This document contains a series of questions which are designed to make polar coordinates clear. We’ll learn what they are and how to use them by solving problems with them. It would be more useful to you if you have a pen and paper handy to draw diagrams and keep track of your thoughts.
The passage from Cartesians to Polars is simply a choice to use new axes. Instead of the i and j axes (which we could otherwise denote e x and e y
) we make a change of axes to e r and e
θ
. We make the decision that e r points in the direction of the position of the particle (which makes an angle θ with the horizontal) so that it points in the direction of increasing radial distance, r . Furthermore we set e
θ to point in the direction of increasing θ which turns out to be orthogonal to e r
.
Draw a diagram to illustrate the basic set-up.
The point is that we could describe a particle’s position either by stating its x and y components as a function of time for use with Cartesian axes or instead by stating its distance from the origin and the angle it makes with the horizontal for use with Polar coordinates.
Question 1: Using your diagram, resolve e r at an angle θ to the horizontal axis.
into its x and y components to show that e r
= cos θ i + sin θ j
In the tutorials I shortened this to e r
= (cos θ, sin θ ). If you wanted to you could also show that e
θ
− sin θ i + cos θ j .
Hint: Recall that we are dealing with unit vectors here - for e r
= draw a vector of length 1
.
Note that what we have done in the above question is to express the new unit vectors in terms of the Cartesian unit vectors – we have written e r in terms of i and j . The results above show that both of our new unit vectors have a direction which depends on θ ; they are not constant. As an objects position changes, so too do the polar axes which are attached to it. But e r will always point from the origin in the direction of the particle. This is why we can write r ( t ) = r ( t ) e r
( θ ( t )) (1) where I have noted that e r depends on the parameter t because it depends on θ (which presumably has some dependence on t if the particle is moving on some arbitrary path). The equation above is simply a statement that to get to the position of the particle we have to go a distance r in the direction e r
, since of course e r points in the direction of the particle. Note above that the r ( t ) on the right hand side is a scalar distance. If we now use the expression we have for e r we can write that r ( t ) = r cos θ i + r sin θ j (2) which I draw attention to for the following reason. If you were to simply draw a vector of length r at an angle θ to the horizontal and then resolve it into its x and y components then the above equation is precisely what you would write down.
I suggest you do this to make sure you understand what I mean.
So the idea of polar coordinates is not entirely unfamiliar; all we are doing here is formalising the ideas and exploring their consequences. The geometry we are considering here is sometimes expressed by writing the transition from Cartesians to Polars as r
2
= x
2
+ y
2 y tan θ = x x y
=
= r r cos cos
θ
θ which encapsulates everything we have discussed up to this point.
This should be clear from your diagram.
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(4)
Question 2: Since the unit vectors are not constant we need to know how they vary with time or position.
Let’s suppose we know how the particle moves with time; i.e. somebody tells us r ( t ) and θ ( t ) (the scalar distance from the origin and the angle it makes with the horizontal). By using the chain rule show that e r
= ˙ e
θ e
θ
= − θ
˙ e r
.
Hint: Use the expressions from Question 1 and differentiate with respect to t using the chain rule. Remember θ depends on t .
Question 3: Now that we have the rate of change of the unit vectors we can find expressions for the velocity and acceleration of the particle written with respect to the polar axes. Find ˙ ( t r ( t ) by starting from ˙ ( t ) = d dt
( r e r
) and applying the product rule. You should find
˙ ( t ) = ˙ e r
+ r θ
˙ e
θ
(5)
¨ ( t ) = ¨ − r θ
˙ 2 e r
+ 2 ˙ θ
˙
+ r
¨ e
θ
(6)
Hint: Use the results of Question 2 and apply the product rule.
In the above equations, the pieces of the velocity and acceleration which are in front of e r are referred to as “radial” components because they are associated with a change which points in the radial direction and the bits in front of e
θ are called “transverse” or “angular” because they are telling us about changes in angle. Now that we’ve found these preliminary results we can do some examples. There are always many ways to answer questions and the trick trying to see which will be quickest. Here we also want to illustrate our results so bear with me whilst we work few some basics.
