Math / Calculus

Questionwhere $m_{E}$ and $d_{E}$ are the mass of the Earth and its distance appraximation of uniform circular motion, determine:
1. The acceleration due to gravity $g_{M}$ at the surface of Mars.
2. The orbital period $T_{M}$ of Mars around the Sun in Earth years.
3. The velocity of a satellite to remain in orbit around Mars at an altitude $h=300 \mathrm{~km}$ . Given: $m_{E}=5.97 \times 10^{24} \mathrm{~kg}, R_{E}=6371 \mathrm{~km}, R_{M}=3396 \mathrm{~km}$ , and $G=6.67 \times 10^{-11} \mathrm{SI}$ . - Exercise 3.5
Alcomsat-1 is an Algerian telecommunications satellite of mass $m=92 \mathrm{~kg}$ , launched on December 10, 2017. The satellite was placed in a geostationary orbit considered to be circular with a radius $r$ .
1. Asuming that the only force acting on the satellite is the gravitational force exerted by the Earth, what are the other forces neglected in this case?
2. Recalling the expression for the vector of the Earth's gravitational force acting on the satellite as a function of its altitude $h$ , show that the motion is uniform (specify the chosen reference frame).
3. Deduce the velocity expression of the satellite and calculate it for $h=300 \mathrm{~km}$ .
4. Find the expression of the angular momentum $\vec{L}_{0}$ of the satellite with respect to the center $O$ of the Earth in the cylindrical coordinate system $\left(\vec{u}_{r}, \vec{u}_{\theta}, \vec{u}_{z}\right)$ . Show that the vector $\vec{L}_{0}$ is constant during the motion. Given: Mass of the Earth $M_{E}=5.97 \times 10^{24} \mathrm{~kg}$ , radius of the Earth $R_{E}=6380 \mathrm{~km}, G=6.67 \times$ $10^{-11} S I$ . e Exercise 3.6 Gravimetry is concerned with the measurement of the gravitational field intensity at a given point. Through gravimetry, the presence of underground cavities can be detected.
1. At the Earth's surface, calculate the gravitational field $g_{0}$ of the Earth without any cavity and the gravitational field $g_{1}$ above a spherical cavity of radius $R_{C}$ with its center located at a depth $d \geq R_{C}$ .
2. Assuming that we can measure the gravitational field with a precision $\delta=\frac{90-91}{90}=10^{-6}$ and we want to detect a cavity just below the ground surface (i.e., $R_{C}=d$ ). What is the radius of the smallest cavity that can be detected? (Radius of the Earth $R_{T}=6370 \mathrm{~km}$ ).
Exercise 3.7

Studdy Solution

STEP 1

1. Mars and Earth are treated as spherical bodies with uniform mass distribution.
2. The gravitational constant $G$ is given as $6.67 \times 10^{-11} \, \text{m}^3/\text{kg}\cdot\text{s}^2$ .
3. The radius of Mars $R_{M}$ is $3396 \, \text{km}$ .
4. The radius of Earth $R_{E}$ is $6371 \, \text{km}$ .
5. The mass of the Earth $m_{E}$ is $5.97 \times 10^{24} \, \text{kg}$ .
6. The mass of Mars $m_{M}$ is not directly given and needs to be calculated or assumed from known data.

STEP 2

1. Calculate the acceleration due to gravity $g_{M}$ at the surface of Mars.
2. Determine the orbital period $T_{M}$ of Mars around the Sun.
3. Calculate the velocity of a satellite to remain in orbit around Mars at an altitude $h = 300 \, \text{km}$ .
4. Discuss the forces neglected for Alcomsat-1.
5. Show that the motion of Alcomsat-1 is uniform.
6. Deduce and calculate the velocity of Alcomsat-1.
7. Find the expression of the angular momentum $\vec{L}_{0}$ of Alcomsat-1.
8. Calculate the gravitational field $g_{0}$ and $g_{1}$ for gravimetry.
9. Determine the smallest detectable cavity radius.

