# The Cambridge Natural Sciences Interview

[ physics ]

In December 2015, just after finishing my IB exams, I went to Cambridge for my (Physical) Natural Sciences Interview. The interview has a reputation for being incredibly demanding and intense, and you’ve probably seen on the internet some of the ‘crazy’ questions that people get asked. Although it is definitely stressful, the Cambridge interview is almost purely academic (and quite reasonable). So without further ado, these are the questions I was asked, along with my thoughts at the time and the interviewer’s reactions.

## Interview 1 – physics and chemistry

Prior to the interview I had thought about this a lot, because I knew they were going to ask me about it. So I just described the experiment I did and what I learnt from it. The interviewers seemed very impressed by this, and they didn’t ask anything more about it.

What does F = ma mean

Of course it’s one of Newton’s laws, so with abounding confidence I said “that’s Newton’s first law!”. Something about that sounded weird, but the damage was done and I will never forgive myself for failing to identify Newton’s 2nd. I was mortified, but the interviewer chuckled and said not to worry about it. I recovered this by giving a half-decent explanation in terms of inertia, i.e reluctance to move. I think this is one of the setup questions to test your fundamental understanding before moving onto harder stuff.

Imagine we have a box, hanging by a light inextensible rope, in an elevator. The box weighs 5kg. The lift is accelerating upwards at 4$ms^{-2}$. What is the tension in the rope?

We will now connect another box, via a light inextensible rope, to the bottom of the box in the previous question. What is the tension in this new piece of rope?

I was familiar with what happens when you put weighing scales in lifts, and this problem was similar. But I got quite confused doing the question, especially with the signs of acceleration, force etc. Frustrating, because I know that if it were an exam I’d be able to do it easily. I now see that the whole question would have easily (and impressively) been done by considering the principle of equivalence. I think I did eke out the correct answer in the end, but it wasn’t pretty.

For the second question, as soon as the question was asked, I made an intuitive observation regarding the system, and it turned out that this is exactly what he wanted. I can’t remember exactly what I said, but I think it was something regarding how we can consider the bottom of the second box as the new ‘roof’ of the elevator, so to speak.

At this point, the second interviewer (who I believe specialised in chemistry) took over. I was really not looking forward to this part, because on my personal statement I said that I had read Why Chemical Reactions Happen – I did read it, but it was a lot harder than I expected it to be, and I wasn’t too sure that I could answer questions about it.

I explained it in terms of electron density. When a hydrogen atom moves near an oxygen atom (for example), the high positive charge density of the oxygen draws toward it the hydrogen electron cloud, leaving the hydrogen atom with a partial positive charge. I think you have to be careful not to rattle off textbook definitions, you have to actually understand what you’re saying.

If I mix some lithium ions in water, what happens?

I said something about coordinate complexes. Although lithium isn’t a transition metal, it has a very high charge density, so I suggested that it could form some kind of ligands. I drew a little diagram with curly arrows going from the lone pairs on $OH$ or $H_2O$ to the lithium ion.

If I mix some aluminium ions in water, what happens?

This one threw me off, because I would have thought that aluminium behaved the same way as lithium. So when they asked me about aluminium after asking me about lithium, I figured something was up. I tried to make up some stuff about Lewis acid-base character (when I say ‘make up’, I mean that I brought up facts that I knew were true, but of questionable relevance).

In the end, after some fidgeting, I decided to be honest. I said “I don’t see how it’s different to the lithium case”. She then asked the next question (very closely related).

Is the aluminium-ligand bond stronger or weaker than the lithium-ligand bond?

I first stated the relevant features. The ionic radius, and the charge. Aluminium obviously has greater charge, but then on the other hand I thought it had a larger radius (I was wrong). I didn’t know how to compare these opposing factors. I reasoned that, because of Coulomb’s law, the radius would be more important (because it is an $r^2$), but she informed me that the charge was much more significant.

In hindsight, I could have worked that out. Aluminium has 3/2 times the charge. For the ionic radius to be more significant, aluminium’s radius would have to be at least $\sqrt{3/2}$ (about 1.2) times greater than that of lithium.

