Circular Motion and Gravitation — AP Physics 1
1. Centripetal Acceleration ★★☆☆☆ ⏱ 4 min
When an object moves at constant speed along a circular path, its velocity changes direction even if speed stays constant. Since acceleration is any change in velocity, the object accelerates toward the center of the circle, called centripetal (center-seeking) acceleration.
a_c = \frac{v^2}{r} = \omega^2 r
Where $a_c$ is centripetal acceleration (m/s²), $v$ is tangential speed (m/s), $r$ is radius of the circular path (m), and $\omega$ is angular speed (rad/s), related by $v = \omega r$.
Exam tip: Examiners frequently test that centripetal acceleration is always perpendicular to tangential velocity, so it never changes the speed of the object, only its direction of motion.
2. Centripetal Force ★★☆☆☆ ⏱ 4 min
From Newton's second law ($F_{net} = ma$), centripetal acceleration requires a net force toward the center of the circle. This net force is called centripetal force. Crucially, centripetal force is not a new fundamental force: it is just the label for the sum of real physical forces pointing toward the center of the circle.
F_c = m a_c = \frac{mv^2}{r} = m\omega^2 r
Where $m$ is the mass of the moving object (kg), and all other variables match those used for centripetal acceleration.
3. Universal Gravitation and Kepler's Laws ★★★☆☆ ⏱ 6 min
Newton's universal law of gravitation unifies terrestrial gravity and celestial orbital motion, stating that every pair of point masses exerts an attractive gravitational force on each other.
F_g = G\frac{Mm}{r^2}
Where $G = 6.67 \times 10^{-11} \text{ N·m}^2/\text{kg}^2$ (universal gravitational constant), $M$ and $m$ are the two masses, and $r$ is the center-to-center distance between the masses.
This formula can be used to find the acceleration due to gravity $g$ on the surface of a planet: equate weight $mg$ to gravitational force, cancel $m$ to get $g = \frac{GM}{r^2}$, where $r$ is the planet's radius.
Kepler derived three empirical laws of planetary motion that are explained by Newton's law of gravitation, all tested on AP Physics 1:
- **Law of Ellipses**: All planets orbit the Sun in ellipses, with the Sun at one focus. Most orbits are nearly circular, so circular approximations work for AP problems.
- **Law of Equal Areas**: A line between an orbiting object and the central mass sweeps equal areas in equal time. Orbiting objects move faster closer to the central mass and slower farther away, due to conserved angular momentum.
- **Law of Periods**: The square of the orbital period $T$ is proportional to the cube of the semi-major axis $a$ of the orbit. For circular orbits, $a=r$, so $\frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3}$ for objects orbiting the same central mass.
4. Apparent Weight in Vertical Circular Motion ★★★☆☆ ⏱ 4 min
Your apparent weight is equal to the normal force exerted on you by a supporting surface, not your actual gravitational weight $mg$. For vertical circular motion (like roller coasters or Ferris wheels), the centripetal net force requirement changes the normal force, changing your apparent weight.
- **Top of the circular path**: Both gravity and normal force point toward the center. Net force: $mg + N = \frac{mv^2}{r}$, so $N = \frac{mv^2}{r} - mg$. If $v = \sqrt{gr}$, $N=0$, which gives the sensation of weightlessness, the minimum speed to stay on the track.
- **Bottom of the circular path**: Gravity points away from the center, normal force points toward the center. Net force: $N - mg = \frac{mv^2}{r}$, so $N = mg + \frac{mv^2}{r}$, making you feel heavier than your actual weight.
- **Orbiting astronauts**: Apparent weight is zero not because gravity is zero, but because gravity provides all the centripetal force for orbit, so there is no normal force acting on the astronaut.
Common Pitfalls
Why: Students mistakenly believe centripetal force is a new fundamental force added to other forces.
Why: Students forget $r$ is defined as the distance between the centers of mass of the two objects.
Why: Students associate all acceleration with changes in speed, not direction.
Why: Students see astronauts floating and assume gravity is zero in orbit.
Why: Students mix up the exponents when recalling the proportionality.