Gravitation

Two ships, each of mass $5.0\cdot 10^7\mathrm{kg}$, have their centers of mass at a distance $d=1.0\cdot 10^2\mathrm{m}$. Treating them as point masses, compute the magnitude of the gravitational force with which they attract each other ($G=6.67\cdot 10^{-11}\mathrm{N}\cdot {\mathrm{m}}^2/k{\mathrm{g}}^2$).

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The Moon has mass $M_L=7.35\cdot 10^{22}\mathrm{kg}$ and radius $R_L=1.74\cdot 10^6\mathrm{m}$. Compute the gravitational acceleration at its surface ($G=6.67\cdot 10^{-11}\mathrm{N}\cdot {\mathrm{m}}^2/k{\mathrm{g}}^2$).

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At what altitude above the Earth’s surface is the gravitational acceleration equal to $9.00m/{\mathrm{s}}^2$? Use $g_0=9.81m/{\mathrm{s}}^2$ and $R_T=6.37\cdot 10^6\mathrm{m}$.

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Two point masses $m_1=4.0\mathrm{kg}$ and $m_2=1.0\mathrm{kg}$ are at a distance $d=0.90\mathrm{m}$. Determine the point of the segment joining them at which the total gravitational field is zero.

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In the model of a spherical Earth of uniform density, with $R_T=6.37\cdot 10^6\mathrm{m}$, at what depth is the gravitational acceleration $90\%$ of its value at the surface?

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A satellite travels on a circular orbit at an altitude $h=1000\mathrm{km}$ above the Earth’s surface. Compute the speed and the period, with $G{M}_T=3.98\cdot 10^{14}{\mathrm{m}}^3/{\mathrm{s}}^2$ and $R_T=6.37\cdot 10^6\mathrm{m}$.

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The Moon travels around the Earth on a nearly circular orbit of radius $3.84\cdot 10^8\mathrm{m}$ in $27.3$ days. Find the mass of the Earth ($G=6.67\cdot 10^{-11}\mathrm{N}\cdot {\mathrm{m}}^2/k{\mathrm{g}}^2$).

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Mars has mass $6.42\cdot 10^{23}\mathrm{kg}$ and rotates about its axis in $88642\mathrm{s}$. Compute the radius of the equatorial circular orbit of a satellite that always stays above the same point of the Martian surface ($G=6.67\cdot 10^{-11}\mathrm{N}\cdot {\mathrm{m}}^2/k{\mathrm{g}}^2$).

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Compute the escape velocity from the surface of Mars, which has mass $6.42\cdot 10^{23}\mathrm{kg}$ and radius $3.39\cdot 10^6\mathrm{m}$, neglecting the atmosphere and the rotation ($G=6.67\cdot 10^{-11}\mathrm{N}\cdot {\mathrm{m}}^2/k{\mathrm{g}}^2$).

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A body is launched vertically from the surface of the Moon with speed $v_0=2.0km/s$. Compute the maximum altitude reached. Data: $M_L=7.35\cdot 10^{22}\mathrm{kg}$, $R_L=1.74\cdot 10^6\mathrm{m}$, $G=6.67\cdot 10^{-11}\mathrm{N}\cdot {\mathrm{m}}^2/k{\mathrm{g}}^2$; the Moon has no atmosphere.

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A satellite of mass $800\mathrm{kg}$ travels on a circular orbit of radius $r=8.0\cdot 10^6\mathrm{m}$ around the Earth ($G{M}_T=3.98\cdot 10^{14}{\mathrm{m}}^3/{\mathrm{s}}^2$). Compute its mechanical energy and the minimum work needed to bring it to infinite distance.

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A comet passes perihelion at $1.0\mathrm{AU}$ from the Sun with speed $40km/s$ and reaches aphelion at $9.0\mathrm{AU}$. Compute its speed at aphelion.

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An asteroid travels its orbit around the Sun in $8.0$ years. Compute the semi-major axis of the orbit in astronomical units.

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