PHYS2250 Chap.4 Circular Motion and Non-inertial Frames
Circular Motion and Non-inertial Frames
Separate turning from speeding up
Curved motion has radial acceleration that changes direction and tangential acceleration that changes speed. Real forces must supply the inward resultant.
This chapter treats static friction as responsive rather than automatically limiting, develops banked-turn reasoning from components, and keeps inertial and rotating-frame descriptions separate so fictitious forces are never counted alongside an inertial analysis.
Reasoning route and exam method
A curved trajectory requires acceleration even when speed is constant, because velocity changes whenever its direction changes.
In uniform circular motion the acceleration points toward the centre and has magnitude v²/r or ω²r; this relationship belongs in the diagnostic checklist for the chapter. If speed also changes, a tangential component dv/dt is present; this is a useful checkpoint before any numerical substitution.
These perpendicular components answer different questions: the radial equation determines what bends the path, while the tangential equation determines what changes the speed; the same result should survive the relevant limiting-case test.
Calling the required inward resultant a new force named centripetal force hides the mechanism; identify the real forces whose radial components provide it; that distinction determines which quantities may be combined.
Static friction takes the value required to prevent slipping until its magnitude reaches μₛN; it is not automatically μₛN; this observation fixes the physical interpretation of the algebra.
Kinetic friction applies after relative sliding begins and has magnitude μₖN in the simple model; the statement supplies a direct check on the final result. On a level turning road, static friction can supply the radial resultant; this condition should remain visible beside the governing equation. On a banked road, the horizontal component of the normal reaction can share or replace that role.
Begin with force components, not a memorised banking formula, because the sign and limiting direction of friction depend on whether the vehicle would otherwise slide up or down the bank.
A 900 kg car turns on a level curve of radius 40 m where μₛ = 0.50. Vertically N = mg. The greatest available inward friction is μₛmg, so mv²/r ≤ μₛmg and v ≤ √(μₛgr). Substitution gives v ≤ √(0.50 × 9.8 × 40) = 14.0 m s⁻¹.
Mass cancels because both the required radial force and the available friction scale with mass. At 16 m s⁻¹ the required coefficient would be v²/(rg) = 0.653, beyond the stated limit.
Prompt. A mass moves in a vertical circle at the top of its path. Which direction is positive if the shortest equation is wanted?
Answer. No. It equals ma while no slip occurs, up to the maximum μₛN.
Equality holds only at impending slip.
FRAME CHOICE Rotating-frame discipline. Either analyse from an inertial frame using real forces, or use a rotating frame and add the appropriate fictitious forces. Mixing the two descriptions counts effects twice.
At the marked point the blue tangent follows the instantaneous velocity while the red radial acceleration points toward the centre.
Constant speed removes tangential acceleration but leaves the inward component intact.
Centripetal describes the direction of the resultant acceleration. It does not add another arrow to the force diagram.
What this chapter covers
- 01
radial acceleration
- 02
tangential acceleration
- 03
static and kinetic friction
- 04
banked motion
- 05
rotating frames
Worked example · free
Find a friction-limited turn speed
- 2Identify friction as the radial resultant.
- 2Apply its limiting value.
- 2Calculate and interpret the mass cancellation.
Key terms
- Radial Acceleration
- The inward acceleration associated with changing velocity direction on a curved path.
- Fictitious Force
- A term introduced in an accelerating frame so Newtonian form can be used there.
- Tangential Acceleration
- The acceleration component along a path that changes speed rather than velocity direction.
Circular Motion and Non-inertial Frames FAQ
Is static friction always equal to its maximum?
No. It adjusts to the value required to prevent slip and equals its maximum only at impending motion, where the direction of likely sliding is also identified.
Can speed be constant while acceleration is nonzero?
Yes. Uniform circular motion changes the direction of velocity continuously, producing inward acceleration at constant speed even though the tangential acceleration component remains zero throughout.
When should I add centrifugal force?
Only in a rotating reference frame. An inertial-frame analysis uses real forces and must not add the same frame effect again, because that would double-count acceleration.
Exam move
For every curved path draw radial and tangential axes at the point of interest. Project real forces and decide whether friction is below or at its limiting value. Practise at the top, side and bottom of a vertical circle so the inward direction changes on the page while the radial rule does not. On banked turns, first solve the frictionless design speed and then decide which way friction points above and below that speed.
Create paired solutions of one rotating-platform problem, once from the laboratory frame with real forces and once from the rotating frame with the required frame terms. Compare the final observable rather than mixing equations midway. Finally, test the limit as curvature vanishes: radial acceleration should tend to zero and the force balance should reduce to straight-line motion.
Test the impending-slip boundary from both sides so the friction direction follows the threatened relative motion rather than an inward or outward rule learned by rote.
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