What happens when $Al_2Cl_6$ reacts with water?

I think a description would have sufficed, but I wrote the chemical equations

$Al_2Cl_6 + 6H_2O \to [Al(H_2O)_6]^{3+} + 6Cl^{-}$ $[Al(H_2O)_6]^{3+} \to [Al(H_2O)_5OH]^{2+} + H^+$

I also mentioned something about the pH decreasing.

In the above equilibrium, how would I produce more of the metal ion

I had no idea what I was being asked. I clarified once, but still didn’t get it. But I figured that it was something to do with Le Chatelier’s principle, so I talked about how adding a strong acid or strong base would affect the equilibrium. This seemed to satisfy them.

Any questions?

I asked “what do you think is the next upcoming field in your subject?”. Physics guy said ‘energy’, chemistry person agreed. Obviously I asked this question because everyone told me that you have to ask something good at the end. I figured I had nothing to lose by asking.

All in all, I felt that the first interview was disastrous, especially regarding chemistry, even though I wasn’t asked anything about Why Chemical Reactions Happen.

## Interview 2 – physics and maths

I felt that I was more prepared for this interview, because it didn’t involve chemistry.

Have you heard of Felix Baumgartner?

Yes, he jumped off some balloon.  I was informed that he had jumped from 40,000m.

Tell me about the forces involved in his jump

Obviously, the force of gravity, and the force of air resistance. I said that I couldn’t find his equation of motion, because I didn’t know about the air resistance. We came back to this shortly.

What value of g are you using?

I perceived that the question was about the variation of g. I wrote down $g=GM/R^2$. I knew that the radius of the earth was about 6400km, so I did a quick mental calculation to note the difference in order of magnitude between 40,000m and 6400km. Therefore, I concluded that the variation of g was not relevant. He agreed, but asked me to quantify it.

I replaced $R$ with $R+h$, ($R$ is the Earth’s radius and $h$ is the height above the surface) and expanded, but didn’t know how to proceed. I think the key point was that the $h^2$ term vanishes in comparison to the $R^2$ and the $2Rh$.

$g = \frac{GM}{(R+h)^2} =\frac{GM}{R^2 + 2Rh + h^2} \approx \frac{GM}{R^2 + 2Rh}$

How does air pressure vary as a function of height?

This was quite a frustrating question. Everyone knows that there is an inverse relationship, but I had never stopped to consider the exact function.I said that I could work probably work it out if I knew the distribution for pressure or density.

It turns out that it can be derived using the ideal gas equation to find the density as a function of height and $dP$, then substituting this into $P = \rho g h$ to get a differential equation.

Have you heard of the Boltzmann distribution?

Yes, it is a distribution for the velocities of particles in an ideal gas. I don’t know the precise form of the equation, except that it has an $\exp(\frac 1 T)$ in it.

This was a weird question, and seemed a bit out of place. But it related to the previous question. Since I didn’t know the form of the Boltzmann distribution, the guy just said “assume that the pressure halves every 5000m”. As he said this I wrote down $\rho = \rho_0 (\frac{1}{2})^{h/5000}$, and he seemed satisfied.

With the above in mind, could Felix have broken the sound barrier?

I thought that he wanted me to use the pressure function from the previous question to do the differential equation. But when I started, the interviewer got a bit impatient and told me that the pressure thing was a separate part of the question. I think that was a bit unfair of him, but never mind.

What they wanted was much more simple. I had already shown that changes in g were pretty much irrelevant. Likewise, because of the exponential relationship of air pressure, air resistance was also negligible. So it reduces to a simple SUVAT equation. I did it mentally and told him the answer, and he was surprised that I got it correct.

At this point, the second interviewer took over. I was thankful to switch interviewers (the first had been a bit grumpy), but I knew that the second interviewer was the author of my IB maths textbook, which is a bit of a daunting prospect.

