6.1. Circular Motion
So we have already looked at a lot of mechanics of motion in a straight line. This chapter begins to introduce the idea of rotational motion. It is really important before starting this stuff that you are fully up to speed with the Mechanics we did in Chapter 2, so make sure you have a look back over that stuff (particularly forces and free body diagrams) if you are rusty.
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When we started learning about forces we learnt that they can do 3 things:

Change an object's speed

Change an object's direction

Change an object's shape
It's the second of these that we will be looking at throughout this chapter. Imagine a bucket being whirled around your head on some string  the bucket's speed may be constant, but its direction is constantly changing. From what we know about vectors, this means its velocity must also be constantly changing.
I quite like this video as a starting point with this topic, with a sprinter trying to complete a loop the loop on foot.
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The chapter has been divided up as follows:

Angular Motion : Introducing the basics of motion in a circle

Centripetal Force : What is it that â€‹is actually causing the motion in a circle.

Motion in a vertical circle : The classic example of the rollercoaster looptheloop
Angular Motion
We are very familiar with our equation for linear velocity, that is:
Now we are looking at motion in a circle we introduce a new term called angular velocity. Angular velocity is a measure of how quickly something moves through a certain angle, measured in Radians per Second, rads1.
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N.B. A radian is another way to measure angle, other than degrees. It is essential you confident converting between degrees and radians for this chapter. If you are unfamiliar with the concept, spend some time brushing up on this topic here.
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Our angular velocity is defined using the following equation:
Crash Course have a nice overview of circular motion which is worth watching.
PHET have a nice simulation on Revolutions (downloading the Java version seems to work best). Select the tab called 'Rotation' and give the turntable an initial angular velocity. The simulation demonstrates quite nicely that the angular velocity is the rate at which the turntable moves through an angle (also notice the position graph at the bottom, and the similarities with SHM).
Some of the terminology we previously used when looking at waves can also be used here:

Frequency  the number of rotations per second, in Hz (though often presented in rpm  revolutions per minute)

Time period  the time in seconds to complete one revolution
These are linked by the equation f = 1/T. An object moving in a complete circle moves through an angle of 2π radians, leading to alternative equations for angular velocity.
Next, we can look at the link between angular and linear velocities. Mathematically, the length of an arc of a circle is linked to the angle through the equation s=rθ.
At this point, it's probably worth spending some time practicing some of the basics. Isaac Physics have a nice selection of testyourself questions to make sure you are confident.
Video Lessons
Chris Doner  Uniform Circular Motion  IB Specific  
Khan Academy  Introduction to Angular Velocity  Relating angular and linear motion  
Science Shorts  Circular Motion  Degrees or Radians  
Study Nova  Circular Motion  Circular Motion (Lecture) 
Resources
IB Physics  Topic 6 Notes  
IBPhysics.net  Chapter 6 Summary  IB Revision Notes  
Isaac Physics  Radians Maths Recap  Circular Kinematics  
Isaac Physics  Circular Kinematics  
Mr. G  6.1 Teaching Notes  6.1 Student Notes  
Physics and Maths Tutor  Further Mechanics Definitions  Further Mechanics Key Points  Further Mechanics Detailed Notes  Further Mechanics Flashcards  A Level Resources  content slightly different 
Questions
Cambridge University Press  Topic 6: MCQs  CUP Website Link  Freely available online  
Dr French's Eclecticon  Uniform Circular Motion  Uniform Circular Motion Solutions  Link to Dr French's Site  Extension: PreUniversity Material  
Grade Gorilla  6.1 (Circular Motion) MCQ  Topic 6 (Gravity/ Rotation) End Quiz  Quick IB Specific Mixed MCQs  
Isaac Physics  Units of Rotary Motion  
Mr. G  6.1 Formative Assessment  Topic 6 Summary Qs  IB Specific Questions  
Physics and Maths Tutor  Circular Motion (AQA 1)  Circular Motion MS (AQA 1)  Circular Motion (AQA 2)  Circular Motion MS (AQA 2)  ALevel Qs: overlapping content 
Centripetal Force and Acceleration
Now we have a few of the basics out the way, let's look at a few key ideas towards motion in a circle.
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Let's start by looking at motion of the Earth around the sun as shown below.
The direction of the linear velocity (shown in blue) is always changing. As velocity is a vector quantity, this means there must be an acceleration. (Note, the velocity is constantly changing, but the speed is constant).
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The linear velocity is always tangential to the circle.
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The only force acting on the Earth as it orbits is the gravitational force of attraction towards the sun. As the resultant force is towards the centre of the circle, the acceleration must also act towards the centre (N2L  the acceleration is always in the same direction as resultant force), and this is known as the CENTRIPETAL ACCELERATION.
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The (centripetal) acceleration always acts towards the centre of the circle (and is perpendicular to the velocity).
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This is a key characteristic of objects moving with circular motion: there is always acceleration, therefore the object can never be in equilibrium.
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If an object is moving in with uniform circular motion, there is always a resultant force acting, directed towards the centre of the circle. This resultant force is known as the centripetal force.
Equations of Circular Motion
The centripetal acceleration is a vector quantity that always acts towards the centre of the circle. Mathematically, the equation for centripetal acceleration as follows (derivation for these can be found here).
By using our equation v = rω, we can derive a second equation for our centripetal acceleration in terms of our angular velocity.
Our centripetal force is linked to our centripetal acceleration by Newton's Second Law, i.e. F = ma. By multiplying through by mass we get:
Worked Example  a banking plane
Once you've got to grips with the basic formulae, it's important to have a bit of practice at applying these. One of the trickiest examples to get your head around involve resolving multiple forces acting on an object to find centripetal force. See a worked example below for a banking plane.
Q. A plane of mass 10 000 kg is banking at 30° in a horizontal circle of radius 500 m. Work out the size of the centripetal force acting and the velocity of the plane.
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For this sort of question, it is essential to start with a diagram. The centripetal force acts towards the centre of the circle, and is the resultant force acting on the plane.
We are looking for the resultant force (i.e. centripetal force), so we must consider the forces acting on the plane. Therefore we must draw a free body diagram. The weight force acts downwards, while the lift force acts perpendicular to the wings.
We know that the resultant force acts towards the centre, so we should now resolve the Lift force into its x and y components.
The resultant force in the ydirection is zero, therefore:
W = L cosθ
10 000 kg x 9.81 = L cos(30)
∴ L = 113 000 N
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The resultant force (i.e. centripetal force) on the plane the same as the xcomponent of the lift, so:
Fcent = L sin θ
= 113 000 cos(30)
∴ = 56 600 N
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We can then substitute this into our equations for centripetal force to calculate the plane's linear velocity:
Fcent = mv²/r
56 600 = 10 000 v² / 500
v = 53 msË‰¹
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Isaac Physics have several questions looking at centripetal force, have a bit of practice applying these ideas.
Video Lessons
Chris Doner  Circular Motion (Centripetal force)  IB Specific  
Khan Academy  Centripetal Force Intuition  Centripetal Force Problems  
Science Shorts  Circular Motion  
Study Nova  Centripetal Acceleration and Force  Circular Motion (Lecture) 
Resources
IB Physics  Topic 6 Notes  
IBPhysics.net  Chapter 6 Summary  IB Revision Notes  
Isaac Physics  Circular Dynamics  
Mr. G  6.1 Teaching Notes  6.1 Student Notes  
Physics and Maths Tutor  Further Mechanics Definitions  Further Mechanics Key Points  Further Mechanics Detailed Notes  Further Mechanics Flashcards  A Level Resources  content slightly different 
Questions
Cambridge University Press  Topic 6: MCQs  CUP Website Link  Freely available online  
Dr French's Eclecticon  Uniform Circular Motion  Uniform Circular Motion Solutions  Link to Dr French's Site  Extension: PreUniversity Material  
Grade Gorilla  6.1 (Circular Motion) MCQ  Topic 6 (Gravity/ Rotation) End Quiz  Quick IB Specific Mixed MCQs  
Isaac Physics  Centripetal Acceleration  
Mr. G  6.1 Formative Assessment  Topic 6 Summary Qs  IB Specific Questions 
Motion in Vertical Circles
Motion in a Vertical Circle
The trickiest questions in this topic involve motion in vertical circles. Things like buckets being whirled on a string, cars going over humpback bridges, or rollercoaster loops as shown below.
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The most important thing to do when solving these problems is to draw a Free Body Diagram, labelling all the forces acting, and remembering that the centripetal force acts towards the centre of the circle.
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In the example below we have Free Body Diagrams for a rollercoaster completing a loop the loop at a constant speed (N.B. in this example some work must be done to maintain a constant speed).
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At the top of the loop, the reaction force acts outwards from the track. This means that the reaction and weight forces act in the same direction. The centripetal force is therefore the sum of the the two forces' magnitudes.

