Uniform Circular Motion and Gravitation

§Text Passage

1Summary 重點摘要

An object moving in a circle needs a centripetal force towards the centre, with acceleration v²/r; gravitation acts on everything with an inverse-square law, and when gravity supplies the centripetal force, planets and satellites orbit stably.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Uniform circular motion and the direction mythAn object moving along a circular path at constant speed is in uniform circular motion. Constant speed does not mean zero acceleration: velocity is a vector with magnitude and direction, and in circular motion the direction changes every instant, so there must be an acceleration directed towards the centre. For example, a cabin on a Ferris wheel moves at a steady rate and riders feel calm, yet the cabin's velocity direction is turning all the time — the change is just smooth and easily missed. If an exam question asks whether uniform circular motion has acceleration, the answer is yes: a classic conceptual trap often paired with Newton's First Law.
2. Step 2: Angular velocity, period and linear speedRotation speed is described by angular velocity: the angle swept per second, in rad s⁻¹. One revolution is 2π radians, so angular velocity equals 2π divided by the period; linear speed equals the circumference 2πr divided by the period, and also equals radius times angular velocity. For example, two points on a rotating dish — one near the centre, one near the rim — share the same period and angular velocity, but the rim point has a larger radius and therefore a larger linear speed. In calculations, first identify whether the question gives linear speed or angular velocity, then pick the matching formula; reading the question carefully is half the battle.
3. Step 3: Centripetal acceleration a = v²/rThe acceleration of circular motion points towards the centre and is called centripetal acceleration, of magnitude speed squared over radius: a = v²/r. The larger the speed or the smaller the radius, the more abruptly the direction changes and the larger the acceleration. For example, a car rounding a sharp bend (small radius) throws passengers outwards far more noticeably than a gentle bend at the same speed, precisely because the smaller radius gives a larger centripetal acceleration. Watch the units: speed in m s⁻¹ and radius in m give acceleration in m s⁻². The formula contains no mass — it is purely kinematic, and it pairs with the dynamical formula in the next step.
4. Step 4: Where the centripetal force comes fromBy Newton's Second Law, acceleration demands a net force: the net force supplying centripetal acceleration is the centripetal force, F = mv²/r, always pointing to the centre. It is not an extra new kind of force — real forces take on the role: string tension for a ball whirled horizontally, friction for a car turning, gravity plus track pressure for a coaster at the loop's top. For example, whirl a stone on a string in a horizontal circle: the instant the string snaps, the tension vanishes and the stone flies straight off along the tangent, obeying Newton's First Law — it never drifts radially outwards. Never draw centripetal force as an additional arrow on top of the real forces in a free-body diagram.
5. Step 5: Newton's law of universal gravitationNewton showed that any two objects in the universe attract each other: the force is proportional to the product of the masses and inversely proportional to the square of the distance between their centres, F = Gm₁m₂/r², with G ≈ 6.67 × 10⁻¹¹ N m² kg⁻². For example, double the separation and the force falls to a quarter; double both masses and the force quadruples. Note that r is the distance between the centres of the two bodies — for a satellite it is the Earth's radius plus the orbital altitude, not the altitude alone. This is the most common slip in calculations. Gravity is feeble in daily life, yet it governs planets, tides and galaxies.
6. Step 6: Gravitational field strength and gGravitational field strength is defined as the gravitational force per unit mass at a point, equal to GM/r², in N kg⁻¹. At the Earth's surface it is about 9.8 N kg⁻¹ — the same number as the free-fall acceleration 9.8 m s⁻². For example, the Moon's much smaller mass gives a surface field of only about 1.6 N kg⁻¹: a 60 kg astronaut weighed about 588 N on Earth but only about 96 N on the Moon, which is why Apollo astronauts could hop so easily; weight changes with location, mass does not. The field weakens with height, so gravity on orbital altitudes is markedly smaller than at the surface.
7. Step 7: Satellite orbits — gravity as the centripetal forceFor a satellite in circular orbit, universal gravitation supplies exactly the centripetal force: GMm/r² = mv²/r; cancelling m gives the orbital speed v = √(GM/r), independent of the satellite's own mass. For example, a satellite skimming the Earth moves at about 7.9 km s⁻¹ and completes a lap in roughly an hour and a half; a geostationary satellite orbits far higher and slower with a period of exactly 24 hours, matching the Earth's rotation so it hangs over the same spot — the basis of communications satellites. In exams, write 'gravity provides the centripetal force', then set up the equation: this framework earns marks almost every year in long questions.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• In uniform circular motion the speed is constant but the velocity direction keeps changing, giving a centripetal acceleration towards the centre
• a = v²/r and F = mv²/r; the centripetal force is supplied by real forces — tension, friction or gravity — not a new force
• Universal gravitation: F = Gm₁m₂/r² with G = 6.67 × 10⁻¹¹ N m² kg⁻², where r is the centre-to-centre distance
• Gravitational field strength g = GM/r² is about 9.8 N kg⁻¹ at the Earth's surface and decreases with height; mass stays constant while weight varies
• In a circular satellite orbit, gravity is the centripetal force and v = √(GM/r); the higher the orbit, the smaller the speed and the longer the period