Feeds:
Posts

## The planets this winter (2015/16)

With winter (in the Northern Hemisphere) approaching, I thought it was about time I gave a summary of which planets are visible over the next few months. The longer nights, enabling easier viewing of the night-time sky, is one of the few pluses about this time of year as far as I am concerned. So, which planets are visible this winter (2015/16)?

The times I will give for various planets rising or setting are for Cardiff, where I live. So, if you are living elsewhere the times will almost certainly be different. Obviously, if you are living in the southern hemisphere you are about to move into summer not winter. But, although times may vary depending on your location; whether a planet is visible or not, and whether it is visible in the evening after sunset or in the morning before sunrise will not be different.

Of the 5 naked-eye planets, all but Saturn are visible this winter. Here is more detail about each.

## Mercury

Mercury is currently in Sagittarius, rising before the Sun and thus setting after the Sun. So, it is currently an evening object. It reaches maximum elongation on the 29 December when it will be $25^{\circ}$ to the East of the Sun, and on this day it will set in Cardiff at 17:43. The Sun sets on this day at 16:11 in Cardiff, giving some 1.5 hours after sunset to see Mercury. Although these setting times will vary depending on your location, what will not vary is the time between sunset and Mercury setting, which will be about 1.5 hours no matter where you live.

1.5 hours between sunset and Mercury setting it very good. Mercury is rarely this far from the Sun; so for those of you who have never seen Mercury, this month of December provides a very good chance. Find a view to the western horizon which is uninterrupted and away from city lights, and use the chart below to find Mercury. It will be reasonably bright, at a magnitude of -0.5.

Mercury just after sunset as seen from Cardiff on 29 December 2015. This month is a good month to see Mercury, as its maximum eastern elongation (the maximum angle between it and the Sun) is nearly as large as it can ever be. There are no bright stars near Mercury at the end of December.

Mercury will reach inferior conjunction on 14 January, whereupon it will reappear as a morning object later in January and February.

## Venus

Venus is currently in Libra. It is a morning object, very bright before sunrise. At a magnitude of -4.1 you cannot fail to see it. It will reach maximum western elongation on 12 January. You can see it in the diagram below of the sky before sunrise, which also shows where to find Mars and Jupiter. Venus will be visible as a morning object throughout this winter and into the spring.

## Mars

Mars is currently in Virgo. It is rising at the end of December just after 2am, so is a morning object. In fact, it can be seen in the morning sky along with Venus and Jupiter throughout much of the winter, as the diagram below shows. At the end of December it has a magnitude of +1.3, fainter than nearby Spica, which is at +1.05. Mars will reach opposition on 22 May, by which time it will have brightened to -2.1, so some 23 times brighter; making April, May and June by far the best time to see this planet.

The morning sky at the end of December as seen from Cardiff. Venus, Mars and Jupiter are all visible in the morning sky this winter. Jupiter and Venus are easy to find as they are so bright. Mars is a little trickier, but will brighten as it approaches opposition in the spring

## Jupiter

Jupiter is in Leo, and is also currently a morning object. At the end of December it rises just before 11pm. It will be at opposition in early March (8 March), and so in late winter and spring it will be an evening object, but for most of this winter it is better seen in the morning before sunrise.

I like it when one can see Jupiter and Venus at the same time, as it allows one to see how much brighter Venus is than Jupiter. Normally Jupiter is the brightest point-like object in the sky, but when Venus is visible it outshines Jupiter by a factor of 6 or so.

## Saturn

Saturn is currently in Ophiuchus, and this winter is not the time to see Saturn. It will reach opposition in early June (3 June), so spring and summer are the best times to see Saturn in 2016 and over the next few years. It will not become a winter object again for another 14 years or so.

## The 100 greatest songwriters – number 20 – Jerry Leiber and Mike Stoller

At number 20 in Rolling Stone Magazine’s list of the 100 greatest songwriters is Jerry Leiber and Mike Stoller. This songwriting pair have written some of the best known songs in rock ‘n’ roll, including “Hound Dog”, “Jailhouse Rock”, “Yakety Yak”, and “Stand By Me” (which they co-wrote with Ben E. King).

