01The mistake
On a free-body diagram for an object in circular motion, students draw "centripetal force" as a separate arrow alongside the real forces. The FBD for a ball on a string ends up with gravity, normal, tension, and a fifth arrow labeled "centripetal," all coexisting. Newton's second law in the radial direction is then mishandled: students sum tension and centripetal together, or count centripetal twice, or treat it as the cause of the radial net force rather than the name for it.
The clearest version of the failure: a puck tied by a string to a peg on a frictionless table, moving in a horizontal circle. The student lists tension, normal, gravity, and centripetal force as the forces on the puck, then writes $T + F_c = ma_c$. The same tension that's supposed to be doing the centripetal job is now sharing the work with a phantom partner, and the equation is wrong.
It's distinct from the centrifugal-force misconception (U2-PH14), where students draw an outward "force" pushing the object away from the center. Centripetal-as-extra-force points the right direction (inward); the failure is the existence of the arrow, not its sign. The two misconceptions can co-exist, and the platform tracks them as separate codes for that reason.
02Why it makes sense to the student
The textbook says "the centripetal force is $F_c = \dfrac{mv^2}{r}$, directed toward the center." That sentence sounds exactly like every other force law the student has learned. $F_g = mg$. $F_s = kx$. $f_k = \mu_k F_N$. Each one is a recipe for an arrow on an FBD. So $F_c = \dfrac{mv^2}{r}$ becomes another arrow on an FBD. The pattern of the chapter trains the student to treat new force laws as new diagram entries.
Worse, the word "force" is in the name. "Centripetal force" means a force, in the same way that "gravitational force" means gravity. A student who sees the word "force" in a name has no syntactic reason to treat it differently from the other forces in the chapter. The semantic load of "this isn't a new force, it's the role played by existing forces" is carried entirely by classroom language and the prefatory paragraph in the textbook, which the student often skips.
Reinforcing this: in early circular-motion problems, the answer often involves setting tension equal to $\dfrac{mv^2}{r}$. The student who treated centripetal as an extra force, then equated it with tension to "make the problem work," gets the right number. The misconception is rewarded by the canonical problem set. It only breaks when the geometry gets harder (vertical loops, banked curves, conical pendulums) and the student needs to identify which components of which real forces are doing the centripetal job.
Searle (1983) and Reif and Allen (1992) document the persistence of this category of failure. diSessa's phenomenological-primitives framework adds a useful frame: the student has a "going-by" intuition (something has to push the object around the circle) and the textbook's "centripetal force" gives that intuition a name and a formula, when the right move would have been to replace the intuition with the slower question "which real force is doing this?"
03The correction
Centripetal force is not a real force. It is a label for the net radial force, whatever combination of real forces happens to point toward the center in a given problem. It belongs on the right side of Newton's second law, never as an arrow on the FBD.
The structure of the equation makes this explicit. Newton's second law in the radial direction reads
$$F_{net,r} = m a_r = \dfrac{m v^2}{r}$$
The left side is the sum of the radial components of the real forces on the object. The right side is what we call "the centripetal requirement." The problem is to figure out which real forces (tension, gravity, normal, friction, spring, applied) contribute to the left side, and then set their sum equal to $\dfrac{mv^2}{r}$. The right side is not a force; it's the radial component of $ma$.
The diagnostic question to ask whenever a student names "centripetal force" on an FBD: "What's the agent? What is providing this force? What is it attached to?" If the answer is "a string," the force is tension and tension belongs on the FBD; centripetal does not. If the answer is "gravity," the force is gravity and gravity belongs on the FBD; centripetal does not. If the student can't name an agent, the arrow is a phantom.
Worked through one scenario: a puck on a frictionless horizontal surface, tied to a peg, moving in a circle. The forces on the puck are gravity (down), the normal force from the surface (up), and tension from the string (toward the peg). That's it. Three arrows on the FBD. Newton's second law in the vertical direction gives $F_N = mg$. In the radial direction, $T = \dfrac{m v^2}{r}$. The tension is the centripetal force. They are the same arrow, named twice for two different purposes (its identity as a real force, and its role as the net radial force). The arrow appears on the FBD once.
04A sample question
A puck of mass $m$ on a frictionless horizontal surface is tied by a string to a fixed peg. The puck moves in a circle of radius $r$ around the peg at constant speed $v$. Which of the following lists all forces acting on the puck?
- AGravity (down), normal force (up), string tension (toward peg), centripetal force (toward peg).
- BGravity (down), normal force (up), string tension (toward peg).
- CGravity (down), normal force (up), centripetal force (toward peg).
- DGravity (down), normal force (up), string tension (toward peg), centrifugal force (away from peg).
05What each wrong answer reveals
- A The target misconception. Student lists every real force correctly and then adds centripetal as an additional FBD entry. The arrow is a phantom, but the rest of the diagram is right. This is the tell that the student is treating "centripetal force" as another force law to draw, in parallel with tension and gravity. The remediation is the agent question: "what is providing the centripetal force?"
- B Correct. Three real forces: gravity, normal, tension. No phantom centripetal. Tension is doing the centripetal job, but it appears on the diagram as tension, not as centripetal force.
- C Replaced the agent. Student knows something is pulling the puck toward the center but identifies it as centripetal force directly, skipping the string. They have the directional intuition right and the phantom-force habit, but they've also forgotten that there's a real tension-providing string in the scenario. Subtler than A and worth flagging separately, because the agent-naming question hits this differently (the student needs to name the string, not just defend the diagram).
- D Centrifugal version. U2-PH14 territory. The student lists the right real forces and adds an outward "centrifugal" force, often with a story about the puck "wanting to fly outward." This is a separate code in the platform with its own remediation, but it shares the underlying FBD-construction failure with A: a phantom force has been added to make the diagram match an intuition. If a student hits A and D in the same section, both codes go active.
A and C both have the centripetal phantom, but they differ on whether the real agent (tension) is also present. A is the "additional arrow" failure; C is the "replaced the real force with a phantom" failure. Both clear with the same agent-naming intervention, but C also requires putting the missing tension back, which is a slightly heavier lift. D is a separate misconception entirely that just happens to live next door.
06Try it in Mistake Master
Topic 2.9 (Circular Motion) is where U2-PH15 is drilled directly. Items there force the student to identify which real force is providing the centripetal requirement before writing Newton's second law in the radial direction. The centrifugal version (U2-PH14) lives in the same topic but is tracked separately, with its own distractor design and its own remediation, so a student who clears one and not the other ends up with a targeted re-drill on whichever phantom is still active. Vertical-loop and banked-curve items re-check the failure under harder geometry, where the centripetal job is shared across components of multiple real forces.