11.2 Gravitation, Escape Velocity, Satellites & Weightlessness

Gravitation, Escape Velocity, Satellites & Weightlessness

Gravitation links the work–energy ideas of Section 10.4 with the circular-motion ideas of Section 11.1. The same few formulas answer questions about falling bodies, rockets, satellites and weightlessness, so it pays to learn how they connect rather than memorising each one separately.

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1. Newton's Law of Universal Gravitation

Every particle attracts every other particle with a force along the line joining them:

$F = \frac{G m_1 m_2}{r^2}, \qquad G = 6.67 \times 10^{-11}\text{ N}\cdot\text{m}^2\text{/kg}^2$

• The force is always attractive and obeys Newton's third law (equal and opposite on the two bodies).
• Doubling the separation reduces the force to one quarter (inverse-square law).
• $G$ is a universal constant; $g$ is not, because it depends on the planet and the position.

Gravitational Field Strength and the Mass of the Earth

At the Earth's surface the weight of a body equals the gravitational force:

$mg = \frac{GMm}{R^2} \implies g = \frac{GM}{R^2}$

With $g = 9.8\text{ m/s}^2$ and $R = 6.4 \times 10^6\text{ m}$, this gives the mass of the Earth:

$M = \frac{gR^2}{G} = \frac{9.8 \times (6.4 \times 10^6)^2}{6.67 \times 10^{-11}} \approx 6.0 \times 10^{24}\text{ kg}$

Variation of $g$

At a height equal to the Earth's radius ($h = R$), $g_h = \frac{g}{4} \approx 2.45\text{ m/s}^2$.

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2. Absolute Gravitational Potential Energy

The familiar $U = mgh$ assumes constant $g$, which is only valid near the surface. For large distances, take the zero of potential energy at infinity. The work done in bringing mass $m$ from infinity to a distance $r$ from the centre of the Earth gives

$U(r) = -\frac{GMm}{r}$

• $U$ is negative everywhere at finite distance, because the gravitational force is attractive: energy must be supplied to separate the bodies.
• The change between $r_1$ and $r_2$ is $\Delta U = GMm\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$, which reduces to $mgh$ when $h \ll R$.
• Gravitational potential (energy per unit mass) is $V = -\frac{GM}{r}$, measured in J/kg.

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3. Escape Velocity

The escape velocity is the minimum launch speed that lets a body reach infinity (with zero final speed) against gravity, neglecting air resistance. Energy conservation gives

$\frac{1}{2}mv_e^2 - \frac{GMm}{R} = 0 \implies v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$

For the Earth: $v_e = \sqrt{2 \times 9.8 \times 6.4 \times 10^6} \approx 1.12 \times 10^4\text{ m/s} = 11.2\text{ km/s}$.

• $v_e$ does not depend on the mass of the body or the direction of launch.
• For the Moon, $v_e \approx 2.4\text{ km/s}$; gas molecules move fast enough to escape over time, which is one reason the Moon has no atmosphere.

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4. Orbital Velocity and Period

For a satellite in a circular orbit of radius $r$, gravity provides the centripetal force:

$\frac{mv_o^2}{r} = \frac{GMm}{r^2} \implies v_o = \sqrt{\frac{GM}{r}}$

• For an orbit just above the surface ($r \approx R$): $v_o = \sqrt{gR} \approx 7.9\text{ km/s}$. A satellite here completes one orbit in about 84 minutes.
• Comparing the two results: $v_e = \sqrt{2}\, v_o$.
• The orbital speed is independent of the satellite's mass and decreases as the orbit radius increases.
• The period is $T = \frac{2\pi r}{v_o} = 2\pi\sqrt{\frac{r^3}{GM}}$, so $T^2 \propto r^3$ (Kepler's third law).

Energy in Orbit

The total energy is negative (the satellite is bound) and $K = -E$. To move a satellite to a higher orbit, energy must be supplied, even though its speed then decreases.

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5. Geostationary Satellites

A geostationary satellite appears fixed above one point on the equator. It must:

• orbit in the equatorial plane,
• move west to east, the same sense as the Earth's rotation, and
• have a period equal to one rotation of the Earth (one sidereal day, about 23 h 56 min, often rounded to 24 h).

Setting $T = 24\text{ h}$ in $T^2 = \frac{4\pi^2 r^3}{GM}$ gives an orbital radius of about $4.2 \times 10^7\text{ m}$, that is an altitude of about $36{,}000\text{ km}$ above the surface, with an orbital speed of about $3.1\text{ km/s}$. Three such satellites spaced $120^\circ$ apart can relay signals over almost the whole Earth, which is why they are used for communications and broadcasting.

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6. Real Weight, Apparent Weight and Weightlessness

The real weight of a body is $W = mg$. The apparent weight is the reading of a scale or the normal force on the body.

An astronaut in an orbiting spacecraft is not outside the Earth's gravity; at a few hundred kilometres $g$ is still almost 90% of its surface value. The astronaut feels weightless because the astronaut and the spacecraft are both in free fall towards the Earth with the same acceleration, so the floor exerts no normal force.

Artificial Gravity

A space station can create artificial gravity by rotating about its axis. Objects pressed against the outer wall need a centripetal force $m\omega^2 R$, supplied by the wall. Setting $\omega^2 R = g$:

$\omega = \sqrt{\frac{g}{R}}, \qquad f = \frac{1}{2\pi}\sqrt{\frac{g}{R}}$

For a station of radius $R = 40\text{ m}$: $\omega = \sqrt{9.8/40} \approx 0.49\text{ rad/s}$, about one rotation every 13 s.