Unlike the songwriting partnership of Lennon and McCartney, where both wrote melodies and lyrics, Leiber and Stoller’s partnership was more traditional, in that Leiber wrote the lyrics and Stoller the melodies (this was also the case for e.g. “Gilbert and Sullivan” and “Rodgers and Hammerstein”). Leiber and Stoller met in Los Angeles in 1950, and found that they shared a love of blues and rhythm and blues music. They began collaborating in the same year, and continued to work together for the best part of 50 years.

At number 20 in Rolling Stone Magazine’s list of the 100 greatest songwriters of all time is Jerry Leiber and Mike Stoller.

The song of theirs that I have decided to share in this blogpost is “Jailhouse Rock”, which they specifically wrote in 1957 as the title song for Elvis Presley’s third movie. The song became a massive hit for Presley, reaching number 1 in the US singles charts. It has sold over 2 million copies world-wide, and is at number 67 in Rolling Stone Magazine’s list of the 500 greatest songs of all time.

The warden threw a party in the county jail.
The prison band was there and they began to wail.
The band was jumpin’ and the joint began to swing.
You should’ve heard those knocked out jailbirds sing.
Let’s rock, everybody, let’s rock.
Everybody in the whole cell block
was dancin’ to the Jailhouse Rock.

Spider Murphy played the tenor saxophone,
Little Joe was blowin’ on the slide trombone.
The drummer boy from Illinois went crash, boom, bang,
the whole rhythm section was the Purple Gang.
Let’s rock, everybody, let’s rock.
Everybody in the whole cell block
was dancin’ to the Jailhouse Rock.

Number forty-seven said to number three:
“You’re the cutest jailbird I ever did see.
I sure would be delighted with your company,
come on and do the Jailhouse Rock with me.”
Let’s rock, everybody, let’s rock.
Everybody in the whole cell block
was dancin’ to the Jailhouse Rock.

The sad sack was a sittin’ on a block of stone
way over in the corner weepin’ all alone.
The warden said, “Hey, buddy, don’t you be no square.
If you can’t find a partner use a wooden chair.”
Let’s rock, everybody, let’s rock.
Everybody in the whole cell block
was dancin’ to the Jailhouse Rock.

Shifty Henry said to Bugs, “For Heaven’s sake,
no one’s lookin’, now’s our chance to make a break.”
Bugsy turned to Shifty and he said, “Nix nix,
I wanna stick around a while and get my kicks.”
Let’s rock, everybody, let’s rock.
Everybody in the whole cell block
was dancin’ to the Jailhouse Rock.

Here is Elvis Presley’s memorable “Jailhouse Rock” video. Enjoy!

Which is your favourite Leiber and Stoller song?

## New Horizons’ latest images of Pluto

NASA has recently released some new images taken by its New Horizons probe which flew past Pluto in mid-July. These new images show unprecedented detail of the varying terrain on this distant world.  At its closest, New Horizons flew within about 12,500 km of the surface of Pluto, giving us our first ever close-up view of this dwarf planet. But, in order to save power, the transmitter on board sends the data back quite slowly, so data will be coming in well into 2016. Below are some of the latest images.

NASA’s New Horizons probe shows unprecedented details on Pluto’s surface, including mountains and planes

NASA’s New Horizons probe shows craters on the surface of Pluto

## The 100 best Beatles songs – number 36 – I Should Have Known Better

At number 36 in Rolling Stone Magazine’s 100 greatest Beatles songs is “I Should Have Known Better”. This wonderful song appears in their first movie A Hard Day’s Night, in a scene which features George Harrison’s future wife Pattie Boyd. In fact, Harrison met her during the filming of the movie, she was a model and actress at the time. The song, credited to Lennon and McCartney but composed by John Lennon on his own, was recorded in February 1964 and released in the DUK in July of the same year on the Hard Day’s Night movie soundtrack, and later on the album of the same name. It was also released as the B-side to the single “A Hard Day’s Night” in the USA.

I wasn’t aware until I read the text from Rolling Stone about this song that it was so influenced by Bob Dylan. I knew that Dylan’s songs influenced The Beatles, in particular Lennon. His flat cap which he was wearing in the Help period was a copy of the flat cap Dylan wore on the cover of his debut album. So, it is interesting to read that the influence started a year or more before then.

