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Dynamics of Circular Motion

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Why this chapter
Why this chapter exists

A body going round a circle accelerates towards the centre, even at a steady speed. The chapter on the kinematics of circular motion showed this. Newton's second law then says something must pull or push it inwards. This chapter asks which force does that job in each case. The answer is never a new kind of force. A string, the ground, a wall, friction or gravity does it. Once you can name the force and write one equation along the radius, much becomes clear. You can explain a car over a hump, a swing ride at a mela and the well of death. You can explain a banked road and the lurch in a turning auto-rickshaw.

You will learn
By the end of this chapter you will
  1. State the centripetal requirement and name the real force that meets it in any circular motion.
  2. Find the normal force on a vehicle over a hump and in a dip, and the speed at which it leaves the road.
  3. Find the angle, tension and period of a conical pendulum, and the least spin of a rotor.
  4. Find the greatest speed on a flat turn and the lean of a cyclist.
  5. Find the ideal speed of a banked road, the range of safe speeds with friction, and the raised outer rail of a railway curve.
  6. Use the centrifugal force correctly in a rotating frame, and only there.
Before you begin
Before you begin

You should know that a body in uniform circular motion has an acceleration \(v^2/r = \omega^2 r\) towards the centre. That is the chapter on the kinematics of circular motion. You should draw free-body diagrams and apply Newton's second law along chosen axes. You should know static friction and its limit \(\mu_s N\), and the pseudo force of an accelerating frame from the chapter on accelerated objects.

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Lessons

1
The Centripetal Requirement
Every circle needs a net inward force of \(mv^2/r\). That force is always a real force, such as tension, gravity, friction or a normal force, doing a job. The stone on a smooth table shows the method, and a cut string shows where the stone goes when the job stops.
The centripetal requirementCentripetal force is a job not a forceStone on a smooth table
15 min0%
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2
Over a Crest, Through a Dip
A car at a steady speed on a road that rises and falls. The road pushes with less than the weight over a hump and more in a dip. Above \(\sqrt{gR}\) the car leaves the top of the hump.
Normal force over a crest and in a dipLeaving the road at a crest
14 min0%
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3
The Conical Pendulum and the Rotor
A bob that swings round in a horizontal circle, whose string opens out as it goes faster while the vertical pull stays equal to the weight. Then the rotor and the well of death, where the wall pushes inwards and friction holds the rider up.
The conical pendulumThe rotor and the well of death
15 min0%
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4
Turning on a Flat Road
On a flat road only static friction can turn a car, which sets a greatest safe speed of \(\sqrt{\mu_s gR}\). A car that is too fast runs wide on a gentler curve. A cyclist or a motorcyclist leans into the turn by \(\tan\theta = v^2/(Rg)\).
Turning on a flat roadLeaning into a turn
14 min0%
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5
Banked Roads and Railway Tracks
Tilt a curved road inwards and the normal force itself turns the car. The one speed that needs no friction, the band of safe speeds that friction adds, and the raised outer rail of a railway curve.
Banking and the ideal speedLeast and greatest speed on a banked roadRaised outer rail of a railway track
15 min0%
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6
Centrifugal Force in a Rotating Frame
Seen from a turning platform or a turning auto-rickshaw, Newton's laws fail unless an outward pseudo force \(m\omega^2 r\) is added. What the centrifugal force is, when it may be used, and why it is never the reaction to the centripetal force.
Centrifugal force in a rotating frameUsing the centrifugal force
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.

angle of friction
The angle between the normal reaction and the total contact force when friction is limiting. Its tangent equals \(\mu_s\).
apparent weight
The reading of a weighing scale under a body, equal to the normal force the scale exerts on it. It differs from mg when the body accelerates up or down.
banking
Tilting a curved road or track so that its outer edge is higher than its inner edge. The normal force then leans towards the centre of the turn.
cant
The height by which the outer rail of a railway curve is raised above the inner rail. It is also called superelevation.
centrifugal force
The outward pseudo force \(m\omega^2 r\) that an observer in a frame rotating at \(\omega\) adds to every body at a distance \(r\) from the axis. It exists only in that rotating frame.
centripetal acceleration
The acceleration of a body on a circle directed towards the centre, of size \(v^2/r = \omega^2 r\).
centripetal force
The name for the net real force towards the centre that keeps a body on a circle. It equals \(mv^2/r\). It is a job done by tension, gravity, friction or a normal force, not a separate force.
conical pendulum
A bob on a string that moves on a horizontal circle, so that the string sweeps out a cone.
free-body diagram
A sketch of one body alone with every real force on it drawn as an arrow from the body.
ideal speed
The one speed at which a vehicle takes a banked turn with no friction at all, given by \(\tan\theta = v^2/(Rg)\).
inertial frame
A frame of reference in which a body with no net force on it keeps a constant velocity. The ground is very nearly one.
limiting friction
The largest value static friction can have, \(\mu_s N\), reached when a body is just about to slide.
non-inertial frame
A frame of reference that accelerates relative to an inertial frame. Newton's laws fail in it unless pseudo forces are added.
pseudo force
A term \(-m\vec a_0\) added to the real forces on a body when its motion is described from a frame that accelerates at \(\vec a_0\). It has no agent and no reaction.
radius of curvature
The radius of the circle that best fits a curved path at a given point. On a circle it is the radius itself.
rotating frame
A frame of reference that turns about an axis, such as a turning platform, a rotor or a car on a curve. It is a non-inertial frame.
rotor
A fairground drum that spins about its vertical axis while riders stand against its wall. Once it spins fast enough, the floor can be lowered and friction from the wall holds the riders up.
tangent
The straight line that touches a curve at one point and has the direction of the curve there. The velocity of a body always lies along the tangent to its path.
uniform circular motion
Motion along a circle at a steady speed. The velocity turns all the time, so the body accelerates towards the centre at \(v^2/r\).