Circular Orbits

Force Balance: A stable circular orbit exists when the gravitational force ($F_g$) equals the required centripetal force ($F_c$). Mathematically, this is expressed as $G \frac{Mm}{r^2} = \frac{mv^2}{r}$.

Orbital Velocity: By solving the force balance equation, we find that the speed $v$ required for a circular orbit is $v = \sqrt{\frac{GM}{r}}$. This shows that the speed depends only on the mass of the central body ($M$) and the radius ($r$), not the mass of the orbiting object ($m$).

Kepler's Third Law: For circular orbits, the square of the orbital period ($T$) is proportional to the cube of the orbital radius ($r$). This relationship, $T^2 \propto r^3$, implies that objects further from the center take significantly longer to complete one revolution.

Determining Orbital Speed: To find the necessary speed for a satellite, identify the mass of the planet and the distance from the planet's center. Use the formula $v = \sqrt{\frac{GM}{r}}$ where $G$ is the universal gravitational constant.

Calculating the Period: Once the orbital speed is known, the time for one full orbit can be found using the circumference of the circle: $T = \frac{2\pi r}{v}$. Alternatively, use the combined form $T = 2\pi \sqrt{\frac{r^3}{GM}}$.

Adjusting Orbits: To move to a higher orbit (larger $r$), an object must actually decrease its long-term orbital speed. Conversely, to maintain a stable orbit closer to the planet, the object must travel at a higher speed to counteract the stronger gravitational pull.

Mass Independence: It is a common misconception that heavier satellites need to move faster. In reality, two objects of different masses at the same radius must travel at the same speed to maintain a circular orbit.

Altitude vs. Radius: In calculations, the radius $r$ is the distance from the center of the planet, not the surface. Always add the planet's radius to the altitude ($r = R_{planet} + h$).

Check the Radius: Examiners often provide the 'altitude' above the surface. You must add the planet's radius to this value before using it in any orbital formulas.

Unit Consistency: Ensure all distances are in meters (m) and masses in kilograms (kg). Orbital periods are often given in hours or days but must be converted to seconds (s) for standard physics equations.

Proportionality Reasoning: If the radius increases by a factor of 4, the orbital speed decreases by a factor of 2 (since $v \propto \frac{1}{\sqrt{r}}$). Using these relationships can help verify if a calculated answer is reasonable.