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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

Angular Motion

We are very familiar with our equation for linear velocity, that is:

Linear Velocity.png

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, rads-1.

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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:

angular velocity.png

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).

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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.

angular velocity 2.png

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.

angular vs linear.png

At this point, it's probably worth spending some time practicing some of the basics. Isaac Physics have a nice selection of test-yourself questions to make sure you are confident.

Video Lessons

Resources

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: Pre-University 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)
A-Level Qs: overlapping content
Centripetal Force

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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Orbital motion.png

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). 

acent1.png

By using our equation v = rω, we can derive a second equation for our centripetal acceleration in terms of our angular velocity.

acent2.png

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:

fcent.png

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.

plane1.png
plane2.png

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.

plane3.png

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 y-direction 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 x-component 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 msˉ¹

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Isaac Physics have several questions looking at centripetal force, have a bit of practice applying these ideas.

Video Lessons

Resources

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: Pre-University 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 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 hump-back 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.

vertcircle.png

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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rollercoasterloop.png

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 msˉ¹

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Video Lessons

Resources

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

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. 

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