Circular dynamics and satellites

If gravitation keeps pulling a satellite down, why does it not hit the ground? And why does an astronaut inside hardly press on the floor, even though Earth clearly acts on the satellite?

Recall a turn: velocity is tangential, while its change points towards the centre. Now ask which interaction produces this acceleration. For a satellite it is gravitation, for a mass on a string it is tension, and for a car on a level road it is friction. In each case, the inward component of the net real force matters. If it is insufficient for the given speed and radius, the body cannot maintain that circular path.

Draw a satellite with one arrow showing Earth's attraction. Must we add a separate centripetal force? No: this names the role of the net force already drawn in uniform circular motion, rather than a new interaction. Two identical arrows would count gravitation twice. A useful check for any force diagram is to name the body or field that produces each force shown.

Kinematics gives the required acceleration aₙ = v²/r. Newton's second law then gives the required radial force Fₙ = maₙ = mv²/r. Using mass m in kilograms, v in m/s and r in metres gives force in newtons. Doubling mass requires twice the force; doubling speed requires four times the force. A wider turn at the same speed requires less force. The force's name varies between problems, but this relationship remains.

Imagine launching an object horizontally from a great height with no air. As it travels forwards, gravity deflects it downwards. Earth's surface also curves away below it. With a suitable tangential speed and enough height, the object keeps missing the surface: it continues falling freely along an orbit. The force does not hold the satellite stationary overhead; it continuously turns its velocity.

Now release an object inside the satellite. Earth gives the object and satellite almost the same acceleration, so the object does not need the floor's support to move along with the spacecraft. The contact force and weight are almost zero: this is weightlessness. The principle is the same as in a falling lift. Gravity remains substantial in orbit; small differences in the field and other effects explain the word 'almost'.

For an ideal circular orbit near the surface, the required acceleration v₁²/R must equal g. Thus v₁ = √(gR), the first cosmic speed. R is the planet's radius and g the acceleration at its surface. The satellite's mass cancels because gravitation and inertia both increase in proportion to mass. Independently doubling g or R increases the speed by a factor of √2 in this formula; on different planets, g and R themselves are linked by the mass distribution.

For Earth, v₁ ≈ 7,9 км/с. A real spacecraft cannot circle the planet at that speed right beside the surface: atmosphere and terrain get in the way. The model assumes a spherical planet and neglects its rotation and air resistance. For a higher circular orbit, use distance from the centre instead of R and the local gravitational acceleration instead of surface g. Height above the ground alone is not the correct distance.

Two satellites of masses 100 and 200 kg move at the same speed along the same circular orbit. Compare their accelerations and gravitational forces.

The orbital radius and speed are the same. Hence aₙ = v²/r is the same for the 100 and 200 kg satellites: their velocity vectors turn at the same rate.

Giving twice the mass the same acceleration requires twice the force: F₂/F₁ = m₂/m₁ = 200/100 = 2.

Earth's gravitational force is proportional to the satellite's mass and provides exactly this ratio. Mass increases both force and inertia, leaving acceleration unchanged.

The acceleration magnitudes are equal; at every point acceleration points towards Earth's centre. The gravitational force on the 200 kg satellite is twice that on the 100 kg satellite.

Centripetal force is the required role of the net real force; there is no separate new interaction.

A satellite falls continuously; the shared falling of spacecraft and astronaut explains weightlessness.

v₁ = √(gR) applies to an ideal circular orbit near the surface; another altitude requires the corresponding distance and field.

For extended bodies at rest, we will also need to consider the turning effect of forces.