STEP 3

Calculate the acceleration due to gravity $g_{M}$ at the surface of Mars using the formula:
$g_{M} = \frac{G \cdot m_{M}}{R_{M}^2}$
Assume or calculate $m_{M}$ based on known data or relationships.
STEP_1.1: Calculate or assume the mass of Mars $m_{M}$ . For example, using the ratio of gravitational accelerations or known data:
$m_{M} \approx 0.107 \times m_{E}$
$m_{M} = 0.107 \times 5.97 \times 10^{24} \, \text{kg}$

STEP 4

Determine the orbital period $T_{M}$ of Mars around the Sun using Kepler's Third Law:
$T_{M}^2 = \frac{4\pi^2}{G \cdot M_{\text{Sun}}} \cdot a_{M}^3$
Where $a_{M}$ is the semi-major axis of Mars' orbit.
STEP_2.1: Assume or use known data for $a_{M}$ and $M_{\text{Sun}}$ .
$a_{M} \approx 1.524 \, \text{AU}$
Convert $a_{M}$ to meters and calculate $T_{M}$ .

STEP 5

Calculate the velocity of a satellite to remain in orbit around Mars at an altitude $h = 300 \, \text{km}$ using:
$v = \sqrt{\frac{G \cdot m_{M}}{R_{M} + h}}$
Substitute the known values and solve for $v$ .

STEP 6

Discuss the forces neglected for Alcomsat-1. Commonly neglected forces include atmospheric drag, solar radiation pressure, and gravitational perturbations from other celestial bodies.

STEP 7

Show that the motion of Alcomsat-1 is uniform by considering the gravitational force as the centripetal force in a circular orbit.
$F_{\text{gravity}} = \frac{G \cdot M_{E} \cdot m}{r^2}$
$F_{\text{centripetal}} = \frac{m \cdot v^2}{r}$
Equate the two and solve for $v$ .

STEP 8

Deduce and calculate the velocity of Alcomsat-1 for $h = 300 \, \text{km}$ .
$v = \sqrt{\frac{G \cdot M_{E}}{R_{E} + h}}$
Substitute the known values and solve for $v$ .

STEP 9

Find the expression of the angular momentum $\vec{L}_{0}$ of Alcomsat-1:
$\vec{L}_{0} = \vec{r} \times \vec{p} = m \cdot \vec{r} \times \vec{v}$
Show that $\vec{L}_{0}$ is constant by demonstrating that $\frac{d\vec{L}_{0}}{dt} = 0$ .

STEP 10

Calculate the gravitational field $g_{0}$ and $g_{1}$ for gravimetry:
$g_{0} = \frac{G \cdot M_{E}}{R_{T}^2}$
$g_{1} = g_{0} - \frac{G \cdot \Delta M}{d^2}$
Where $\Delta M$ is the mass of the cavity.

STEP 11

Determine the smallest detectable cavity radius using the precision $\delta$ .
$\delta = \frac{g_{0} - g_{1}}{g_{0}}$
Solve for $R_{C}$ when $R_{C} = d$ .

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Studdy Solution

STEP 1

STEP 2

STEP 3

STEP 4

STEP 5

Calculate the velocity of a satellite to remain in orbit around Mars at an altitude $h = 300 \, \text{km}$ using:
$v = \sqrt{\frac{G \cdot m_{M}}{R_{M} + h}}$
Substitute the known values and solve for $v$ .

STEP 6

Discuss the forces neglected for Alcomsat-1. Commonly neglected forces include atmospheric drag, solar radiation pressure, and gravitational perturbations from other celestial bodies.

STEP 7

Show that the motion of Alcomsat-1 is uniform by considering the gravitational force as the centripetal force in a circular orbit.
$F_{\text{gravity}} = \frac{G \cdot M_{E} \cdot m}{r^2}$
$F_{\text{centripetal}} = \frac{m \cdot v^2}{r}$
Equate the two and solve for $v$ .

