OCR Physics B Synoptic Insert

[Tim Alfred]

I saw that someone else was asking if anyone had resources for this
year's insert and was wondering the same. My class was given a load of
questions that supposedly might come up but no answers and since I'm on
study leave this has limited use.

One particularly hard question refers to the 3 body problem and
'Lagrange points'. It asks 'At what distance from the Earth would the
gravitational fields of the Earth and the Sun exactly cancel?'. At
first I thought this was easy because it would be just be where GM /
r^2 was the same for the Earth and the Sun but I think this gives an
answer of 2.60 x 10^8 m whilst the actual position is 1.5 x 10^9 m. The
next question dos ask 'Why is it necessary for SOHO to be slightly
closer to the Earth than the point calculated?' so I know my answer
must be wrong.

Researching on the intenet suggested that this problem was very
complicated, coming down to having to solve a quintic equation, clearly
something we won't be asked. So I don't know if there is a simple way
to do the question or if the person who wrote it didn't realise how
difficult it was.

Does anyone have any ideas for this question? Or any resources for
revising this part of the Synoptic paper?

[Dr A. N. Walker]

Post by Tim Alfred
I saw that someone else was asking if anyone had resources for this
year's insert and was wondering the same. My class was given a load of
questions that supposedly might come up but no answers and since I'm on
study leave this has limited use.

OK, I know nothing at all about the "OCR Physics B Synoptic
Insert" in particular, but the idea of a synoptic paper in general
is to test "across the field" rather than specific topics. In maths
and physics [in particular] this tends to mean "back of the envelope"
calculations. So you tend to need to know things like "How big is
the Earth?", "What is a typical wind speed?", "How much does a mouse
weigh?" and/or to be able to estimate these things easily and use
the result in an order-of-magnitude calculation.

Post by Tim Alfred
One particularly hard question refers to the 3 body problem and
'Lagrange points'. It asks 'At what distance from the Earth would the
gravitational fields of the Earth and the Sun exactly cancel?'. At
first I thought this was easy because it would be just be where GM /
r^2 was the same for the Earth and the Sun but I think this gives an
answer of 2.60 x 10^8 m

The Earth-Sun distance is around 93M miles, and the year is
about 30M seconds, so the acceleration of the Earth towards the Sun
is 93Mx5280x(2pi/30M)^2 ft/s^2, which is, give-or-take, 1/1600 g
[where g ~ 32 ft/s^2], so you need [inverse square] to be about 40
Earth-radiuses or 160000 miles away from Earth for the gravitational
forces to balance; that is indeed about 260000 km.

Post by Tim Alfred
whilst the actual position is 1.5 x 10^9 m.

See below.

Post by Tim Alfred
The
next question dos ask 'Why is it necessary for SOHO to be slightly
closer to the Earth than the point calculated?' so I know my answer
must be wrong.

1.5M km is not "slightly closer" than 0.26M km, so that's
not the point the Q is trying to make! There are several things
they may have had in mind, such as that the Earth-Sun system is
rotating about the centre of mass of the system rather than the
centre of the Sun, or that SOHO is rotating about the Earth once
per year rather than "stationary", or they may have been thinking
about radiation pressure or tides or the effects of the Moon or
relativity or something else daft. I suspect the question is a
botch.

Post by Tim Alfred
Researching on the intenet suggested that this problem was very
complicated, coming down to having to solve a quintic equation, clearly
something we won't be asked. So I don't know if there is a simple way
to do the question or if the person who wrote it didn't realise how
difficult it was.

Well, it's not *that* complicated. They *asked* about
gravity; but there is another major effect, namely centrifugal
force. Indeed, by definition, centrifugal force is sufficient
at the Earth's orbit to keep the Earth "up" against the Sun's
gravity with no help at all from the Earth's gravity. So, at the
point you [and I] just calculated, the gravitational forces may
balance, but almost the full centrifugal force is still acting,
and SOHO would zoom "out" [relative to rotating co-ordinates in
which the Earth is stationary with respect to the Sun]. So you
need to move further from Earth, so that the total gravity towards
the Sun balances the centrifugal force. This is indeed given by
a quintic equation, though one that's fairly easy to write down,
and fairly easy to solve numerically, esp since the Earth is much
less massive than the Sun.

Alternatively: at the point calculated, gravity forces
balance, so in "true" Newtonian mechanics the satellite has zero
acceleration; but the Earth is accelerating towards the Sun, so
the satellite is accelerating relative to the Earth.

[Tim Alfred]

Thank you very much your very full answer, good to see that you got the
same one! I agree with you that the question is probably wrong as other
questions from the same set have some mistakes in them.

The points you give about the complications of the question are what I
had kind of worked out but stated more clearly than I could find. But I
don't see how a quintic equation can be 'fairly easy to solve
numerically.' Do you mean that you could basically use trial and error
if you have the equation in the form f(x) = 0 with x being the distance
of the satellite from the Earth?

[Dr A. N. Walker]

[...]. But I
don't see how a quintic equation can be 'fairly easy to solve
numerically.' Do you mean that you could basically use trial and error
if you have the equation in the form f(x) = 0 with x being the distance
of the satellite from the Earth?

Not exactly "trial and error". If you have done a Numerical
Methods module as part of your A-levels, you will/should have heard
of the "Newton-Raphson" method, amongst others. If not, then Google
is your friend! N-R makes it very easy [in this case] to get full
calculator accuracy [say, 12 significant figures] in the answer in
something of the order of ten minutes; if you want more, or faster,
you'll need to use a computer ....

[Tim Alfred]

OK, I see your point. I learnt Newton-Raphson method in Further Maths
AS level. This still shows the question is wrong though as there is no
way Newton-Raphson method would be required in A level Physics.