doPhysics · Circular Motion
Kinematics of Circular Motion
Motion along a curve, described with angles. Why a path bends, angular position and angular velocity, uniform circular motion and the centripetal acceleration \(v^2/r\), speeding up on a circle, the equations of angular motion, and the radius of curvature of any path, from a projectile to a straight line seen from a point.
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Why this chapter
Why this chapter exists
A ceiling fan, a car on a roundabout, the Moon round the Earth and a cricket ball in flight all move along curves. Motion in Two Dimensions showed that the part of the acceleration across the velocity bends a path. This chapter follows that idea to its simplest case, the circle. It describes circular motion with one angle and finds the acceleration of a body going round at a steady speed. Then it adds a changing speed, and uses the circle to measure how sharply any path bends.
You will learn
By the end of this chapter you will
- Say when a path stays straight and when it bends, and split an acceleration into tangential and normal parts.
- Describe motion on a circle by angular position, angular displacement, angular velocity, period and frequency, and use \(v = \omega r\).
- Derive and use the centripetal acceleration \(a_c = v^2/r = \omega^2 r\), directed towards the centre.
- Find the total acceleration on a circle when the speed changes, and use \(\vec v = \vec\omega \times \vec r\) and \(\vec a = \vec\alpha \times \vec r + \vec\omega \times \vec v\).
- Solve problems of angular motion with constant angular acceleration, with graphs, and with calculus when \(\alpha\) varies.
- Find the radius of curvature of any path, including a projectile's, and the angular velocity of a particle about any point.
Before you begin
Before you begin
You should know Motion in Two Dimensions, especially two facts. The velocity lies along the tangent to the path, and an acceleration can change the speed, the direction or both. You should be able to use the equations of motion on a line and read slopes and areas of graphs. You should also split a vector into components and differentiate and integrate simple powers of \(t\).
Lessons
1
Why a Path Curves
A path stays straight only when the acceleration lies along the velocity. The part of the acceleration across the velocity bends the path, the part along it changes the speed, and an acceleration kept across the velocity at every instant gives motion on a circle.
When a path stays straightTangential and normal accelerationCircular motion the simplest curve
2
Angular Position and Angular Velocity
Describing motion on a circle by one angle. The radian, angular position and angular displacement, why a finite rotation is not a vector, angular velocity with its right-hand direction, the period and the frequency, and the link \(v = \omega r\).
Radian and angular positionAngular displacementAngular velocity and its directionPeriod frequency and v equals omega r
3
Uniform Circular Motion and Centripetal Acceleration
Motion on a circle at a steady speed. The speed stays constant while the velocity does not, the triangle of velocities shows that the change in velocity points at the centre, and the centripetal acceleration is \(v^2/r = \omega^2 r\). Three standard examples.
Uniform circular motionCentripetal accelerationThree standard examples
4
Speeding Up on a Circle
Motion on a circle with a changing speed. Angular acceleration, the tangential acceleration \(a_t = \alpha r\), the total acceleration and its angle with the radius, and the vector forms \(v = \omega \times r\) and \(a = \alpha \times r + \omega \times v\).
Angular accelerationTotal acceleration in non uniform circular motionVector form of circular kinematics
5
Equations of Angular Motion
The equations for constant angular acceleration as the twins of the equations of motion on a line, the angle turned in the nth second, the slope and the area of the \(\omega\)-t graph, and calculus for an angular acceleration that varies with time or with angle.
Equations for constant angular accelerationGraphs of angular motionVariable angular acceleration
6
The Radius of Curvature
Circular motion as a tool for any curved path. The osculating circle, the radius of curvature \(R = v^2/a_\perp\), the curvature of a projectile's path, smallest at the top, and the angular velocity of a particle about any point, even for motion along a straight line.
Osculating circle and radius of curvatureCurvature of a projectile pathAngular velocity about any point
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- angular acceleration
- The rate of change of angular velocity, \(\alpha = d\omega/dt\), in rad/s².
- angular displacement
- The change in angular position, \(\Delta\theta = \theta_2 - \theta_1\), positive for an anticlockwise turn. A finite angular displacement is not a vector.
- angular position
- The angle between a fixed reference line through the centre and the line from the centre to the body, measured anticlockwise.
- angular velocity
- The rate of change of angular position, \(\omega = d\theta/dt\), in rad/s. Its direction lies along the axis of rotation.
- centre of curvature
- The centre of the osculating circle, on the normal to the path, on the inner side of the bend.
- centripetal acceleration
- The acceleration of a body on a circle directed towards the centre, of size \(v^2/r = \omega^2 r\). It turns the velocity.
- circular motion
- Motion along a circle, a path whose every point is the same distance from one centre.
- frequency
- The number of revolutions in one second, \(f = 1/T\), measured in hertz.
- line of sight
- The line from a chosen fixed point to a moving particle. The rate at which it turns is the angular velocity of the particle about that point.
- non-uniform circular motion
- Motion on a circle with a changing speed. The body has both a tangential and a centripetal acceleration.
- normal acceleration
- The part of the acceleration at right angles to the velocity, \(a_{\perp}\). It turns the velocity and points to the inside of the bend.
- osculating circle
- The circle that fits a curve best at a point: it touches the curve there, shares its tangent and bends at the same rate.
- radian
- The unit of angle in which an angle is the arc length divided by the radius. A full turn is \(2\pi\,\text{rad}\), and \(1\,\text{rad} \approx 57.3°\).
- radius of curvature
- The radius of the osculating circle at a point of a path, \(R = v^2/a_{\perp}\). A small radius means a sharp bend; a straight line has an infinite radius.
- right-hand rule
- Curl the fingers of the right hand in the sense of turning; the thumb then points along the angular velocity.
- rigid body
- A body whose shape and size do not change, so the distance between any two of its points stays fixed. All its points turn with the same angular velocity.
- tangential acceleration
- The part of the acceleration along the velocity, \(a_t = dv/dt\). It changes the speed and not the direction.
- time period
- The time for one full revolution, \(T = 2\pi/\omega\) for steady turning.
- uniform circular motion
- Motion on a circle at constant speed. The velocity turns steadily, so the body has a centripetal acceleration.