At number 36 in Rolling Stone Magazine’s list of the 100 greatest Beatles songs is “I Should Have Known Better”.

I should have known better with a girl like you
That I would love everything that you do
And I do, hey, hey, hey, and I do

Whoa, oh, I never realized what a kiss could be
This could only happen to me
Can’t you see, can’t you see

That when I tell you that I love you, oh
You’re gonna say you love me too, oh
And when I ask you to be mine
You’re gonna say you love me too

So I should have realized a lot of things before
If this is love you’ve got to give me more
Give me more, hey hey hey, give me more

Whoa, oh, I never realized what a kiss could be
This could only happen to me
Can’t you see, can’t you see

That when I tell you that I love you, oh
You’re gonna say you love me too, oh
And when I ask you to be mine
You’re gonna say you love me too
You love me too, you love me too
You love me too

This video is from the movie, Pattie Boyd is one of the school girls. She was 19 when they met, and they married in. The song features Lennon’s voice being double-tracked, McCartney does not sing on it despite miming to the song in the movie. Pattie Boyd is the blonde schoolgirl, the one who features most prominently in the video.

Which is your favourite song in the movie A Hard Day’s Night?

## 25 years of rock

Several months ago I blogged about the song “Oh Well”, a Fleetwood Mac song with which I thought I was unfamiliar, but when I heard it on a countdown of the best guitar riffs I realised the song had been used at the end of each episode of a radio series 25 Years of Rock which I had listened to and recorded as a teenager.

25 Years of Rock was a great mixture of news of the year, interspersed with songs from the same year. It was a fantastic way  for me to not only discover some new music, but also to learn something about the events of those years.

Someone came across my blog about this, and sent me an email with a link to the entire series! Here it is – 25 Years of Rock. You can see from the file names that the series was broadcast in 1980, once a week running from June until the 7th of December (ironically, 35 years ago yesterday and also the day before John Lennon was murdered). I have just discovered in preparing this blog that the series is also now being repeated on BBC Radio 6, but this may not be accessible to those outside of the Disunited Kingdom. You can also see that the series was extended in 1985 to 30 Years of Rock, with 1980-1984 being added.

25 Years of Rock was first broadcast on BBC Radio 1 in 1980. It is currently being repeated on BBC Radio 6

I love such programmes, even as a teenager (yes, I was a geek back then too 😉 ), so I recorded the series on cassette tape. But, I only recorded the years which covered The Beatles, 1963-1970, and over the years I have lost the tape with the 1970 episode on it, but several years ago I transferred my copies of the 1963-1969 shows into MP3 files to listen to anywhere.

Having this link has enabled me to download the whole series, as I now have more interest in the music before and after the Beatles than I did as a teenager. The Beatles era is still my preferred period, but I have grown more eclectic in my old age 😉

In addition to including the link to the whole series, I have also created this video below (as today is the 35th anniversary of John Lennon’s death) of the part of the 30 Years of Rock episode from 1980 which has a few clips from Andy Peeble’s famous interview which he did with Lennon and Yoko two days before he was murdered. The clip at the beginning, where he is wishing everyone a happy Christmas, was broadcast on the 7th of December and I remember hearing it as I was at home revising for an exam. Little did anyone know what would happen the following day.

The Lennon Tapes is a book of the entire interview conducted by the BBC’s Andy Peebles with John Lennon and Yoko Ono on the 6th of December 1980, two days before he was shot

I hope you enjoy listening to these programmes as much as I did and still do.

## Does centrifugal force exist?

For several weeks now I have been planning to write a blog about centrifugal force, mainly prompted by seeing a post by John Gribbin on Facebook of the xkcd cartoon about it. In the cartoon James Bond is threatened with torture on a centrifuge. Here is a link to the original cartoon.

The xkcd cartoon about centrifugal force involves James Bond being tortured on a centrifuge

I have taught mechanics many times to physics undergraduates, and they are often confused about centripetal force and centrifugal force, and what the difference is between them. Some have heard that centrifugal force doesn’t really exist, just as Bond states in this cartoon. What is the real story?

