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Motion in a Vertical Circle

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Why this chapter
Why this chapter exists

Think of a bucket of water swung overhead, or a coaster car in a loop. Think of a swing that goes too high, or an ice cube sliding off a bowl. Each moves on a vertical circle. On such a circle the weight speeds the body up and slows it down, so the speed changes at every point. The chapter on Dynamics of Circular Motion gave the radial equation for a steady speed. The chapter on Potential Energy and Conservation of Energy gave the speed from the height. This chapter joins the two ideas. It finds the pull of a string at every point and the least speed that keeps a body on its circle. It also finds what happens when that speed is not reached.

You will learn
By the end of this chapter you will
  1. Split the weight into radial and tangential parts on a vertical circle, and say which part changes the speed.
  2. Find the speed at any angle from energy, \(v^2 = u^2 - 2gr(1 - \cos\theta)\), and the tension from the radial equation.
  3. Derive the critical speed \(\sqrt{gr}\) at the top and \(\sqrt{5gr}\) at the bottom, the \(6mg\) rule and the height \(5r/2\) for a loop-the-loop.
  4. Find where a string goes slack for \(\sqrt{2gr} < u < \sqrt{5gr}\), the speed there, and the parabola that follows.
  5. Compare a rod, a tube, a bead on a ring and a track, and use \(\sqrt{4gr}\) for a rod.
  6. Find where a body sliding on a smooth sphere leaves it, \(\cos\varphi = 2/3\), with and without a starting speed.
Before you begin
Before you begin

You should know the centripetal requirement, that the net inward force on a body moving on a circle is \(mv^2/r\), and the car over a crest. You should be able to use conservation of mechanical energy with \(mgh\) and \(\tfrac12 mv^2\). You should also split a force into components and handle a projectile.

Already know this?

Lessons

1
Forces Around a Vertical Circle
A bob whirled on a string in a vertical circle. Only the parts of the forces along the string meet the centripetal requirement, while the weight's part along the tangent changes the speed. Energy gives the speed at every angle, and the tension follows from it.
Radial and tangential parts of the weightSpeed at any point from energyTension at any point of the circle
15 min0%
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2
The Critical Speed
A string can pull but never push. So a bob stays on its circle at the top only if it moves at least as fast as \(\sqrt{gr}\) there, which needs \(\sqrt{5gr}\) at the bottom. The tension at the bottom always exceeds the tension at the top by \(6mg\), and a loop fed from a smooth ramp needs a start \(5r/2\) high.
Least speed at the topLeast speed at the bottom and the 6mg ruleLoop the loop from a height
15 min0%
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3
When the String Goes Slack
Launched at between \(\sqrt{2gr}\) and \(\sqrt{5gr}\), a bob on a string rises above the level of the centre. There the tension falls to zero while the bob is still moving. Where the string goes slack, how fast the bob is moving there, and the parabola it follows afterwards. Below \(\sqrt{2gr}\) the bob only swings.
Three ranges of launch speedWhere the string goes slackThe projectile after the string goes slack
15 min0%
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4
Rods, Tubes and Tracks
A rod, a tube or a ring can push as well as pull. So a body held by one needs no speed at the top and only \(\sqrt{4gr}\) at the bottom. A track pushes one way only, inwards on the inside of a loop and outwards on a hump, and the direction decides the condition.
A rod can pushBeads rings and tubesInside and outside of a track
13 min0%
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5
Leaving a Smooth Sphere
A particle sliding down the outside of a smooth sphere or dome. The normal force falls as \(mg(3\cos\varphi - 2)\), so the particle leaves where \(\cos\varphi = 2/3\), at a height \(2R/3\) above the centre, whatever its mass. With a starting speed it leaves sooner, and at \(\sqrt{gR}\) it leaves at once, like a car over a crest.
Normal force on a smooth domeWhere a particle leaves a sphereLeaving with a starting speed
13 min0%
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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

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The Workbook
Questions to solve on paper, marked by you.
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Glossary

The key words of this chapter, with the meaning this chapter gives them.

bead on a ring
A small bead with a hole through which a wire ring passes. The ring can push the bead towards its centre or away from it.
centripetal force
The net force towards the centre on a body moving on a circle, equal to \(mv^2/r\). It is the job done by real forces such as tension, the normal force or gravity, not an extra force.
critical speed
The least speed, \(\sqrt{gr}\), that a body on a string or on the inside of a track needs at the top of a vertical circle. At this speed the weight alone supplies \(mv^2/r\).
loop-the-loop
A vertical loop of track with the body riding on its inside, upside down at the top. Roller coasters and toy car tracks have them.
mechanical energy
The sum of the kinetic energy and the potential energy of a body or a system. It stays constant when only conservative forces, such as gravity, do work.
normal force
The push that a surface exerts on a body in contact with it, at right angles to the surface. A surface can push but cannot pull, so the normal force is never negative.
projectile
A body moving under gravity alone. Near the Earth its path is a parabola.
rigid rod
A rod that does not bend or stretch. A light rigid rod can pull or push a body along its own length.
slack
The state of a string that carries no tension. A string goes slack when keeping it straight would need a push.
smooth
Without friction. A smooth surface exerts only a normal force.
tension
The pull that a string or rope exerts on a body tied to it, directed along the string. A string can pull but never push, so its tension is never negative.
vertical circle
A circular path in a vertical plane. The weight of the moving body lies in the plane of the circle and changes its speed.