Integrate $f(x) = \arcsin(x)$

Standard trick, use integration by parts with $u=\arcsin(x)$ and $dv = 1$.

Differentiate $f(x) = \arcsin (\cos x)$

I was very happy to get some calculus, because I thought it was my strong suit.

I started off by saying that I would simplify the function, but this is actually a mistake because it can’t be simplified – I got $f(x)$ confused with something similar, $\cos(\arcsin x)$ (which can be simplified). So I had to backtrack and just differentiated it normally.

Easy application of the chain rule, so as usual I skipped a few steps (which I later regretted).

$\frac{d}{dx} \arcsin (\cos x) = -\frac{\sin x}{\sqrt{1-\cos^2 x}} = -1$

(Do you spot the mistake here? I didn’t)

Find $\int_0^1 f(x)dx$

I had absolutely no clue how to integrate it (at first). I started off by suggesting substitutions, but knew that these wouldn’t work.Luckily I had a mini ‘eureka’ moment – the derivative is constant!. This means that the $f(x)$ ALWAYS has the same negative gradient – it’s just a straight line.

So I knew that $f(x)$ could be rewritten as $–x + c$. I didn’t know how to proceed at first, but the interviewer suggested “how might you find c?” and I worked it out. $c = f(0) = \pi/2$. The integral is then just the area of the triangle. The interviewer seemed very pleased that I had managed to spot this method. However, he then smiled and asked the next question.

That’s a nice method. But the answer is wrong. What did you miss?

As soon as he said I missed a step, I knew exactly where my error was - I felt that I had messed up the square root of $(1 - \cos^2 x)$ in the derivative. He agreed, but after this it took me a while to figure out exactly what it was. I suggested that I had forgot a +/-, but it wasn’t that. We then had a really sad discussion (sad and embarrassing for me) about square roots.

What is the square root of 16?

+/- 4, but the sqrt sign technically means only the positive root, so 4.

What is the square root of 4?

2.

What is the square root of $( -2)^2$?

I got really flustered, said -2, then quickly corrected to 2.

What is the square root of $x^2$?

Thank goodness I realised what he was getting to, the answer is clearly $|x|$. I’m ashamed it took me so long. Having figured this out, I quickly realised that the function would look like a sawtooth, and said so.

Overall, I was pleased that I figured out the integral, but very humbled by the square root discussion. Of course, I missed an obvious trick to instantly figure out the integral:

$\int_0^1 \arcsin (\cos x)dx = \int_0^1 \arcsin (\sin(x + \pi/2))dx$

What is the factorial of a half?

I smiled and said $\sqrt{\pi/2}$. (I mentioned the Gamma function in my personal statement).

In your maths exploration, how did you show that the gamma function worked as a factorial?

Repeated integration by parts. I asked if he wanted me to do it, he said no and seemed to be content with this answer.

Is the Gamma function the only function that can model a factorial?

I said that it wasn’t: given any (finite) set of coordinates, there are an infinite number of functions that can model it (polynomials).

However, I said that I had proved that the gamma function satisfies the recursion principle, and that $\Gamma(1)=0$. This means that it is at worst, a superset of the factorial. I also said that I was aware that the Gamma function had an extra property which meant that it was the only function that satisfactorily extended the factorial to non-integers, but I was honest and said I didn’t know exactly what it was.

He told me that the name of the property was analytic continuation. Thanks mate.

Why do you want to study natural sciences?

This is a question that everyone should prepare for, but I mucked it up.  In my personal statement, I had already talked about the philosophical aspects – I love to find simple explanations, reductionism etc. I hinted at that in this answer, but ended up saying “these are where my strengths lie”. The grumpy interviewer said “so you’re basically saying that you’re good at this subject”. I gave a cautious “yeah”

How are you going to keep your academics up to scratch prior to entering?

I said that I had already planned which books I would study, mentioning the Feynman lectures. I think this was a smart choice, because I know that physicists and mathematicians love Feynman. They were happy, and said that it was a good book.

Any questions? You don’t have to try to impress us with good questions, we’re genuinely asking if you have any questions.