Circular Motion & Gravitation — Print · Grade 12 Science

Circular Motion & Gravitation

Learning goals

• I can calculate centripetal acceleration (a = v²/r) and centripetal force (F = mv²/r) for uniform circular motion.
• I can apply Newton's law of universal gravitation (F = Gm₁m₂/r²) to find the gravitational force between two masses.
• I can explain how gravity supplies the centripetal force that holds a satellite in orbit.

Readings for this unit

• The force on the box is not the force on the floor

An object can move at a perfectly constant speed and still be accelerating — a fact that feels wrong until you remember that acceleration means any change in velocity, and velocity includes direction. A car rounding a curve at a steady 50 km/h is constantly changing direction, so it is constantly accelerating. This is uniform circular motion: constant speed, continuously turning. The acceleration always points toward the center of the circle and is called centripetal acceleration, given by the clean formula a = v²/r, where v is the speed and r is the radius. Notice that tighter turns (smaller r) and higher speeds (larger v) both demand more acceleration — which is exactly why a sharp turn at high speed feels so violent.

By Newton's second law, an acceleration requires a net force in the same direction, so circular motion demands a net inward force called the centripetal force, F = ma = mv²/r. Here is the single most important idea in this topic: *centripetal is not a new kind of force.* The word simply means 'center-seeking' and describes the *direction* of whatever real force is doing the job. For a ball on a string it is tension; for a car on a curve it is friction between tires and road; for the Moon circling Earth it is gravity. Whenever you see circular motion, ask: which real force is pointing toward the center and supplying the mv²/r that the motion requires?

This clears up the most stubborn misconception in the whole unit — the so-called centrifugal 'force.' When a car turns left, you feel flung to the right and swear something is pushing you outward. Nothing is. Your body, obeying the first law, simply wants to continue straight ahead while the car turns underneath you; the door then pushes you inward to make you turn with it. The outward 'force' is an illusion of riding in an accelerating frame. The only real force in play points *inward*, toward the center.

Newton's greatest leap was realizing that the same force pulling an apple to the ground also holds the Moon in its orbit — and that gravity is universal, acting between *every* pair of masses in the cosmos. He captured it in Newton's law of universal gravitation: F = Gm₁m₂/r². The force grows with each mass and shrinks with the square of the distance r between their centers, and G is the gravitational constant, a very small fixed number, 6.67 × 10⁻¹¹ N·m²/kg². Because G is so tiny, the gravitational pull between everyday objects is utterly negligible; it takes a body as massive as a planet for gravity to become the dominant force we feel every second of our lives.

The r² in the denominator makes gravity an inverse-square law, and its consequences are dramatic. Double the distance between two objects and the force drops not to half but to one-quarter; triple the distance and it falls to one-ninth. This same equation, evaluated at Earth's surface, reproduces the familiar weight W = mg — plug in Earth's mass and radius and out comes an acceleration of about 9.8 m/s² for every object, regardless of its own mass. Universal gravitation and everyday weight are not two separate ideas; they are the same law seen at two different scales.

Orbits are the grand payoff, and Newton explained them with a famous thought experiment: imagine a cannon on a mountaintop firing a ball ever faster. Fired gently, the ball arcs down and lands; fired fast enough, the ground curves away beneath it just as quickly as it falls, and the ball 'falls around' the planet forever. That is an orbit — continuous free-fall in which gravity supplies exactly the centripetal force the circular path requires, so Gm₁m₂/r² = mv²/r. This is also why astronauts on the space station float: they are not beyond gravity's reach (Earth's gravity is still nearly as strong up there) but are in constant free-fall together with their station, endlessly falling around the Earth.

Model check: Centripetal force is the net inward force, not an extra interaction added to gravity or tension. ac=v²/r=8 m/s². Speed can be constant while velocity direction changes.

Centripetal Acceleration

a = v² ÷ r

v is speed in m/s; r is the radius of the circular path in meters; a points toward the center.

Centripetal Force

F = mv² ÷ r

m is mass in kg; this is the net inward force required for circular motion, supplied by a real force like tension, friction, or gravity.

Universal Gravitation

F = G m₁m₂ ÷ r²

m₁ and m₂ are the two masses in kg; r is the distance between their centers in meters; G = 6.67 × 10⁻¹¹ N·m²/kg².

Worked examples

Avoid these mistakes

• Centripetal force is the net inward force, not an extra interaction added to gravity or tension.

Words to know

centripetal acceleration

The acceleration of an object moving in a circle, always directed toward the center; a = v²/r.

centripetal force

The net inward force required to keep an object moving in a circle; F = mv²/r. It is supplied by a real force such as tension, friction, or gravity.

period (T)

The time it takes an object to complete one full revolution around a circle.

Newton's law of universal gravitation

Every pair of masses attracts with a force F = Gm₁m₂/r², where r is the distance between their centers.

gravitational constant (G)

The fixed number in the gravitation equation, G = 6.67 × 10⁻¹¹ N·m²/kg², measured first by Henry Cavendish.

Newton's apple and the unification of the heavens

The famous story of an apple falling in Newton's garden captures a genuine breakthrough: Newton realized that the force pulling the apple down might be the same force holding the Moon in its orbit. This was a stunning claim, because for centuries the heavens and the Earth were thought to obey entirely different rules. His law of universal gravitation, F = Gm₁m₂/r², proposed that every pair of masses in the universe attracts, from apples to moons to distant galaxies. Using it, Newton could explain Kepler's laws of planetary motion, the tides, and the paths of comets with a single equation — one of the greatest unifications in the history of thought.

Cavendish 'weighs the Earth': measuring G

Newton's equation contained the constant G, but he never knew its value. In 1798, more than a century later, the British scientist Henry Cavendish measured it using an exquisitely sensitive torsion balance: two small lead balls on a suspended rod, twisted ever so slightly by the gravitational pull of larger lead spheres brought nearby. From the tiny twist he extracted G ≈ 6.67 × 10⁻¹¹ N·m²/kg². Once G was known, scientists could rearrange the surface-gravity relationship g = GM/r² to calculate Earth's mass for the first time — which is why Cavendish's delicate experiment earned the nickname 'weighing the Earth.'

There is no centrifugal force

When a car turns sharply, you feel thrown against the door and might swear an outward force is pushing you — the so-called centrifugal force. Physically, no such force exists. Your body, obeying Newton's first law, simply tries to continue in a straight line while the car curves underneath you; the door then pushes you inward to make you turn with the car. The outward sensation is an artifact of observing motion from within an accelerating (rotating) frame of reference. In an inertial frame, the only real horizontal force in a turn is the centripetal one, pointing toward the center — supplied here by friction and the car's door.

Orbits, GPS, and Newton's cannonball

Newton explained orbits with a thought experiment: a cannon fired from a tall mountain shoots a ball that arcs to the ground, but fired fast enough, the ground curves away beneath it as quickly as it falls, so the ball circles the planet — an orbit is simply perpetual free-fall. Real satellites embody this: the International Space Station circles about 400 km up at roughly 7.7 km/s, completing an orbit every 90 minutes, while GPS satellites sit much higher and time their signals so precisely that engineers must even correct for effects of relativity. In every case, gravity supplies exactly the centripetal force the circular path demands, setting Gm₁m₂/r² equal to mv²/r.

In the real world

GPS satellites stay in orbit because Earth's gravity supplies precisely the centripetal force their circular paths require — the same physics that keeps the Moon circling Earth.

Exit check

v=4 m/s,r=2 m. Find inward acceleration.