 10^{37} \mathrm{~kg} \mathrm{~m}^{2}, M=\) \(5.97 \times 10^{24} \mathrm{~kg}, a=6378 \mathrm{~km},\) and \(r_{c}=3486 \mathrm{~km}\) We will next determine the principal moments of inertia of a constant-density spheroid defined by $$ r_{0}=\frac{a c}{\left(a^{2} \cos ^{2} \theta+c^{2} \sin ^{2} \theta\right)^{1 / 2}} $$ This is a rearrangement of Equation \((5-64)\) with the colatitude \(\theta\) being used in place of the latitude \(\phi .\) By substituting Equations \((5-6)\) and \((5-7)\) into Equations \((5-26)\) and \((5-29),\) we can write the polar and equatorial moments of inertia as $$ \begin{aligned} C=& \rho \int_{0}^{2 \pi} \int_{0}^{r_{0}} \int_{0}^{\pi} r^{\prime 4} \sin ^{3} \theta^{\prime} d \theta^{\prime} d r^{\prime} d \psi^{\prime} \\ A=& \rho \int_{0}^{2 \pi} \int_{0}^{r_{0}} \int_{0}^{\pi} r^{\prime 4} \sin \theta^{\prime} \\ & \times\left(\sin ^{2} \theta^{\prime} \sin ^{2} \psi^{\prime}+\cos ^{2} \theta^{\prime}\right) d \theta^{\prime} d r^{\prime} d \psi^{\prime}, \quad(5-84) \end{aligned} $$ where the upper limit on the integral over \(r^{\prime}\) is given by Equation \((5-82)\) and \(B=A\) for this axisymmetric body. The integrations over \(\psi^{\prime}\) and \(r^{\prime}\) are straightforward and yield $$ \begin{aligned} C &=\frac{2}{5} \pi \rho a^{5} c^{5} \int_{0}^{\pi} \frac{\sin ^{3} \theta^{\prime} d \theta^{\prime}}{\left(a^{2} \cos ^{2} \theta^{\prime}+c^{2} \sin ^{2} \theta^{\prime}\right)^{5 / 2}} \quad(5-85) \\ A &=\frac{1}{2} C+\frac{2}{5} \pi \rho a^{5} c^{5} \int_{0}^{\pi} \frac{\cos ^{2} \theta^{\prime} \sin \theta^{\prime} d \theta^{\prime}}{\left(a^{2} \cos ^{2} \theta^{\prime}+c^{2} \sin ^{2} \theta^{\prime}\right)^{5 / 2}} \\ (5-86) \end{aligned} $$ The integrals over \(\theta^{\prime}\) can be simplified by introducing the variable \(x=\cos \theta^{\prime}\left(d x=-\sin \theta^{\prime} d \theta^{\prime}, \sin \theta^{\prime}=\right.\) \(\left.\left(1-x^{2}\right)^{1 / 2}\right)\) with the result $$ \begin{array}{l} C=\frac{2}{5} \pi \rho a^{5} c^{5} \int_{-1}^{1} \frac{\left(1-x^{2}\right) d x}{\left[c^{2}+\left(a^{2}-c^{2}\right) x^{2}\right]^{5 / 2}} \quad(5-87) \\ A=\frac{1}{2} C+\frac{2}{5} \pi \rho a^{5} c^{5} \int_{-1}^{1} \frac{x^{2} d x}{\left[c^{2}+\left(a^{2}-c^{2}\right) x^{2}\right]^{5 / 2}} \end{array} $$ From a comprehensive tabulation of integrals we find $$ \int_{-1}^{1} \frac{d x}{\left\\{c^{2}+\left(a^{2}-c^{2}\right) x^{2}\right\\}^{5 / 2}}=\frac{2}{3} \frac{\left(2 a^{2}+c^{2}\right)}{c^{4} a^{3}} $$ $$ \int_{-1}^{1} \frac{x^{2} d x}{\left\\{c^{2}+\left(a^{2}-c^{2}\right) x^{2}\right\\}^{5 / 2}}=\frac{2}{3} \frac{1}{c^{2} a^{3}} $$ By substituting Equations \((5-89)\) and \((5-90)\) into Equations \((5-87)\) and \((5-88)\), we obtain $$ \begin{array}{l} C=\frac{8}{15} \pi \rho a^{4} c \\ A=\frac{4}{15} \pi \rho a^{2} c\left(a^{2}+c^{2}\right) \end{array} $$ These expressions for the moments of inertia can be used to determine \(J_{2}\) for the spheroid. The substitution of Equations \((5-91)\) and \((5-92)\) into the definition of \(J_{2}\) given in Equation \((5-43)\), together with the equation for the mass of a constant-density spheroid $$ M=\frac{4 \pi}{3} \rho a^{2} c $$ yields $$ J_{2}=\frac{1}{5}\left(1-\frac{c^{2}}{a^{2}}\right) $$

PROBLEM \(5-9\) Assuming that the difference in moments of inertia \(C-A\) is associated with a nearsurface density \(\rho_{m}\) and the mass \(M\) is associated with a mean planetary density \(\bar{\rho}\), show that $$ J_{2}=\frac{2}{5} \frac{\rho_{m}}{\bar{\rho}} f $$ Determine the value of \(\rho_{m}\) for the Earth by using the measured values of \(J_{2}, \bar{\rho},\) and \(f\). Discuss the value obtained.