At the bottom of the loop, the weight still acts downwards, though now the reaction force acts in the opposite direction (outwards from the track). The centripetal force still must act towards the centre, therefore the reaction force must be much larger than the weight force.
Worked example  minimum speed to complete loop
One common question that the IB likes to ask is how slow does the rollercoaster need to travel in order to just lose contact with the track.
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Q. A rollercoaster has a loop diameter of 30 m. Calculate the minimum speed at which the rollercoaster will complete the loop.
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The rollercoaster will lose contact when upside down at the top of the loop (left picture above). The slower it travels, the lower the reaction force from the track keeping the coaster moving in a circle. At the point where it just loses contact, the reaction force will be equal to ZERO.
Therefore at this point Fcent = W.
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By equating our weight to our equation for centripetal force we can solve for v.
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v = √(gr)
= √(9.81 x 15)
= 12 msË‰¹
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Video Lessons
Chris Doner  Problem Solving in Circular Motion  IB Specific  
Khan Academy  Vertical Bowling Ball Loop  Vertical YoYo  Mass in a Horizontal Circle  
Science Shorts  Circular Motion (Vertical)  
Study Nova  Vertical Circle Example 
Resources
IB Physics  Topic 6 Notes  
IBPhysics.net  Chapter 6 Summary  IB Revision Notes  
Mr. G  6.1 Teaching Notes  6.1 Student Notes  
Physics and Maths Tutor  Further Mechanics Definitions  Further Mechanics Key Points  Further Mechanics Detailed Notes  Further Mechanics Flashcards  A Level Resources  content slightly different 
Questions
Cambridge University Press  Topic 6: MCQs  CUP Website Link  Freely available online  
Grade Gorilla  6.1 (Circular Motion) MCQ  Topic 6 (Gravity/ Rotation) End Quiz  Quick IB Specific Mixed MCQs  
Mr. G  6.1 Formative Assessment  Topic 6 Summary Qs  IB Specific Questions 
Additional Resources
IB Questions
A question by question breakdown of the IB papers by year is shown below to allow you to filter questions by topic. Hopefully you have access to many of these papers through your school system. If available, there may be some links to online sources of questions, though please be patient if the links are broken! (DrR: If you do find some broken links, please contact me through the site)
Questions on this topic (Section 6) are shown in violet.
Use this grid to practice past IB questions topic by topic. You can see from the colours how similar the question topic breakdown is year by year. The more you can familiarise yourself with the IB question style the better  eventually you will come to spot those tricks and types of questions that reappear each year.