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7.1Circular motion and gravitational fields (A Level)

Motion IN A Circle

5 min read 9 parts Lesson 1 of 14 1 quiz

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Lesson summary

7.1 Motion in a Circle — AS PHYSICS 9702 notes: Express angular displacement in radians, use angular speed and explain uniform circular motion with centripetal acceleration and force.Radian — the angle subtended at the centre of a circle by an arc equal in length to the radius.Eq…

  • Full circle
  • Conversion
  • Calculator
  • Definition
  • Uniform motion
  • What the symbols mean

Section 1 of 9Radians and angular displacement

Radian
  • Radian the angle subtended at the centre of a circle by an arc equal in length to the radius.
Equation
  • θ = s/r, with arc length s and radius r in the same units.
Full circle
  • 2π rad = 360°.
Conversion
  • Conversion radians = degrees × π/180.
Angular displacement
  • Angular displacement the change in angle, in radians, of a body as it rotates around a circle.
Calculator
  • Calculator it must be in the correct degree or radian mode.

Section 2 of 9Angular speed

Definition
  • Definition the rate of change of angular displacement, ω = Δθ/Δt, in rad s−1.
Uniform motion
  • ω = 2π/T = 2πf.
    What the symbols mean: ω = angular frequency (rad s−1) · f = frequency (Hz) · T = period (s)
Linear speed
  • Linear speed v = rω.
Rigid body
  • Rigid body all points share the same ω, but v grows with distance from the axis.
Angular velocity
  • Angular velocity the same as angular speed, but a vector.

Section 3 of 9Why circular motion involves acceleration

Velocity
  • Velocity a vector, so changing direction means changing velocity.
Constant speed
  • Constant speed the object still accelerates because its direction constantly changes.
Force and speed
  • Force and speed a force of constant magnitude always perpendicular to the velocity changes the direction but not the speed.
Work
  • Work the force does no work, so the kinetic energy is constant.

Section 4 of 9Centripetal acceleration and force

Acceleration
  • Acceleration a = v2/r = rω2, directed towards the centre.
    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)
Force
  • F = ma gives F = mv2/r = mrω2.
Meaning of centripetal
  • Meaning of centripetal it describes the direction of the resultant force towards the centre.
Supplied by real forces
  • Supplied by real forces tension, friction, weight, normal contact or gravity.

Section 5 of 9Forces providing the centripetal force

SituationForce towards the centre
Planet orbiting the SunGravitational force
Car on a level bendFriction
Mass whirled on a string in a horizontal circleTension
Electron in a magnetic fieldMagnetic force
Conical pendulumHorizontal component of tension

Section 6 of 9Horizontal and vertical circles

Car on a level bend
  • Car on a level bend friction supplies the centripetal force.
Conical pendulum
  • Conical pendulum the horizontal component of tension supplies it while the vertical component balances the weight.
Vertical circle
  • Vertical circle speed and contact forces vary round the circle.
  • At the top, mg + N = mv2/r.
  • At the bottom, N − mg = mv2/r.
  • The minimum speed at the top to keep contact is v = √(gr).

Section 7 of 9Solving circular motion problems

1
Draw a free-body diagram of the object.
2
Identify the direction towards the centre of the circle.
3
Resolve the forces along that direction.
4
Set the resultant towards the centre equal to mv2/r or mrω2.
5
Solve, checking that ω is in rad s−1.

Section 8 of 9Key terms

TermMeaning
RadianAngle subtended at the centre of a circle by an arc equal in length to the radius.
Angular displacementAngle through which a body turns about the centre of its circular path.
Angular speedRate of change of angular displacement.
Centripetal accelerationAcceleration of a body in circular motion, directed towards the centre.
Centripetal forceResultant force towards the centre of the circle causing circular motion.
PeriodTime for one complete revolution.

Section 9 of 9Exam technique

1
Leaving angles in degrees in θ = s/r — angles must be in radians.
2
Calling centripetal force a separate extra force — it is the resultant provided by real forces.
3
Thinking constant speed means no acceleration — direction changes.
4
Saying the centripetal force does work — it is perpendicular to the motion, so it does none.
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/s2 downwards (air resistance negligible).