## Rotating frames of reference

Everyone reading this (apart from a few “flat-Earth adherents” maybe) knows that we live on the surface of a planet which is rotating on its axis once a day. This means that we do not live in an inertial frame of reference (an inertial frame is one which is not accelerating), as clearly being on the surface of a spinning planet means that we are experiencing acceleration all the time; as we are not travelling in a straight line. That acceleration is provided by the force of gravity, and it stops us from going off in a straight line into space!

Because we are living in a non-inertial frame of reference we need to modify Newton’s laws of motion to properly describe such a non-inertial frame (which I am going to call a “rotating frame” from now on, although a rotating frame is just one example of a non-inertial frame but it is the one relevant to us on the surface of a rotating Earth).

Let us consider our usual Cartesian coordinate system. The unit vector in the x-direction is usually written as $\hat{\imath}$, the one in the y-direction as $\hat{\jmath}$, and the one in the z-direction as $\hat{k}$. We are going to consider an object rotating about the $\hat{k}$ (z-axis) direction.

We will consider two reference frames, one which stays fixed (the inertial reference frame), denoted by $(\hat{\imath},\hat{\jmath},\hat{k})$, and a second reference frame which rotates with the rotation, denoted by $(\hat{\imath}_{r} ,\hat{\jmath}_{r} ,\hat{k}_{r})$, where the subscript $r$ reminds us that this is the rotating frame of reference.

For the derivation below I am going to assume that we are considering motion with a constant radius $r$. I want to illustrate how centrifugal force arrises in a rotating frame such as being on the surface of our Earth. Our Earth is not spherical, but at any given point the size of the radius does not change, so this is a reasonable simplification.

As I showed in this blog on angular velocity, we can write the linear velocity $\vec{v}$ of an object moving in a circle as

$\vec{v} = \frac{ d \vec{r} }{ dt } = \vec{\omega} \times \vec{r}$

where $\vec{r}$ is the radius vector and $\vec{\omega}$ is the angular velocity.

Writing $\vec{r}$ in terms of its x,y and z-components in our inertial (non-rotating) frame, $\vec{r}=(\hat{\imath},\hat{\jmath},\hat{k})$, so in general we then have

$\vec{v} = \frac{ d \vec{r} }{ dt } \rightarrow \frac{ d \hat{\imath} }{dt} = \vec{\omega} \times \hat{\imath}, \; \; \frac{ d \hat{\jmath} }{dt} = \vec{\omega} \times \hat{\jmath} , \; \; \frac{ d \hat{k} }{dt} = \vec{\omega} \times \hat{k}$

Let us consider the specific case of a small rotation $\delta \theta$ about the $\hat{k}$ axis, as shown in the figure below. As the figure shows, in our inertial (fixed) frame of reference, the new direction of the x-axis is now $\hat{\imath} + \delta \hat{\imath}$, and the new direction of the y-axis is $\hat{\jmath} + \delta \hat{\jmath}$. The direction of the $\hat{k}$ axis is unchanged.

We are going to rotate about the z-axis ($\hat{k}$ direction) by an angle $\delta \theta$

Because we are rotating about the $\hat{k}$ axis, the angular velocity is in this direction, and so we can write (using the right-hand rule for vector products as I blogged about here)

$\vec{\omega} \times \hat{\imath} = \omega \hat{\jmath}, \; \; \vec{\omega} \times \hat{\jmath} = -\omega \hat{\imath}, \; \; \vec{\omega} \times \hat{k} =0$

Let us now consider some vector $\vec{a}$, which we will write in the rotating frame of reference as

$\vec{a} = a_{x} \hat{\imath}_{r} + a_{y} \hat{\jmath}_{r} + a_{z} \hat{k}_{r}$

If we now look at the rate of change of this vector in the rotating frame we have

$\left( \frac{d \vec{a} }{dt} \right)_{r} = \frac{d}{dt}(a_{x}\hat{\imath}_{r}) + \frac{d}{dt}(a_{y}\hat{\jmath}_{r}) + \frac{d}{dt}(a_{z}\hat{k}_{r})$

In the rotating frame of reference, $\hat{\imath}_{r}, \hat{\jmath}_{r}$ and $\hat{k}_{r}$ do not change with time, so we can write