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*Question 4: It’s quite interesting to consider the consequences of the equations for acceleration. Suppose that you are a stationary observer and that you are watching a particle whiz around under the influence of some force. Then Newton’s law in your frame tells you that F = m ¨ . Let’s work out what exactly the particle feels in its frame: imagine that the only force acting is radial so that F = f e r only. Use Netwon’s law to show that the particle’s distance from the origin – r ( t ) – can be described by an effective force
˜
= f + mr θ
˙ 2 . This is the fictitious centrifugal force that we feel going around a roundabout.
Hint: Use the radial part of the acceleration in Question 3; use Newton’s law and rearrange for ¨ .
Question 5: A particle is attached to a spring and undergoes simple harmonic motion: r ( t ) = 2 cos ω
0 t and θ ( t ) = 0. Use this information to write the particle’s position vector in Cartesians to show that the motion is simple harmonic on the x -axis. Use the Polar coordinate expression for velocity and acceleration to write down the radial and transverse components.
Question 6: A particle moves on the trajectory r ( t ) = ln t and θ ( t ) = 2 π sin t for 1 6 t 6 π
2
. Write the motion in Cartesian components. Work out the transverse and radial velocity and acceleration.
Question 7: Given the Cartesian expression for position r ( t ) = (2 cos ωt, 3 sin ωt ) work out the radial distance r ( t ) and the angle it makes with the horizontal θ ( t ). Show also that the particle moves on an ellipse with major / minor axes of length 3 and 2.
Hint: Use the transition functions to transform the x and y components into Polar coordinates r and θ . Recall the equation of an ellipse from the last homework.
Question 8: In this question we will work out the angular momentum a given motion. First consider a bike wheel spinning about its natural axis which we will take to be aligned along the z -axis so that the wheel spins in the x − y plane. Explain why a point on its rim moves with polar components r ( t ) = R
0
, a constant, and θ ( t ) = ωt . Convert the motion to Cartesian coordinates and work out the velocity. Hence work out the angular momentum L = m r × ˙r .
Important Hint: The cross product is only well defined with three dimensional vectors. Once you’ve worked out the x and y components of the position and velocity you can promote both vectors to 3d by simply setting their z -components equal to zero. The angular momentum will end up pointing in the z -direction – can you understand why using the right hand rule for the cross product?
Question 9: In the same way as the previous question we will work out the angular momentum for the motion r ( t ) = arctan t and θ ( t ) = ωt (1 − e − t ) for t > 0. Try to get a rough picture of the motion in your head. First convert to Cartesians and then work out the angular momentum. What happens for very small and very large t ?
Question 10: The spring in Question 5 is now threaded to a pivot which allows it to rotate in the x y plane.
The thread is rotated at a constant rate ω (where initially ω < ω
0
) so that the motion can be described by θ ( t ) = ωt . This also has an effect on the radial motion so that in this case r ( t ) = 2 cos p
ω 2
0
− ω 2 t .
Repeat the analysis of Question 4 – in particular show that the particle now has transverse components of velocity and acceleration. Is the radial acceleration greater than or less than in the previous question?
What happens if we increase ω until ω = ω
0 or beyond?
Hint: The first part of the question requires the same differentiation as the previous one. When investigating the effect of increasing ω it may be worthwhile to look first at the position vector and to then consider the effective force in the same style as Question 4.
Question 11: Show that the magnitude of the velocity in Polars is equal to ˙ 2 + r 2 θ 2 . Thus write down an expression for the kinetic energy ( E =
1
2 m | ˙ |
2
). Convert this back to Cartesians to show that you get the correct expression written in terms of x ( t ) and y ( t ).
Hint: Recall that the unit vectors e r and e
θ are orthogonal and use the expression for velocity in Polars. Use the transition functions to convert r , r ˙ and
θ
˙ into Cartesians.
Plane polars work very well for systems which have a symmetry in the x − y plane or are confined to that plane.
But to really describe proper three dimensional motion we need to consider spherical polar coordinates. Again we are simply making a change of axes from { i , j , k } to { e r
, e
θ
, e
φ
} . The transition is given by r
2
= x
2
+ y
2
+ z
2 cos θ = z r y tan φ = x x y z
=
=
= r r r sin sin cos
θ
θ
θ cos sin
φ
φ
(7)
(8)
(9)
2

z h
θ
φ r sin θ r e
φ e r
θ e
θ y x
Now it is extremely important to try to understand this. Have a look at the diagram and let me take you through it.
