Radial and tangential acceleration
You will be able to: Separate changes in direction from changes in speed and identify the real forces producing them.
How can constant speed still require a net force?
A toy car travels around a circle at steady speed. Its velocity keeps turning, so it accelerates toward the center even though its speedometer reading stays the same.
A useful starting point: Gravity, drag and terminal velocity →
Words and symbols before equations
- Radius r
- Distance from the circle center to the object, in m.
- Radial acceleration a_r
- Inward component v²/r, in m/s².
- Tangential acceleration a_t
- Rate of change of speed, dv/dt, along the path tangent.
- Period T and frequency f
- Time per revolution in s and revolutions per second in Hz; f = 1/T.
What this picture assumes
Instantaneous circular-motion snapshot at the rightmost point, moving upward. Acceleration arrows are not a free-body diagram. A nonzero tangential component changes speed.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- a_r = 9 m/s² inward, a_t = 2 m/s² along motion; |a| = 9.22 m/s². Speed is changing; the constant-speed period formula does not describe a full accelerating revolution.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
For uniform circular motion, write r(t) = r cos(ωt)î + r sin(ωt)ĵ with constant ω. Differentiating twice gives a = −ω²r(t). Since speed is v = rω, the inward magnitude is v²/r. The derivative of a turning vector can be nonzero even when its magnitude is fixed.
When speed changes, acceleration also has a tangent component a_t = dv/dt. Radial and tangential components are perpendicular, so |a| = √[(v²/r)² + a_t²]. The net force has the corresponding components ma_r and ma_t.
“Centripetal” describes the required inward net force component, not an extra physical force. Tension, friction, gravity or a combination supplies it. At constant speed T = 2πr/v and f = v/(2πr).
A worked example, step by step
A 2 kg object moves on a 4 m radius circle at 6 m/s, increasing speed at 2 m/s². Find radial acceleration and net force magnitude.
- Inward a_r = v²/r = 36/4 = 9 m/s².
- Tangential a_t = 2 m/s².
- Total acceleration magnitude = √(9² + 2²) = √85 ≈ 9.22 m/s².
- Net force magnitude = 2√85 ≈ 18.44 N, tilted away from purely inward toward the direction of speed increase.
Do not add a separate “centripetal force” arrow to a diagram that already includes the real interactions.
At constant speed is tangential acceleration zero?
Compare with an explanation
Yes, but radial acceleration remains v²/r as the velocity direction changes.
Predict. Change one thing. Explain.
Hold radius fixed and double speed. Predict radial acceleration. Then add a tangential acceleration and compare the total acceleration direction.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
a_r = 9 m/s² inward, a_t = 2 m/s² along motion; |a| = 9.22 m/s². Speed is changing; the constant-speed period formula does not describe a full accelerating revolution.
Instantaneous circular-motion snapshot at the rightmost point, moving upward. Acceleration arrows are not a free-body diagram. A nonzero tangential component changes speed.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant force, system boundary, acceleration or calculus relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA 1 kg object travels at 4 m/s on a 2 m radius circle with tangential acceleration 6 m/s². Find radial acceleration, total acceleration, net force magnitude and describe why velocity is not constant.
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Compare with the answer and four-point rubric
- 1 point: a_r = 16/2 = 8 m/s² inward.
- 1 point: |a| = √(8² + 6²) = 10 m/s².
- 1 point: Net force magnitude is 10 N.
- 1 point: Velocity changes both direction and speed here.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Does constant speed mean zero acceleration?
No, a curved path changes velocity direction.
RECALL 2What does centripetal mean?
Directed toward the center.
RECALL 3Which component changes speed?
Tangential acceleration.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Radial and tangential acceleration
- ΣF_inward = mv²/r.
- a_t = dv/dt; |a| = √(a_r² + a_t²).
- Uniform circular motion: T = 2πr/v and f = 1/T.
Remember: Do not add a separate “centripetal force” arrow to a diagram that already includes the real interactions.
Conditions: Instantaneous circular-motion snapshot at the rightmost point, moving upward. Acceleration arrows are not a free-body diagram. A nonzero tangential component changes speed.
Refresh Kid · AP Physics C: Mechanics Unit 2 (official Unit 2) · Objectives 2.10.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.10, objectives 2.10.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026 alongside the Fall 2026 clarifications. This is Mechanics Unit 2: Force and Translational Dynamics. The unit covers Topics 2.1–2.10. Calculus is introduced where it is needed for continuous mass and velocity-dependent forces. Shell theorem is applied without requiring a proof; spring combinations are purely series or purely parallel. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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