$\left( \frac{d \vec{a} }{dt} \right)_{r} = \frac{ da_{x} }{dt} \hat{\imath}_{r} + \frac{ da_{y} }{dt} \hat{\jmath}_{r} + \frac{ da_{z} }{dt} \hat{k}_{r}$

In the inertial frame of reference $\hat{\imath}_{r}, \hat{\jmath}_{r}$ and $\hat{k}_{r}$ move, so

$\left( \frac{d \vec{a} }{dt} \right)_{i} = \frac{d}{dt} (a_{x} \hat{\imath}_{r}) + \frac{d}{dt} (a_{y} \hat{\jmath}_{r}) + \frac{d}{dt} (a_{z} \hat{k}_{r})$

$\left( \frac{d \vec{a} }{dt} \right)_{i} = \frac{ da_{x} }{dt}\hat{\imath}_{r} + \frac{ da_{y} }{dt}\hat{\jmath}_{r} + \frac{ da_{z} }{dt}\hat{k}_{r} + a_{x} \frac{d \hat{\imath}_{r} }{dt} + a_{y} \frac{d \hat{\jmath}_{r} }{dt} + a_{z} \frac{d \hat{k}_{r} }{dt}$

But, we can write (see above) that

$\frac{d\hat{\imath}_{r} }{dt} = \vec{\omega} \times \hat{\imath}_{r}, \; \; \frac{d\hat{\jmath}_{r} }{dt} = \vec{\omega} \times \hat{\jmath}_{r}, \; \; \frac{d\hat{k}_{r} }{dt} = \vec{\omega} \times \hat{k}_{r}$

and so

$\left( \frac{d \vec{a} }{dt} \right)_{i} = \frac{ da_{x} }{dt}\hat{\imath}_{r} + \frac{ da_{y} }{dt}\hat{\jmath}_{r} + \frac{ da_{z} }{dt}\hat{k}_{r} + a_{x} \vec{\omega} \times \hat{\imath}_{r} + + a_{y} \vec{\omega} \times \hat{\jmath}_{r} + + a_{z} \vec{\omega} \times \hat{k}_{r}$

$\left( \frac{d \vec{a} }{dt} \right)_{i} = \frac{ da_{x} }{dt}\hat{\imath}_{r} + \frac{ da_{y} }{dt}\hat{\jmath}_{r} + \frac{ da_{z} }{dt}\hat{k}_{r} + \vec{\omega} \times \vec{a}$

$\boxed{ \left( \frac{d \vec{a} }{dt} \right)_{i} = \left( \frac{d \vec{a} }{dt} \right)_{r} + (\vec{\omega} \times \vec{a}) }$

## A fixed point on the Earth’s surface

Let us now consider the point $\vec{a} = \vec{r}$, where $\vec{r}$ is a fixed point on the Earth’s surface. We can write

$\left( \frac{d \vec{r} }{dt} \right)_{i} = \left( \frac{d \vec{r} }{dt} \right)_{r} + (\vec{\omega} \times \vec{r})$

But, in the rotating frame of reference this point does not change with time, so

$\left( \frac{d \vec{r} }{dt} \right)_{r} = 0$

and so

$\left( \frac{d \vec{r} }{dt} \right)_{i} = (\vec{\omega} \times \vec{r}) = \omega r \sin(\theta)$

where $\theta$ is the angle between the Earth’s rotation axis and the latitude of the point (so $\theta = 90^{\circ} - \text{ latitude}$).

Let us now calculate the acceleration in an inertial frame in terms of acceleration in a rotating frame. Writing $\vec{a}$ as $\vec{r}$ as above, we now have

$\left( \frac{d \vec{r} }{dt} \right)_{i} = \left( \frac{d \vec{r} }{dt} \right)_{r} + (\vec{\omega} \times \vec{r})$

To make things easier to write, we will re-write

$\left( \frac{d \vec{r} }{dt} \right)_{i} = \frac{d \vec{r}_{i} }{dt} \text{ and } \left( \frac{d \vec{r} }{dt} \right)_{r} = \frac{d \vec{r}_{r} }{dt}$

so

$\frac{d \vec{r}_{i} }{dt} = \frac{d \vec{r}_{r} }{dt} + (\vec{\omega} \times \vec{r})$

