Quick questions on Newton's law of universal gravitation and gravitational fields (QCE Physics Unit 3)

Force is inversely proportional to the square of the distance. Doubling $r$ reduces $F$ to one quarter. Halving $r$ quadruples $F$. This rapid fall-off explains why Earth's gravity dominates near the surface but becomes negligible far from the planet, while still extending to infinity in principle.

The gravitational field strength $g$ at a point is the gravitational force per unit mass on a test mass placed there:

For a freely falling object of mass $m$ in a gravitational field $g$, the acceleration is $a = g$ (independent of $m$, because $F = m g$ and $a = F / m$). All objects fall with the same acceleration in a given gravitational field, in the absence of air resistance. This is why $g$ appears in projectile-motion equations as the constant downward acceleration.

Expect a table of $g$ at various altitudes and a question asking you to extract $M$ of the planet, or a single-line problem asking you to compute $g$ at a given altitude and discuss the apparent weight of an astronaut. Markers focus on candidates who substitute altitude $h$ for $r$ instead of $R_E + h$.

A common IA2 measures local $g$ by timing a simple pendulum at different lengths and linearising $T^2$ against $L$ (slope $= 4 \pi^2 / g$). The Unit 3 universal-gravitation theory provides the framework in the justification section: the value of $g$ extracted should agree with $G M_E / R_E^2$ at the school's altitude.

State Newton's law of universal gravitation. [2 marks]

Calculate $g$ at $1.0 \times 10^3 \text{ km}$ altitude, given $M_E = 5.97 \times 10^{24} \text{ kg}$, $R_E = 6.37 \times 10^6 \text{ m}$. [3 marks]

An ANSTO low-Earth-orbit (LEO) target satellite at $h = 500 \text{ km}$. (a) Calculate $g$ at that altitude. (b) Determine the gravitational force on a $250 \text{ kg}$ satellite.