Theme A — Space, Time and Motion
A.2.19 Non-Uniform Circular Motion — IB Physics HL Revision Notes & Topical Questions
5 min read 11 parts Lesson 30 of 63 1 quiz
Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution
pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
Word Equation & Conditions: الرقم الهيدروجيني = -لوغاريتم تركيز أيون الهيدروجين | pH + pOH = 14
Location & Specifications: الاتزان الأيوني، كيمياء 3 ثانوي، التحصيلي، و AP Chemistry.
Lesson summary
A.2.19 Non-Uniform Circular Motion — IB PHYSICS SL notes: Analyse circular motion in which the speed changes, including vertical circles, using radial and tangential components.Non-uniform circular motion = circular motion in which the speed changes because the resultant force ch…
- Radial (centripetal) component
- Total acceleration
- Newton's second law
- Tension changes
- Maximum at the bottom
- Minimum at the top
Section 1 of 11Uniform and non-uniform circular motion
Uniform and non-uniform circular motion
Non-uniform circular motion
Non-uniform circular motion circular motion in which the speed changes because the resultant force changes.
Vertical circle
- Vertical circle a ball swung on a string in a vertical circle is the standard example.
Weight
- Weight it always points downwards, so the resultant force varies with position.
Section 2 of 11Two components of acceleration
Two components of acceleration
Radial (centripetal) component
- Radial (centripetal) component ar = v2/r, directed to the centre; it accounts for the change in direction.
Tangential component
- Tangential component at = Δv/Δt, along or against the motion; it accounts for the change in speed.
Total acceleration
- Total acceleration its magnitude is √(ar2 + at2), and it points to the centre only if at = 0.
Newton's second law
- Newton's second law apply F = ma separately along the radius (net force mv2/r) and along the tangent (net force mat).
Work
- Work a tangential force changes the speed and does work, unlike a purely centripetal force.
Section 3 of 11Forces in a vertical circle
Tension changes
- Tension changes the direction and size of the tension vary continuously.
Maximum at the bottom
- Maximum at the bottom the tension must overcome the weight, Tmax = mv2/r + mg.
Minimum at the top
- Minimum at the top the tension and weight act in the same direction, Tmin = mv2/r − mg.
Speed
- Speed the ball is slower at the top and faster at the bottom.
Sides of the circle
- Sides of the circle the weight is tangential, so it changes the speed but supplies no centripetal force.
Section 4 of 11Condition at the top
Minimum speed at the top
- Minimum speed at the top the tension is zero, so mg = mv2/r and v = √(gr).
Bucket of water
- Bucket of water this speed is needed to keep the water from spilling.
Taut string
- Taut string it stays taut at the top if vt2 ≥ gr.
Section 5 of 11Energy in a vertical circle
1Conservation
- Conservation with only gravity and tension doing work, mechanical energy is conserved.
2Bottom to top
- ½mvb2 = ½mvt2 + mg(2r), so vb2 = vt2 + 4gr.
3Minimum bottom speed
- Minimum bottom speed vb = √(5gr) to just complete the circle.
4Tensions then
- Tensions then zero at the top and 6mg at the bottom.
Section 6 of 11Angle-dependent relations
Angle-dependent relations
Angle θ from the downward vertical
- Angle θ from the downward vertical along the string, T − mg cos θ = mv2/r.
Tangential acceleration
- Tangential acceleration the weight component mg sin θ gives at = −g sin θ.
Speed
- Speed v2 = u2 − 2gr(1 − cos θ) for launch speed u at the bottom.
Tension
- Tension it is largest at the bottom and falls as the mass rises.
Hump or dip
- Hump or dip a car over a hump or through a dip follows the same ideas, with the normal force changing from point to point.
Section 7 of 11Other examples
Other examples
Examples
- Examples a car speeding up round a bend, a roller-coaster loop, a pendulum bob, a ball in a bowl and a spinning-up object.
Car accelerating in a bend
- Car accelerating in a bend friction needs a tangential part to speed it up and a radial part to hold the curve.
- The total friction cannot exceed μR, so skidding can occur below the steady-motion limit.
Elliptical orbit
- Elliptical orbit a satellite changes speed and has a tangential component because gravity is not perpendicular to the velocity.
Section 8 of 11Method
1
Decide which positions matter and sketch the forces at each.
2
Apply Newton's second law along the radius to find the force or tension.
3
Use conservation of energy to link speeds at different heights.
4
Check conditions such as a taut string or contact with a track.
5
Use F = mat along the tangent if the change of speed is needed.
Section 9 of 11More from the specification
1Non-Uniform Circular Motion Dynamics
- When an object experiences both radial centripetal acceleration and tangential acceleration, its angular velocity is not constant,
- causing tangential linear speed and momentum to vary continuously.
2Tangential Force and Rate of Change of Momentum
- The tangential component of net force acting on the body changes the magnitude of linear momentum according to Newton's equation Ft = m * at = dp / dt.
3Linear Impulse Definition
- The impulse of a force acting over a time interval equals the change in momentum produced, measured in N * s or kg * m / s.
- (Impulse = integral F dt = Delta p)
4Energy and Momentum in Collisions
- During a collision, total momentum is conserved in all directions,
- whereas total kinetic energy is conserved only in perfectly elastic collisions, with equal action-reaction impulses exchanged between colliding bodies.
5Static friction
- Friction between surfaces at rest relative to each other; fs ≤ μsFN.
6Kinetic energy
- K = ½mv2, energy of motion.
7Angular speed
- Magnitude of the angular velocity.
8Normal reaction
- The contact force perpendicular to a surface.
9Reaction force
- Force from a support pushing up on an object.
10Fluid resistance
- Drag force from a fluid opposing motion.
11Momentum
- P = mv, a vector.
12Constant
- Not changing.
13Collision
- Short interaction with large internal forces.
14Linear
- A graph that is a straight line through the origin, so resistance is constant.
15Impulse
- J = FΔt = Δp.What the symbols mean: F = force (N) · Δt = time the force acts (s) · Δp = change in momentum (kg m s−1); FΔt is the impulse (N s)
16Buoyancy
- Another name for upthrust.
Section 10 of 11Key terms
| Term | Meaning |
|---|---|
| Non-uniform circular motion | Circular motion in which the speed changes |
| Radial acceleration | Component of acceleration towards the centre, v2/r |
| Tangential acceleration | Component of acceleration along the path, Δv/Δt |
| Vertical circle | Circular motion in a vertical plane, in which gravity does work |
| Minimum speed at top | Speed v = √(gr) at which the tension at the top is zero |
| Taut string | String that remains under tension |
Section 11 of 11Exam technique
1
Assuming the tension is constant around the circle — it varies with position.
2
Thinking the tension is largest at the top — it is largest at the bottom.
3
Forgetting weight in the radial equation — T + mg at the top, T − mg at the bottom.
4
Saying the minimum speed at the top is zero — it is √(gr).
5
Treating the total acceleration as always pointing to the centre — it does only when the speed is constant.
Marked practiceQuizzes for this lesson
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Physics 0625 & 9702 SUVAT Mastery
Graph Interpretation
Constant Acceleration Equations (SUVAT) & Slopes
v = u + at | s = ut + ½at2 | v2 = u2 + 2as | s = ½(u + v)t
- Displacement-Time Graph: Gradient = Velocity (curve = accelerating).
- Velocity-Time Graph: Gradient = Acceleration; Area under curve = Distance travelled / Displacement.
- Free Fall in vacuum:
a = g ≈ 9.81 m/s2downwards (air resistance negligible).