$\vec{v}_{i} = \vec{v}_{r} + (\vec{\omega} \times \vec{r})$

If we now differentiate $\vec{v}_{i}$ with respect to time, we will have the acceleration in the inertial frame

$\left( \frac{d \vec{v}_{i} }{dt} \right)_{i} = \left( \frac{d \vec{v}_{i} }{dt} \right)_{r} + (\vec{\omega} \times \vec{v}_{i})$

But, $\vec{v}_{i} = \vec{v}_{r} + (\vec{\omega} \times \vec{r})$

so

$\left( \frac{d \vec{v}_{i} }{dt} \right)_{i} = \frac{d}{dt}(\vec{v}_{r} + \vec{\omega} \times \vec{r})_{r} + \vec{\omega} \times (\vec{v}_{r} + \vec{\omega} \times \vec{r})$

Expanding this out we get

$\left( \frac{d \vec{v}_{i} }{dt} \right)_{i} = \left( \frac{d \vec{v}_{r} }{dt} \right)_{r} + \frac{d}{dt}(\vec{\omega} \times \vec{r}_{r}) + \vec{\omega} \times \vec{v}_{r} + \vec{\omega} \times (\vec{\omega} \times \vec{r}_{r})$

$\vec{a}_{i} = \vec{a}_{r} + 2\vec{\omega} \times \vec{v}_{r} + \vec{\omega} \times (\vec{\omega} \times \vec{r}_{r})$
Multiplying the acceleration by the mass $m$ to get a force

$m\vec{a}_{i} = m\vec{a}_{r} + 2m\vec{\omega} \times \vec{v}_{r} + m\vec{\omega} \times (\vec{\omega} \times \vec{r}_{r})$

So, writing the force in the rotating frame in terms of the force in the inertial frame, we have

$\boxed{ m\vec{a}_{r} = m\vec{a}_{i} - 2m\vec{\omega} \times \vec{v}_{r} - m\vec{\omega} \times (\vec{\omega} \times \vec{r}_{r}) }$

So,

$\boxed{\vec{F}_{r} = \vec{F}_{i} - 2m\vec{\omega} \times \vec{v}_{r} - m\vec{\omega} \times (\vec{\omega} \times \vec{r}_{r}) }$

Notice that there are two extra terms (Term A and Term B) in the equation on the right, I have highlighted them below.

If we compare the force in a rotating frame to an inertial frame, two extra terms (Term A and Term B) arise. Term A is the Coriolis force, Term B is the centrifugal force

Term A is what we call the Coriolis force, which depends on the velocity in the rotating frame $v_{r}$. It is the force which causes water going down a plughole to rotate about the hole and to move anti-clockwise in the northern hemisphere and clockwise in the southern hemisphere. It is also the force which determines the direction of rotation of low pressure systems in the atmosphere. I will discuss the coriolis force more in a future blog.

Term B is the centrifugal force, the force we were aiming to derive in this blogpost. The strength of the centrifugal force depends on the position of the object in the rotating frame – $r_{r}$.

## What is the direction of the centrifugal force

The direction of $(\vec{\omega} \times \vec{r}_{r})$ can be found using the right-hand rule for the vector product, which I blogged about here. Remembering that the direction of $\vec{r}_{r}$\$ is radially outwards from the centre of the Earth, and the direction of $\vec{\omega}$ is the direction of the Earth’s axis (pointing north), then the direction of $\vec{\omega} \times \vec{r}_{r}$ is towards the east (right if looking at the Earth with the North pole up).

We now need to take the vector produce of $\vec{\omega}$ with a vector in this eastwards direction, and again using the right-hand rule gives us that the direction of $(\vec{\omega} \times \vec{r}_{r})$ is outwards (not radially from the centre of the Earth, but at right angles to the axis of the Earth). But, notice the centrifugal force has a minus sign in front of it, so the direction of the centrifugal force is outwards, away and at right angles to the Earth’s axis.

