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2.8.2Force and Translational Dynamics

Acceleration in Circular Motion — AP PHYSICS 1 Revision Notes & Topical Questions

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2.8.2 Acceleration in Circular Motion — AP PHYSICS 1 notes: Use ac = v2/r and Fnet = mv2/r for centripetal acceleration and force, including tangential and net acceleration.ac = v2/r — magnitude of the acceleration component toward the centre.ac in m/s2, v tangential speed in m/s…

  • Fc = mac = mv2/r
  • Single force
  • More than one force
  • Components
  • Tangential acceleration
  • Circular orbit

Section 1 of 10Centripetal acceleration

ac
ac v2/r
  • magnitude of the acceleration component toward the centre.
  • ac in m/s2, v tangential speed in m/s, r radius in m.
  • With v = 2πr/T the same acceleration is 4π2r/T2.
Direction
  • Direction always toward the centre along the radius, perpendicular to the velocity.
Effect
  • Effect it changes only the direction of motion.

Section 2 of 10Centripetal force

Centripetal force
Fc
Fc mac = mv2/r
  • directed toward the centre, in the direction of the acceleration.
Source
  • Source it can be one force, several forces or components of forces.
Single force
  • Single force tension on a ball in a horizontal circle;
    • gravity on the Moon;
    • friction on a turning car;
    • electrostatic force on an electron.
More than one force
  • More than one force tension and gravity in a vertical circle;
    • gravity and normal force on a loop-the-loop carriage.
Components
  • Components static friction and weight components on a banked curve; tension component in a conical pendulum.

Section 3 of 10Solving radial problems

1
Draw a free-body diagram and take inward as positive.
2
Add force components toward the centre and subtract those away from it.
3
Set the sum equal to mv2/r.
4
Forces perpendicular to the plane of the circle cancel.

Section 4 of 10Scaling relationships

ChangeEffect on ac and Fc
Double speed, same rFour times larger
Double radius, same vHalved
Double massF doubles, ac unchanged
Double period, same rac one quarter
Graphs
  • F against v2 is a line through the origin with gradient m/r; F against 1/r is linear with gradient mv2.
Safety
  • Safety sharp bends at high speed need friction growing as v2.

Section 5 of 10Tangential and net acceleration

Tangential acceleration
  • Tangential acceleration at = Δv/Δt, the rate of change of speed, directed along the tangent.
Net acceleration
  • Net acceleration anet = ac + at as a vector sum.
Magnitude
  • Magnitude ac and at are perpendicular, so anet2 = ac2 + at2.

Section 6 of 10Gravity as centripetal force

Circular orbit
  • GMm/r2 = mv2/r.
Orbital speed
  • Orbital speed v = √(GM/r), independent of the satellite's mass.
Larger orbit
  • Larger orbit smaller speed.

Section 7 of 10Reasoning patterns

1Which force provides it
  • Which force provides it name the actual interaction.
    • (tension, friction, gravity, normal force)
2Changing v or r
  • Changing v or r rewrite F = mv2/r with the new multiples of the old values.
    • (twice the speed on half the length needs 8F)
3Period given
  • Period given use F = 4π2mr/T2.
4Horizontal circle on a string
  • Horizontal circle on a string tension alone provides mv2/r; weight is balanced by a vertical force.

Section 8 of 10More from the specification

Uniform
  • Same density throughout, so the weight acts at the centre.
Frequency
  • Revolutions per second, f = 1/T.
    What the symbols mean: T = period, the time for one complete wave (s) · f = frequency (Hz)
Constant
  • Not changing.
Gravitational force
  • Attractive force between masses.
Minimum speed
  • Smallest top speed that keeps contact, √(gr).
Reaction force
  • Force from a support pushing up on an object.

Section 9 of 10Key terms

TermMeaning
Centripetal accelerationAcceleration toward the centre, v2/r.
Centripetal forceNet inward force, mv2/r.
Tangential accelerationRate of change of speed along the tangent.
Radial directionAlong the line joining the object to the centre.
Orbital speedSpeed of an orbiting object, √(GM/r).

Section 10 of 10Exam technique

1
Using v instead of v2 in the centripetal formula.
2
Including centripetal force as an additional force in the free-body diagram.
3
Pointing the acceleration along the velocity.
4
Using the diameter instead of the radius.
5
Thinking heavier objects have larger centripetal acceleration at the same v and r — ac is independent of mass.

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