STEP 8

Deduce and calculate the velocity of Alcomsat-1 for $h = 300 \, \text{km}$ .
$v = \sqrt{\frac{G \cdot M_{E}}{R_{E} + h}}$
Substitute the known values and solve for $v$ .

STEP 9

Find the expression of the angular momentum $\vec{L}_{0}$ of Alcomsat-1:
$\vec{L}_{0} = \vec{r} \times \vec{p} = m \cdot \vec{r} \times \vec{v}$
Show that $\vec{L}_{0}$ is constant by demonstrating that $\frac{d\vec{L}_{0}}{dt} = 0$ .

STEP 10

Calculate the gravitational field $g_{0}$ and $g_{1}$ for gravimetry:
$g_{0} = \frac{G \cdot M_{E}}{R_{T}^2}$
$g_{1} = g_{0} - \frac{G \cdot \Delta M}{d^2}$
Where $\Delta M$ is the mass of the cavity.

STEP 11

Determine the smallest detectable cavity radius using the precision $\delta$ .
$\delta = \frac{g_{0} - g_{1}}{g_{0}}$
Solve for $R_{C}$ when $R_{C} = d$ .

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Studdy Solution

STEP 1

STEP 2

STEP 3

STEP 4

STEP 5

Calculate the velocity of a satellite to remain in orbit around Mars at an altitude h=300 km h = 300 \, \text{km} h=300km using:v=G⋅mMRM+h v = \sqrt{\frac{G \cdot m_{M}}{R_{M} + h}} v=RM​+hG⋅mM​​​Substitute the known values and solve for v v v.

STEP 6

Discuss the forces neglected for Alcomsat-1. Commonly neglected forces include atmospheric drag, solar radiation pressure, and gravitational perturbations from other celestial bodies.

STEP 7

STEP 8

Deduce and calculate the velocity of Alcomsat-1 for h=300 km h = 300 \, \text{km} h=300km.v=G⋅MERE+h v = \sqrt{\frac{G \cdot M_{E}}{R_{E} + h}} v=RE​+hG⋅ME​​​Substitute the known values and solve for v v v.

STEP 9

STEP 10

Calculate the gravitational field g0 g_{0} g0​ and g1 g_{1} g1​ for gravimetry:g0=G⋅MERT2 g_{0} = \frac{G \cdot M_{E}}{R_{T}^2} g0​=RT2​G⋅ME​​g1=g0−G⋅ΔMd2 g_{1} = g_{0} - \frac{G \cdot \Delta M}{d^2} g1​=g0​−d2G⋅ΔM​Where ΔM \Delta M ΔM is the mass of the cavity.

STEP 11

Determine the smallest detectable cavity radius using the precision δ \delta δ.δ=g0−g1g0 \delta = \frac{g_{0} - g_{1}}{g_{0}} δ=g0​g0​−g1​​Solve for RC R_{C} RC​ when RC=d R_{C} = d RC​=d.

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Calculate the velocity of a satellite to remain in orbit around Mars at an altitude $h = 300 \, \text{km}$ using:
$v = \sqrt{\frac{G \cdot m_{M}}{R_{M} + h}}$
Substitute the known values and solve for $v$ .

Deduce and calculate the velocity of Alcomsat-1 for $h = 300 \, \text{km}$ .
$v = \sqrt{\frac{G \cdot M_{E}}{R_{E} + h}}$
Substitute the known values and solve for $v$ .

Calculate the gravitational field $g_{0}$ and $g_{1}$ for gravimetry:
$g_{0} = \frac{G \cdot M_{E}}{R_{T}^2}$
$g_{1} = g_{0} - \frac{G \cdot \Delta M}{d^2}$
Where $\Delta M$ is the mass of the cavity.

Determine the smallest detectable cavity radius using the precision $\delta$ .
$\delta = \frac{g_{0} - g_{1}}{g_{0}}$
Solve for $R_{C}$ when $R_{C} = d$ .