The direction of the centrifugal force is away from the axis of rotation, as shown in this diagram

This means that it acts to reduce the force of gravity which keeps us on the Earth’s surface. It also depends on the angle between where you are and the Earth’s axis, so is greatest at the equator and goes to zero at the pole. It means that you will weight slightly less than if the Earth were not rotating, but the effect is quite small and you would not notice such a difference going from the pole to the equator.

## What is the strength of the centrifugal acceleration due to Earth’s rotation?

Let us calculate the centrifugal force at the Earth’s equator, where it is at its greatest.

At the equator, we can write that the centrifugal acceleration has a value of

$\omega^{2} r \text{ as } \theta = 90^{\circ}$

We can calculate $\omega$ for the Earth by remembering that it takes 24 hours to rotate once, and $\omega$ is related to the period $T$ of rotation via

$\omega = \frac{2 \pi }{ T}$

We need to convert the period $T$ to seconds, so $T = 24 \times 60 \times 60 = 86400 \; s$. This gives that

$\omega = 7.272 \times 10^{-5} \text{ rad/s }$

If we take the Earth’s radius to be 6,378.1 km (this is the radius at the equator), then we have that

$\omega^{2} r = 0.0337 \text{ m/s/s}$

Compare this to the acceleration due to gravity which pulls us towards the Earth’s surface, which is 9.81 m/s/s and we can see that the centrifugal force at its greatest is only $0.34 \%$ of the acceleration due to gravity. Tiny.

It is, however, noticeable when you are on a roundabout, and is used on fairground rides where you spin inside a drum and the floor moves away leaving you pinned to the wall of the drum. The force you feel pushing against this wall is the centrifugal force, and it is very real for you in that rotating frame!

So, there we have it, centrifugal force does exist in a rotating frame of reference, but does not exist from the perspective of someone in an inertial frame of reference.

## The 100 greatest songwriters – number 21 – Lou Reed

At number 21 in Rolling Stone Magazine’s list of the 100 greatest songwriters is Lou Reed. I blogged about Lou Reed in October 2013 when he died, and shared in that week two of his songs, “Walk On the Wild Side” and “Perfect Day”.

Reed was born in New York City in 1942, and became the lead singer of Velvet Underground, who formed in 1964 and were briefly managed by Andy Warhol. Although they did not achieve much commercial success, the Velvet Underground are now generally recognised as one of the most influential bands in history. I blogged about their album The Velvet Underground here, it is at number 13 in Rolling Stone Magazine’s list of the 500 greatest albums.

The Velvet Underground broke up in 1973, and Reed embarked on a solo career which became more commercially successful. In fact, his song “Walk On the Wild Side” was a hit in 1972, when Velvet Underground were still together.

At number 21 in Rolling Stone Magazine’s list of the 100 greatest songwriters of all time is Lou Reed.

As there are so many Lou Reed written songs that I like, I had some difficulty in choosing one for this blogpost. I eventually chose “Sweet Jane”, which was written by Reed and originally recorded by the Velvet Underground on their album Loaded. In 1974 Reed released a solo version of the song as a single, taken from his album Rock ‘n’ Roll Animal.

Standin’ on a corner,
Suitcase in my hand.
Jack’s in his corset, Jane she’s in her vest,
Me, babe, I’m in a rock n’ roll band.
Riding that Stutz Bearcat Jim,
Those were different times.
All the poets studied rules of verse,
And the ladies, they rolled their eyes

Sweet Jane, Sweet Jane, Sweet Jane

Now, Jack, he is a banker,
And Jane, she is a clerk.
And the both of them save their money…
And when they come home from work.
Sittin’ by the fire…
Radio just played that classical music for all you protest kids,
The march of the wooden soldiers
And you can hear Jack say

Sweet Jane, Sweet Jane, Sweet Jane

Some people like to go out dancing
And other people, (like me) they gotta work
And there’s even some evil mothers
They’ll tell you life is just made out of dirt.
That women never really faint,
That villans always blink their eyes.
And that children are the only ones who blush.
And that life is just to die.
But, anyone who has a heart
Wouldn’t turn around and break it
And anyone who ever played a part
He wouldn’t turn around and hate it

Here is a video of Reed performing “Sweet Jane” live. Enjoy!

Which is your favourite Lou Reed song?