The radius (r) is the distance from the center of a circle or sphere to any point on its edge, equal to half the diameter. In AP Physics 1, r sets the size of a circular path and appears in centripetal acceleration (a = v²/r), gravitational field strength (g = GM/r²), torque, and angular momentum.
Geometrically, the radius is simple. It is the distance from the center of a circle or sphere out to the edge, and it is half the diameter. In AP Physics 1, though, radius is rarely just a shape measurement. It is the variable that controls almost everything about rotational and circular motion.
The same letter r shows up wearing different hats. In circular motion, r is the size of the path, and it sits in the denominator of a = v²/r. In gravitation, r is the distance from the center of a planet to the object, and it gets squared in g = GM/r². In rotation, r connects angular and linear quantities through v = rω, acts as the lever arm in torque, and appears in angular momentum (L = mvr for a point mass). Every time you see r, your first job is to identify what center it is measured from and what role it is playing in that equation.
Radius threads through Topics 3.4 (Gravitational Field/Acceleration Due to Gravity on Different Planets), 3.6 (Centripetal Acceleration and Centripetal Force), 7.3 (Angular Momentum and Torque), and 7.4 (Conservation of Angular Momentum). It also feeds energy reasoning under AP Physics 1 Revised 3.4.B, because gravitational potential energy and circular-motion kinetic energy both depend on how far an object sits from the center of its path or its planet. The exam loves r because it tests proportional reasoning. Double the orbital radius and gravitational field strength drops to one quarter. Double the radius of a circular path at the same speed and centripetal acceleration halves. If you can track how r scales through an equation, you can answer a huge slice of MCQs without a calculator.
Keep studying AP Physics 1 Unit Udr75ggSrboDhyB7
Centripetal Acceleration and Centripetal Force (Unit 3)
In a = v²/r, the radius tells you how tight the turn is. A bigger radius at the same speed means a gentler curve and less acceleration toward the center. Watch out though, because if angular speed ω is what stays constant instead, then a = ω²r and a bigger radius means MORE acceleration.
Acceleration due to gravity (Unit 3)
In g = GM/r², the radius is the distance from the planet's center, and it gets squared. This is the inverse-square law in action. An object orbiting at twice Earth's radius feels one quarter the gravitational field. The 2018 FRQ about a spacecraft in a circular orbit of radius R is exactly this setup.
Angular Momentum and Torque (Unit 7)
Radius is the lever arm. Torque grows with the distance from the axis where the force is applied, and angular momentum of a point mass is L = mvr. This is why an ice skater spins faster when she pulls her arms in. Shrinking r forces ω up to keep L conserved (Topic 7.4).
Linear Velocity and Angular Velocity (Unit 7)
The equation v = rω is the bridge between rotational and linear motion. Two pulleys glued together on the same axle share one ω, but the point on the bigger pulley moves faster because its r is bigger. The 2021 short FRQ about two pulleys with different radii tests exactly this idea.
Radius shows up constantly in released FRQs. The 2018 short answer put a spacecraft in a circular orbit of radius R around Earth and asked you to reason about its motion. The 2021 short FRQ attached two pulleys with different radii to a common axle, forcing you to use the fact that they share angular speed but not linear speed. The 2023 long FRQ spun a spring-mounted block in a circle, where the radius of the path itself changes as the spring stretches. In MCQs, expect proportional-reasoning stems like "if the radius doubles, the centripetal force becomes..." Your jobs are to (1) state what point r is measured from, (2) decide whether v or ω is held constant before predicting how a or F changes with r, and (3) square r correctly in gravitational field problems.
In gravitation and orbit problems, r is the distance from the planet's CENTER, not from the surface. A satellite at an altitude of one Earth radius is actually at r = 2R_Earth, so it feels g/4, not g/2. Plugging altitude into g = GM/r² instead of center-to-center distance is one of the most common point-killers on orbit questions.
Radius is the distance from the center of a circle or sphere to its edge, and it equals half the diameter.
In gravitation, r is always measured from the planet's center, and field strength falls off as 1/r², so doubling the distance cuts g to one quarter.
In centripetal motion, a = v²/r means a larger radius at constant speed gives a smaller acceleration, but a = ω²r means a larger radius at constant angular speed gives a larger one.
The equation v = rω links rotation to linear motion, so points farther from the axis move faster even though everything shares the same angular velocity.
Radius acts as the lever arm in torque and appears in angular momentum (L = mvr), which is why pulling mass closer to the axis makes a spinning object speed up.
It is the distance from the center of a circle or sphere to any point on its edge, half the diameter. On the exam it appears as the size of a circular path (a = v²/r), the distance from a planet's center (g = GM/r²), and the lever arm in torque and angular momentum.
No. Orbital radius is measured from the planet's center, so r equals the planet's radius plus the altitude. A satellite one Earth radius above the surface sits at r = 2R_Earth and experiences g/4.
No, it depends on what is held constant. With constant speed v, force shrinks as r grows (F = mv²/r). With constant angular speed ω, force grows with r (F = mω²r). Always check which quantity the problem fixes.
The diameter is the full distance across a circle through its center, and the radius is half of that. AP Physics 1 equations almost always use the radius, so if a problem hands you a diameter, divide by 2 before plugging in.
They rotate together, so they share one angular speed ω, but linear speed is v = rω. The edge of the larger pulley moves faster because its r is bigger. The 2021 short FRQ tested exactly this with two pulleys of different radii on a common axle.