Physics › 🚀 Mechanics

Circular Motion & Gravitation

Why moving in a circle at constant speed still counts as accelerating — and how that same idea keeps the Moon from ever hitting the ground.

High schoolAP Physics 1 levelUni Year 1
💡
The big idea: Anything moving in a circle at constant speed is still accelerating — not because it's speeding up, but because its direction keeps changing, and that acceleration always points toward the center. Whatever real force creates that inward pull — tension, friction, or gravity — we call centripetal force. Newton showed that gravity follows the exact same inverse-square law whether it's holding a satellite in orbit or pulling an apple off a tree.
🎯 By the end, you'll be able to
  • explain why an object moving in a circle at constant speed is still accelerating, and in which direction
  • calculate centripetal acceleration (a = v²/r) and centripetal force (F = mv²/r) for a moving object
  • identify which real force — friction, tension, gravity, or a normal force — is playing the role of centripetal force in a given situation
  • apply Newton's law of universal gravitation (F = Gm₁m₂/r²) to find the force between two masses and connect it to orbital speed
📎 You should already know
  • Newton's Laws of Motion
  • Vectors & Basic Trigonometry
  • Speed, Velocity & Acceleration

Going in circles is still accelerating

Picture a car cruising around a roundabout at a perfectly steady 30 km/h. The speedometer never moves. Is the car accelerating?

Yes — and this trips people up constantly. Acceleration is the rate of change of velocity, and velocity is a vector: it has both a size (speed) and a direction. Even though the car's speed is frozen, its direction is changing every single instant as it curves around the circle. That changing direction is an acceleration, and it points straight toward the center of the circle.

🔑 Centripetal force isn't a new kind of force

"Centripetal" just means "center-seeking" — it's a label for direction, not a new physical force like gravity or friction. In every real situation, some familiar force is doing the job: friction between tires and road on a curve, tension in a string swinging a ball, the normal force on a rider in a loop, or gravity holding a satellite in orbit. If nothing pulls or pushes the object toward the center, it simply moves off in a straight line — which is exactly what Newton's first law predicts.

\[ a_c = \dfrac{v^2}{r} \]
Centripetal acceleration: how fast the direction of velocity is changing, always pointing toward the circle's center. Bigger speed or a tighter (smaller) radius means a sharper turn and more acceleration.
\[ F_c = \dfrac{mv^2}{r} = m a_c \]
Centripetal force: just Newton's second law (F = ma) applied to circular motion. It's the net inward force needed to keep an object of mass m on its circular path.
\[ F = \dfrac{Gm_1 m_2}{r^2} \]
Newton's law of universal gravitation: every pair of masses attracts each other with a force that grows with both masses and shrinks with the square of the distance between them. G = 6.674\times10^{-11}\,\text{N·m}^2/\text{kg}^2.
🎮 Interactive: Circular Motion Lab LIVE
Drag the speed and radius sliders and watch the centripetal acceleration and force respond. Notice that doubling the speed changes things far more dramatically than doubling the radius — that's the squared v² at work.

Orbit is just falling and missing

Newton's real leap wasn't discovering gravity — people already knew things fell. His leap was realizing that the force pulling an apple to the ground and the force holding the Moon in orbit are the same force, obeying the same law.

A satellite doesn't defy gravity — it's constantly falling toward Earth. But it's also moving sideways so fast that as it falls, the curved surface of the Earth drops away beneath it at the same rate. It keeps falling and missing, forever. That endless "falling and missing" is what we call an orbit, and gravity itself is the centripetal force keeping it there.

📝 Worked example: A 1,200 kg car rounds a flat, unbanked curve of radius 50 m at a steady 15 m/s. What centripetal force must friction between the tires and the road supply?
  1. Centripetal acceleration: \(a_c = \dfrac{v^2}{r} = \dfrac{(15\,\text{m/s})^2}{50\,\text{m}} = \dfrac{225}{50} = 4.5\,\text{m/s}^2\)
  2. Centripetal force: \(F_c = m a_c = 1200\,\text{kg} \times 4.5\,\text{m/s}^2 = 5400\,\text{N}\)
  3. This entire 5,400 N has to come from static friction between tires and road — there's no other horizontal force available on a flat curve.
✓ 5,400 N of static friction, directed toward the center of the curve.
📝 Worked example: The International Space Station orbits about 400 km above Earth's surface. Using Earth's mass (5.97×10²⁴ kg) and radius (6.37×10⁶ m), estimate the ISS's orbital speed.
  1. Orbital radius from Earth's center: \(r = 6.37\times10^6 + 4.00\times10^5 = 6.77\times10^6\,\text{m}\)
  2. Gravity supplies the centripetal force: \(\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}\), so the mass of the ISS cancels and \(v = \sqrt{\dfrac{GM}{r}}\)
  3. \(GM = (6.674\times10^{-11})(5.97\times10^{24}) \approx 3.98\times10^{14}\,\text{m}^3/\text{s}^2\)
  4. \(v = \sqrt{\dfrac{3.98\times10^{14}}{6.77\times10^6}} = \sqrt{5.88\times10^7} \approx 7,670\,\text{m/s}\)
✓ About 7.7 km/s (roughly 27,600 km/h) — very close to the ISS's real orbital speed.
⚠️ There's no such thing as 'centrifugal force' flinging you outward

When a car takes a sharp turn and you feel thrown against the door, it's tempting to imagine some outward force pushing you. There isn't one — at least not a real one. What's actually happening is your own inertia: your body "wants" to keep moving in a straight line (Newton's first law), while the car curves underneath you. The door pushes inward on you to force you along the curve. The only real force in the story is centripetal, and it points toward the center — the outward sensation is just your body resisting the turn.

Check your understanding

1. In uniform circular motion, which statement is correct?
Velocity is a vector. Even with constant speed, a continuously changing direction means the velocity vector is changing — and that change is an acceleration, directed toward the center of the circle.
2. A 2 kg ball swings on a string in a horizontal circle of radius 0.5 m at a constant 4 m/s. What is the tension in the string (ignore gravity's effect on the string's angle)?
a_c = v²/r = (4)²/0.5 = 16/0.5 = 32 m/s². Then F = ma = 2 kg × 32 m/s² = 64 N. The string tension supplies this entire centripetal force.
3. A car's speed around a curve of fixed radius doubles. How does the centripetal force required change?
Centripetal force is F = mv²/r, proportional to v². Doubling v multiplies the force by 2² = 4 — which is exactly why fast turns feel so much more violent than slow ones.
4. A satellite is moved to a new orbit at three times its original distance from Earth's center. How does the gravitational force on it change?
Gravity follows an inverse-square law, F = Gm₁m₂/r². Tripling the distance r means dividing the force by 3² = 9.
✅ Key takeaways
  • Uniform circular motion has constant speed but constantly changing direction, so it's always accelerating — toward the center.
  • Centripetal acceleration is a = v²/r and centripetal force is F = mv²/r; doubling speed quadruples the force needed, but doubling radius only halves it.
  • Centripetal force isn't a separate force of nature — it's whatever real force (tension, friction, gravity, normal force) happens to point toward the center.
  • Newton's law of gravitation, F = Gm₁m₂/r², shows why orbits work: a satellite is continuously falling, just moving sideways fast enough to keep missing the ground.