First note that the new axes point in the direction of increasing r , θ and φ respectively and are still orthogonal, though once again are position dependent. Varying r corresponds to moving radially inwards and outwards which is easy to understand. Changing θ corresponds to changing the dip from the vertical – this is shown by the vertical dashed arc.
Changing φ corresponds to rotating in the x − y plane and is shown by the dashed horizontal ellipse.
Can you now understand why these variables naturally fill out a sphere?
We must also note that we restrict r > 0, 0 < θ < π and
0 6 φ < 2 π to ensure that we only cover the sphere once. Try considering what happens if you let θ range up to 2 π ; can you see that you would cover the sphere twice?
The easiest transformation to understand is the z transformation. The horizontal dashed line from the position of the particle to the z -axis finishes a right-angle triangle. Simple trigonometry shows that the height of the particle along the z-axis (labelled h) is given by z = r cos θ . For the x and y transformations we will apply similar trigonometry.
However, as I have labelled on the diagram, the length that the vector has in the x − y plane is given by r sin θ .
This can be derived by noting the right angled triangle made by joining the particle vertically to the x − y plane and seeing that the angle θ also appears in this triangle as labelled. Once we have the length of the vector it is easy to then resolve this length onto the x and y axes much as we did for plane polars – for the x component we multiply the length by cos φ and for the y component we multiply it by sin φ . We have thus found that x = r sin θ cos φ and y = r sin θ sin φ which completes the transformations.
Now we know the x , y and z components of the transformation we know how to write the position of the particle in terms of these axes: r = r sin θ cos φ i + r sin θ sin φ j + r cos θ k (10) and we can use this to work out the expression for e r with respect to the Cartesian axes. Since this vector must point in the direction from the origin to the particle, just as in polar coordinates, it lies in the direction r . Furthermore it must be of unit magnitude so we simply need to normalise the above expression.
Question 1: Show that | r | = r so that if we normalise it we find that e r
= (sin θ cos φ, sin θ sin φ, cos θ ) (11)
There are similar expressions for the other two unit vectors which you should have been given in your lectures. Now at this point, if we so wished, we could follow the tutorial sheet (Question 2) and work out how the vectors change with time by differentiating them. If you wish to do so then please do follow through the working from that question.
However, it is a little less instructive to do so and is rather tedious so instead we will focus on actually using the above information to carry out some calculations.
Question 2: A particle moves on the surface of the sphere (so the vertical, rotating in the x − y r = R
0
, a constant) at an angle of plane at a constant rate. Explain why θ =
π
4
π
4 to and φ = ωt . Use the transformation functions to re-write this with respect to Cartesian axes.
Hint: Use the expression for r ( t ) written in terms of i , j , and k and substitute the values of r , θ and φ .
Question 3: A particle’s trajectory is given as follows: r ( t ) = R
0
1 − 1
2 sin ωt ; θ ( t ) = π (1 − e − ωt ) ; φ =
ωt for t > 0. Can you imagine what this trajectory looks like? Use the transformation functions to express this in terms of x , y and z components – look how much more complicated it is in this case!
Question 4: A helix is described by circular motion in the x − y plane and a constant velocity in the z -direction: x = R cos ωt , y = R sin ωt and z = vt . Use the transition functions to express this in spherical polars – that is, find r , θ and φ . Can you understand the evolution of these variables – what happens for very small or very large times?
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Question 5: A circular orbit is given by r = 10 cos t, 10
√
2 sin t, 10 cos t . Work out the radial and spherical angular components of the motion. Back in Cartesians now, work out the velocity and acceleration to demonstrate that it is indeed motion under a central force. Furthermore work out the angular momentum. Interpret your result to determine the orientation of the circle.
Hint: There are two vectors orthogonal to L which will tell you the orientation of the circle.
That’s pretty much it for now. Further questions from the homework sheet may help you if you’re struggling; look at
Questions 20, 21 and 27 but note that they are not really related the polars. The most important thing is to get your head around the geometry of the transformations and try to imagine what is going on. If you work through this sheet and your lectures notes you should be ok. Please let me know if you have any questions.
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