A conical pendulum is a mass on a string (or rod) that swings in a horizontal circle while the string traces out a cone, so the string makes a constant angle θ with the vertical. The horizontal component of tension supplies the centripetal force, while the vertical component balances gravity.
Picture a ball on a string, but instead of swinging back and forth, it whips around in a flat horizontal circle. The string sweeps out the surface of a cone, which is where the name comes from. The mass moves at constant speed, the angle θ between the string and the vertical stays fixed, and the radius of the circle is r = L sin θ.
The whole problem comes down to splitting the tension into components. The vertical component, T cos θ, holds the mass up against gravity, so T cos θ = mg. The horizontal component, T sin θ, points toward the center of the circle and acts as the net centripetal force, so T sin θ = mv²/r. Divide those two equations and the tension cancels, leaving the result the AP exam loves: a_c = g tan θ. That one line connects the geometry of the cone directly to the circular motion, no tension required.
The conical pendulum lives in Topic 2.10 (Circular Motion) in AP Physics C: Mechanics, and it's one of the cleanest tests of whether you actually understand uniform circular motion. The trap is treating 'centripetal force' as its own force you can add to a free-body diagram. It isn't. The centripetal force here is just the horizontal component of tension, and the conical pendulum forces you to prove you know that.
It also drills the core Unit 2 skill of resolving a single force into components that do two different jobs at once. Tension simultaneously cancels gravity (vertical) and bends the path into a circle (horizontal). The same component logic shows up in banked curves, so mastering the conical pendulum pays off twice on the exam.
Keep studying AP® Physics C: Mechanics Unit 2
Tension (Unit 2)
Tension is the only force besides gravity acting on the mass, and it does double duty. Its vertical component holds the mass up while its horizontal component is the entire centripetal force. If you can decompose tension correctly here, you've got the heart of the problem.
Banked curves (Unit 2)
A car on a frictionless banked curve is mathematically the same problem with tension swapped for the normal force. Both setups give a_c = g tan θ, so solving one means you've basically solved the other.
Orbital period (Units 2 and 7)
The conical pendulum has a period too, T = 2π√(L cos θ / g), found the same way you find an orbital period. You set the centripetal force equal to the real force causing it and solve for the time around. It's a preview of the gravitation playbook with tension standing in for gravity.
Tangential acceleration (Unit 2)
Because the speed is constant, the tangential acceleration of a conical pendulum is zero. All the acceleration is centripetal, pointing toward the center of the horizontal circle. Recognizing that is often worth a quick MCQ point.
The conical pendulum is a multiple-choice favorite. Typical stems give you mass, string length L, and angle θ, then ask for the centripetal acceleration, the speed, or the relationship tying v, L, g, and θ together. Practice questions ask things like finding the centripetal acceleration of a 2.0 kg mass on a 1.5 m string at 30° from vertical. The fastest route is almost always a_c = g tan θ, derived by dividing T sin θ = mv²/r by T cos θ = mg. No released FRQ has used the term verbatim, but the exam regularly asks free-response questions where one component of a force balances gravity while another provides centripetal force, and the conical pendulum is the template for that move. Two things to nail: draw the free-body diagram with only tension and weight (never a separate 'centripetal force' arrow), and use r = L sin θ, not r = L.
A simple pendulum swings back and forth in a vertical plane and (for small angles) is a simple harmonic motion problem with period 2π√(L/g). A conical pendulum sweeps a horizontal circle at constant speed and is a uniform circular motion problem solved with Newton's second law in components. Same string-and-mass setup, completely different physics. If the problem says 'horizontal circle' or 'constant angle with the vertical,' you're doing circular motion, not SHM.
In a conical pendulum, the vertical component of tension balances gravity (T cos θ = mg) and the horizontal component provides the centripetal force (T sin θ = mv²/r).
Dividing those two equations gives the most exam-useful result, a_c = g tan θ, with the tension canceling out entirely.
The radius of the circular path is r = L sin θ, the horizontal distance from the axis to the mass, not the full string length L.
Never draw 'centripetal force' as its own arrow on the free-body diagram; the only forces on the mass are tension and gravity.
Because the speed is constant, the tangential acceleration is zero and the net force points horizontally toward the center of the circle.
A conical pendulum is a uniform circular motion problem, not simple harmonic motion, so the simple-pendulum period formula 2π√(L/g) does not apply.
It's a mass on a string that moves in a horizontal circle at constant speed while the string sweeps out a cone at a fixed angle θ from the vertical. You analyze it with Newton's second law, where T cos θ = mg vertically and T sin θ = mv²/r horizontally.
No. The centripetal force is just the horizontal component of the string's tension, T sin θ. The only forces on the mass are tension and gravity, and adding a third 'centripetal force' arrow to a free-body diagram is a classic error that costs FRQ points.
A simple pendulum swings back and forth in a vertical plane and is treated as simple harmonic motion for small angles. A conical pendulum moves in a horizontal circle at constant speed, so it's solved as a circular motion problem with force components, and the SHM period formula 2π√(L/g) doesn't apply.
a_c = g tan θ, where θ is the angle the string makes with the vertical. You get it by dividing T sin θ = ma_c by T cos θ = mg, which makes the tension cancel.
No. The radius of the horizontal circle is r = L sin θ, the horizontal component of the string. Plugging in L instead of L sin θ is one of the most common mistakes on conical pendulum multiple-choice questions.
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