Circular Motion and Gravitation

Driving question

How can an object accelerate continuously while its speed remains constant, and what interaction keeps an orbit curved?

Before learning: direction prediction

A ball moves counterclockwise around a circle at constant speed. At the rightmost point, draw its velocity and acceleration vectors. Then list the actual forces that might produce the acceleration.

Hook: constant speed can still mean acceleration

In uniform circular motion, velocity changes direction every instant. Acceleration points toward the circle's center even when speed is constant.

Circular kinematics

For radius $r$ and speed $v$,

With period $T$,

Velocity is tangent to the path; radial acceleration points inward and is perpendicular to velocity in uniform circular motion.

Worked example: rotating platform

A marker moves in a circle of radius $0.80\text{ m}$ with period $2.0\text{ s}$:

“Centripetal force” is the inward net force

Newton's second law radially is

Do not add a new force labeled “centripetal” to a free-body diagram. Tension, gravity, friction, normal force, or a combination supplies the inward net force.

Level curve

For a car turning on a flat road without slipping, static friction points toward the center:

Since $f_s\le\mu_smg$, the ideal maximum speed is $v_{max}=\sqrt{\mu_sgr}$.

Vertical circle

At the bottom of a vertical circle, inward is upward:

At the top, inward is downward. Radial signs must follow the local inward direction.

Universal gravitation

Two pointlike or spherically symmetric masses attract with magnitude

$r$ is center-to-center separation. The gravitational field due to mass $M$ is

Doubling distance from the center reduces field magnitude to one-fourth.

Circular orbits

For a small satellite in a circular orbit around mass $M$, gravity supplies radial net force:

Thus

Satellite mass cancels. Higher circular orbits have lower speed but longer period.

Worked comparison

If orbital radius becomes four times larger around the same central mass, speed becomes one-half and period becomes $4^{3/2}=8$ times larger.

Apparent weight and weightlessness

Scale reading is usually a normal force, not gravitational force. In orbit, astronauts and spacecraft are both freely falling; gravity is not zero. Their apparent weight is near zero because support force is near zero.

During learning: radial audit

1. Mark the circle center and local inward direction.
2. Draw only real external forces.
3. Resolve the inward components and write $\sum F_r=mv^2/r$.
4. Keep tangential and radial equations separate.
5. For gravity use center-to-center radius and state spherical/point-mass assumptions.

Misconception clinic

“Constant speed means no acceleration.” Direction change produces radial acceleration.

“Centripetal force is an extra force.” It names the inward net force requirement.

“Circular motion needs an outward force.” In an inertial frame, the net force is inward.

“Astronauts float because gravity is absent.” They float because spacecraft and occupants fall together.

“Higher orbit means faster satellite.” Higher circular orbit means lower speed and longer period.

After learning: retrieval and transfer

1. How are velocity and acceleration oriented in uniform circular motion?
2. What real force turns a car on a flat road?
3. Why must center-to-center distance be used in gravitation?
4. How do circular orbital speed and period change with radius?
5. Why does an orbiting scale read nearly zero?

AP-style evidence routine

1. Define object/system, center, radius, and instant.
2. Draw a free-body diagram with real forces only.
3. Choose inward-positive radial and tangential axes.
4. Apply Newton's second law by direction.
5. State gravity/orbit assumptions.
6. Check dimensions, limits, directions, and physical contact constraints.

Key takeaway

Circular motion requires inward acceleration supplied by real interactions. Gravity can provide that inward net force, producing continuous free fall around a central body.

Further learning and alignment

• College Board, “AP Physics 1: Algebra-Based Course,” https://apcentral.collegeboard.org/courses/ap-physics-1 (retrieved 2026-08-19). Alignment only; no endorsement.
• OpenStax, “6.2 Uniform Circular Motion” and “6.3 Centripetal Force,” https://openstax.org/books/physics/pages/6-2-uniform-circular-motion and https://openstax.org/books/physics/pages/6-3-centripetal-force (retrieved 2026-08-19). CC BY verification; no exercises copied.
• NASA Glenn Research Center, “The Mass of the Earth,” https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/mass-of-the-earth/ (retrieved 2026-08-19). Government gravitation/orbit reference; no media reused.