PROBLEM 5-13 Show that the gravity anomaly of an infinitely long horizontal cylinder of radius \(R\) with anomalous density \(\Delta \rho\) buried at depth \(b\) beneath the surface is $$ \Delta g=\frac{2 \pi G R^{2} \Delta \rho b}{\left(x^{2}+b^{2}\right)} $$ where \(x\) is the horizontal distance from the surface measurement point to the point on the surface directly over the cylinder axis. What is the maximum gravity anomaly caused by a long horizontal underground tunnel of circular cross section with a \(10-\mathrm{m}\) radius driven through rock of density \(2800 \mathrm{~kg} \mathrm{~m}^{-3}\) if the axis of the tunnel lies \(50 \mathrm{~m}\) below the surface?

A volcanic plug of diameter \(10 \mathrm{~km}\) has a gravity anomaly of \(0.3 \mathrm{~mm} \mathrm{~s}^{-2}\). Estimate the depth of the plug assuming that it can be modeled by a vertical cylinder whose top is at the surface. Assume that the plug has density of \(3000 \mathrm{~kg} \mathrm{~m}^{-3}\) and the rock it intrudes has a density of \(2800 \mathrm{~kg} \mathrm{~m}^{-3}\).

PROBLEM 5-4 Assume a large geoid anomaly with a horizontal scale of several thousand kilometers has a mantle origin and its location does not change. Because of continental drift the passive margin of a continent passes through the anomaly. Is there a significant change in sea level associated with the passage of the margin through the geoid anomaly? Explain your answer. The anomaly in the potential of the gravity field measured on the reference geoid \(\Delta U\) can be related directly to the geoid anomaly \(\Delta N\). The potential anomaly is defined by $$ \Delta U=U_{m 0}-U_{0} $$ where \(U_{m 0}\) is the measured potential at the location of the reference geoid and \(U_{0}\) is the reference value of the potential defined by Equation \((5-54)\). The potential on the measured geoid is \(U_{0}\), as shown in Figure \(5-6 .\) It can be seen from the figure that \(U_{0}, U_{m 0},\) and \(\Delta N\) are related by $$ U_{0}=U_{m 0}+\left(\frac{\partial U}{\partial r}\right)_{r=r_{0}} \Delta N $$ because \(\Delta N / a \ll 1 .\) Recall from the derivation of Equation \((5-53)\) that we obtained the potential by integrating the acceleration of gravity. Therefore, the radial derivative of the potential in Equation \((5-69)\) is the acceleration of gravity on the reference geoid. To the required accuracy we can write $$ \left(\frac{\partial U}{\partial r}\right)_{r=r_{0}}=g_{0} $$ where \(g_{0}\) is the reference acceleration of gravity on the reference geoid. Just as the measured potential on the reference geoid differs from \(U_{0}\), the measured acceleration of gravity on the reference geoid differs from \(g_{0}\). However, for our purposes we can use \(g_{0}\) in Equation \((5-69)\) for \((\partial U / \partial r)_{r=r_{0}}\) because this term is multiplied by a small quantity \(\Delta N\). Substitution of Equations \((5-69)\) and \((5-70)\) into Equation \((5-68)\) gives $$ \Delta U=-g_{0} \Delta N $$ A local mass excess produces an outward warp of gravity equipotentials and therefore a positive \(\Delta N\) and a negative \(\Delta U\). Note that the measured geoid essentially defines sea level. Deviations of sea level from the equipotential surface are due to lunar and solar tides, winds, and ocean currents. These effects are generally a few meters. The reference acceleration of gravity on the reference geoid is found by substituting the expression for \(r_{0}\) given by Equation \((5-62)\) into Equation \((5-50)\) and simplifying the result by neglecting quadratic and higher order terms in \(J_{2}\) and \(a^{3} \omega^{2} / G M\). One finds $$ g_{0}=\frac{G M}{a^{2}}\left(1+\frac{3}{2} J_{2} \cos ^{2} \phi\right)+a \omega^{2}\left(\sin ^{2} \phi-\cos ^{2} \phi\right) $$ To provide a standard reference acceleration of gravity against which gravity anomalies are measured, we must retain higher order terms in the equation for \(g_{0}\). Gravity anomalies are the differences between measured values of \(g\) on the reference geoid and \(g_{0} .\) By international agreement in 1980 the reference gravity field was defined to be $$ \begin{aligned} g_{0}=& 9.7803267715\left(1+0.0052790414 \sin ^{2} \phi\right.\\\ &+0.0000232718 \sin ^{4} \phi \\ &+0.0000001262 \sin ^{6} \phi \\ &\left.+0.0000000007 \sin ^{8} \phi\right) \end{aligned} $$ with \(g_{0}\) in \(\mathrm{m} \mathrm{s}^{-2}\). This is known as the 1980 Geodetic Reference System (GRS) (80) Formula. The standard reference gravity field given by Equation \((5-73)\) is of higher order in \(\phi\) than is the consistent quadratic approximation used to specify both \(g_{0}\) in Equation ( \(5-72\) ) and \(r_{0}\) in Equation \((5-67)\). The suitable SI unit for gravity anomalies is \(\mathrm{mm} \mathrm{s}^{-2}\).
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