Circular Motion and Gravitation
Circular motion isn't a new force, just a new way of applying F=ma when the acceleration points toward a center instead of along a line -- the free-body-diagram skills from Unit 2 transfer directly. Gravitation extends that same idea to orbits: once students see that gravity *is* the centripetal force in an orbit rather than something separate, vertical circular motion, banked curves, and satellite motion all become variations on one recurring free-body diagram.
What you'll learn
- Define centripetal acceleration and relate it to speed and radius.
- Identify the real force(s) providing centripetal force in different scenarios.
- Apply Newton's law of universal gravitation between two masses.
- Analyze circular orbits by equating gravitational force to centripetal force.
- Relate orbital period, speed, and radius (Kepler's third law, qualitatively).
- Distinguish real centripetal force from the "centrifugal force" misconception.
- Solve vertical circular motion problems (loops, satellites).
1. Uniform Circular Motion: Period, Frequency, and Speed
An object in uniform circular motion travels at constant speed around a circle. The period \(T\) is the time for one complete revolution; the frequency \(f = 1/T\) is the number of revolutions per second. Speed relates to these through the circumference: \(v = \dfrac{2\pi r}{T} = 2\pi r f\).
"Uniform" refers only to speed being constant — the velocity vector is still changing every instant, since its direction constantly rotates. That changing direction is exactly what produces the centripetal acceleration discussed next.
2. Centripetal Acceleration and Centripetal Force
Even at constant speed, an object moving in a circle accelerates, because its velocity direction is always changing. This centripetal ("center-seeking") acceleration points toward the center of the circle and has magnitude \(a_c = v^2/r\).
Centripetal force is not a new, separate type of force — it is simply the name for whatever net force happens to point toward the center and cause the circular motion. On an exam, never add a "centripetal force" arrow to a free-body diagram alongside the real forces; instead, identify which real force (tension, gravity, friction, normal force) is supplying the center-pointing net force, then write \(\Sigma F_{\text{toward center}} = \dfrac{mv^2}{r}\).
3. Newton's Law of Universal Gravitation
Newton's insight was that the same force pulling an apple to the ground also holds the Moon in orbit: every pair of masses attracts each other with a force \(F_g = \dfrac{Gm_1 m_2}{r^2}\), where \(G \approx 6.67\times10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2\) and \(r\) is the distance between the objects' centers.
This is an inverse-square law: doubling the distance between two masses reduces the force to one quarter, not one half — a frequent point of confusion. Gravity between everyday-sized objects is also astonishingly weak; \(G\) is tiny, and it takes planet-sized masses for gravity to become a force you can feel.
4. Gravitational Field Strength and Orbital Speed
The gravitational field strength at distance \(r\) from a mass \(M\) is \(g = \dfrac{GM}{r^2}\) — this is the acceleration any small test mass would experience there, independent of its own mass (which is why all objects fall at the same rate near Earth's surface).
For a satellite in circular orbit, gravity supplies the entire centripetal force: \(\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}\). The orbiting mass \(m\) cancels completely, giving \(v = \sqrt{\dfrac{GM}{r}}\) — orbital speed depends only on the central mass and orbital radius, never on the satellite's own mass.
5. Kepler's Third Law
Combining \(F_g = F_c\) with \(v = 2\pi r/T\) shows that the orbital period squared is proportional to the orbital radius cubed for objects orbiting the same central body: \(T^2 \propto r^3\). This is Kepler's third law, and it means outer orbits have disproportionately longer periods — doubling the radius doesn't double the period, it multiplies it by \(2^{3/2} \approx 2.83\).
This relationship is especially useful for comparing two orbits around the same planet or star without needing to know \(G\) or the central mass at all.
6. Vertical Circular Motion
When circular motion happens in a vertical plane, gravity contributes to (or opposes) the centripetal force depending on where the object is in the loop. At the top of a loop, both gravity and the normal force point toward the center (downward), so \(N + mg = \dfrac{mv^2}{r}\). The minimum speed to maintain contact occurs when \(N = 0\): \(v_{\min} = \sqrt{gr}\).
At the bottom of a loop (or dip), gravity points away from the center while the normal force points toward it, so \(N - mg = \dfrac{mv^2}{r} \Rightarrow N = mg + \dfrac{mv^2}{r}\) — this is why you feel pressed into your seat at the bottom of a dip or a roller-coaster valley: the normal force is greater than your weight.
7. Banked Curves
On a frictionless banked curve, the normal force is tilted, so it has both a vertical component (supporting the object's weight) and a horizontal component (supplying the centripetal force). Resolving: \(N\cos\theta = mg\) (vertical) and \(N\sin\theta = \dfrac{mv^2}{r}\) (horizontal, toward the center).
Dividing the second equation by the first eliminates both \(N\) and \(m\): \(\tan\theta = \dfrac{v^2}{rg}\). This is the "design speed" for a banked curve — the one speed at which a car can round the curve with zero reliance on friction. Real curves are banked for a specific typical speed but still rely on some friction to accommodate cars going faster or slower than that design speed.
Key equations
- ac = v^2/r — Any uniform circular motion problem
- Fc = mv^2/r — Relating force to circular motion
- Fg = Gm1m2/r^2 — Gravitational force between two masses
- g = GM/r^2 — Gravitational field strength
- T = 2πr/v — Relating period to speed and radius