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Circular motion and gravitational fields (A Level)

12. Motion in a Circle

5 min read 7 parts Lesson 13 of 14 Unit quiz

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12. Motion in a Circle — Cambridge AS & A Level Physics 9702 — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Use radians, angular speed, centripetal acceleration and centripetal force to analyse objects moving in uniform circular motion.Radian = the angle subtended …

  • Angular displacement θ
  • Conversion
  • Calculator mode
  • Angular speed ω
  • Angular velocity
  • Full cycle

Section 1 of 7Radians and angular displacement

  • Radian = the angle subtended at the centre of a circle by an arc equal in length to the radius.
    • A full circle is 2πr ÷ r = 2π rad, which equals 360°.
    • Half a circle is π rad, a quarter is π/2 rad and three-quarters is 3π/2 rad.
  • Angular displacement θ = the angle, in radians, through which an object turns, θ = s/r.
    • s is the arc length (distance travelled around the circle) and r is the radius, in the same unit.
  • Conversion
    • radians = degrees × π/180 and degrees = radians × 180/π.
DegreesRadians
3602π
2703π/2
180π
90π/2
60π/3
  • Calculator mode
    • check Deg or Rad before using sin, cos or tan.

Section 2 of 7Angular speed

  • Angular speed ω = the rate of change of angular displacement, ω = Δθ/Δt, measured in rad s−1.
    • A greater rotation angle in a given time gives a greater angular speed.
  • Angular velocity
    • the same quantity as a vector, with a direction.
  • Full cycle
    • one revolution is 2π rad in a period T, so ω = 2π/T = 2πf.
  • Linear speed
    • v = ωr, where v is the speed along the circle.
      • (tangential speed)
    • For one angular speed, a point further from the centre moves faster along the circle.
    • Every point on a turning body has the same angular speed, but linear speed depends on r.

Section 3 of 7Why circular motion involves acceleration

  • Constant speed, changing velocity
    • velocity is a vector and its direction changes continually in a circle.
  • Acceleration
    • because the velocity changes, the object accelerates even though its speed is constant.
  • Direction of velocity
    • it is always along the tangent, perpendicular to the radius.

Section 4 of 7Centripetal force

  • Centripetal force = the resultant force towards the centre of the circle that keeps a body in uniform circular motion.
    • It is not a separate force but is supplied by other forces.
    • It acts perpendicular to the direction of motion, so it does no work and the speed does not change.
  • Direction
    • the centripetal force and the centripetal acceleration both point towards the centre.
SituationForce providing the centripetal force
Car on a roundaboutFriction between tyres and road
Ball on a stringTension in the string
Earth orbiting the SunGravitational attraction

Section 5 of 7Centripetal acceleration

  • Centripetal acceleration = the acceleration towards the centre of the circle of an object moving at constant speed.
  • Equations
    • a = v2/r,
      • and with v = ωr this becomes a = ω2r.
        What the symbols mean: a = centripetal acceleration towards the centre (m s−2) · v = speed (m s−1) · ω = angular speed (rad s−1) · r = radius (m)
    • a is in m s−2, v in m s−1, ω in rad s−1 and r in metres.
  • Speed
    • doubling the speed at a fixed radius quadruples the acceleration.

Section 6 of 7Centripetal force equations

  • Newton's second law
    • F = ma gives the centripetal force F = mv2/r.
  • Angular form
    • substituting v = ωr gives F = mω2r.
  • Units
    • F in newtons, m in kilograms, v in m s−1, r in metres and ω in rad s−1.
Quantity changedEffect on F
Speed doubledF × 4
Radius doubled at the same speedF ÷ 2
Mass doubledF × 2
Angular speed doubled at the same rF × 4

Section 7 of 7Solving circular-motion problems

  1. Identify the forces pointing towards the centre; their resultant is the centripetal force.
  2. Choose the equation that fits the data: F mv2/r, F = mω2r, a = v2/r or a = ω2r.
    What the symbols mean: F = centripetal force towards the centre (N) · m = mass (kg) · v = speed (m s−1) · ω = angular speed (rad s−1) · r = radius (m)
  3. Convert frequency or period to ω using ω 2πf or ω = 2π/T before substituting.
  • Sign check
    • a force or acceleration pointing away from the centre signals an error.
  • Vertical circle
    • at the top, weight points towards the centre, so the minimum speed occurs when mg alone supplies the centripetal force.
    • mg = mv2/r, giving v = √(gr).

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