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2.8.3Force and Translational Dynamics

Circular Motion in a Vertical Loop — AP PHYSICS 1 Revision Notes & Topical Questions

5 min read 8 parts Lesson 33 of 36 1 quiz

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Lesson summary

2.8.3 Circular Motion in a Vertical Loop — AP PHYSICS 1 notes: Analyse forces and minimum speeds for objects moving in vertical circles at the top, bottom and sides.Gravity — always acts straight down, but its direction relative to the centre changes.Support force — tension or th…

  • Tension for a ball on a string
  • Centre above the object
  • Centre below the object
  • Contact lost
  • Energy conservation
  • Minimum bottom speed

Part 1 of 6Why vertical circles differ

1Gravity
  • Gravity always acts straight down, but its direction relative to the centre changes.
2Support force
  • Support force tension or the normal force varies in size and direction to keep Fnet,in = mv2/r.
3Tension for a ball on a string
  • Tension for a ball on a string maximum at the bottom, minimum at the top.
4Speed
  • Speed changes as kinetic and gravitational potential energy exchange, but the radial equation holds with the instantaneous speed.

Part 2 of 6Bottom of the circle

Centre above the object
  • Centre above the object inward is upward.
Equation
  • N − mg = mv2/r, so N = mg + mv2/r.
Result
  • Result the support force exceeds the weight, so riders feel heavy.
String
  • T = mg + mv2/r is the largest tension, so strings tend to break here.

Part 3 of 6Top of the circle

Centre below the object
  • Centre below the object inward is downward and both weight and N (or T) point inward.
Equation
  • N + mg = mv2/r, so N = mv2/r − mg.
Contact lost
  • Contact lost if mv2/r < mg the object leaves the circle and moves as a projectile.
Minimum speed
  • Minimum speed set N = 0 (or T = 0): mg = mv2/r, so vmin = √(gr).
  • Gravity alone then supplies the centripetal force, a = g.
  • vmin is independent of mass.
  • The rider feels weightless for an instant.

Part 4 of 6Energy and the sides

Energy conservation
  • Energy conservation on a frictionless track ½mvbottom2 = ½mvtop2 + mg(2r).
Minimum bottom speed
  • Minimum bottom speed with vtop ≥ √(gr), vbottom ≥ √(5gr).
Release height
  • Release height a coaster starting from rest needs at least 2.5r above the loop bottom.
Side points
  • Side points the normal force or tension is horizontal and supplies all of mv2/r;
    • weight is tangent and changes the speed.

Part 5 of 6Apparent weight in loops

Apparent weight
  • Apparent weight equals the normal force.
Values
  • Values greater than mg at the bottom, less than mg at the top, zero at the minimum speed.
g-force
  • N/(mg), the number of g experienced.
Pull-out from a dive
  • Pull-out from a dive at the bottom N/(mg) = 1 + v2/(gr).

Part 6 of 6More from the specification

More from the specification
Static friction
  • Friction between surfaces at rest relative to each other; fs ≤ μsFN.
Conical pendulum
  • Mass on a string moving in a horizontal circle
Orbital period
  • Time for one orbit.
Reaction force
  • Force from a support pushing up on an object.
Acceleration
  • Rate of change of velocity.
Period
  • Time for one revolution.
Radius
  • Distance from the centre to the circular path.
Linear
  • A graph that is a straight line through the origin, so resistance is constant.
Uniform
  • Same density throughout, so the weight acts at the centre.
Component
  • Part of a vector along one axis.
Pendulum
  • A weight swinging on a string, exchanging GPE and KE.
Frequency
  • Revolutions per second, f = 1/T.
    What the symbols mean: T = period, the time for one complete wave (s) · f = frequency (Hz)

Key terms

TermMeaning
Vertical loopCircular path in a vertical plane.
Minimum speedSmallest top speed that keeps contact, √(gr).
Apparent weightNormal force on a rider.
Mechanical energySum of kinetic and potential energy.
g-forceApparent weight divided by true weight.

Avoid these mistakes

1
Using N = mg at the top or bottom of a loop.
2
Ignoring weight in the radial equation.
3
Thinking the minimum speed at the top depends on mass.
4
Using the same sign for the weight term at top and bottom — it is opposite.
5
Assuming constant speed in a vertical loop — speed changes with height.

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