How do the degrees of freedom change

I came across the following problem in an old exam:

How many degrees of freedom does a system of 4 mass points (A,B,C,D) have, if the distances AB, BC and CD are given?

So my attempt was to say particle A can move freely and has therefore 3 DOF, particle B is then constrained by the distance AB can only rotate around A, and has therefore 2 DOF. The same then applies for C and D. So in total I would have 3+2+2+2=9 degrees of freedom, is this a correct approach?

I would really appreciate your help and advice on this particular problem (but maybe also from a more general perspective).

First of all yes, your answer seems correct (if we talk about a system in a 3 dimensional space). In general, you can consider all the possible dof for the system. So, 4 particles each with 3 dof make it 12 dof and then you simply subtract the amount of constraints you have (in your case this would mean 12-3=9 dof. (with the given distances as constraints).

Hopefully this helps you!

• Thank you for your help!
– user307828
Jul 25, 2021 at 14:56
• You talked about 3 particles but question had 4, I am a bit lost. Is it that considering four particles the total DOF would become $(4\times3)-3=9$ DOF? Jul 25, 2021 at 16:00
• Yes you are right, I completly overlooked the 4. one and just wrote it for 3, but it works the same way as I explained. I will edit it to the correct amount of particles shortly
– user292868
Jul 25, 2021 at 16:22
• So just again for everyone who might wonder: If you have n particles and m deegres of freefom in general you have m times n DOF. If you then consider some x conditions of restraint (not sure if you call it so in eglish, I translated it from russian) you get a total of (nm)-x DOF
– user292868
Jul 25, 2021 at 16:29
• Thank you both again, I undestood what you meant but now it is clear for everyone
– user307828
Jul 25, 2021 at 16:32