Inspired by the hard integral thread !!!
Topics of discussion : Hard questions from the A level Physics/ Mathematics (Mechanics) and the firstyear undergraduate classical mechanics topics including Special Relativity.
The second Question:
The two components of a binary star are observed to move in circles of radii r_{1} and r_{2}. What is the ratio of their masses?
(Note : Binary stars orbits around the combined centre of mass.)
If you have some interesting questions please post them.
Hard Mechanics Discussion Thread

 Follow
 1
 30092016 22:35
Post rating:2 
 Follow
 2
 01102016 10:45
This time a simpler one from the projectile motion :
Please feel free to post any questions, interesting/confusing ideas or anything interesting related to mechanics you may have.
Feel free to tag any members who you may think might be interested in this. 
 Follow
 3
 01102016 11:47
(Original post by tangotangopapa2)
Inspired by the hard integral thread !!!
Topics of discussion : Hard questions from the A level Physics/ Mathematics (Mechanics) and the firstyear undergraduate classical mechanics topics including Special Relativity.
The second Question:
The two components of a binary star are observed to move in circles of radii r_{1} and r_{2}. What is the ratio of their masses?
(Note : Binary stars orbits around the combined centre of mass.)
If you have some interesting questions please post them.Post rating:1 
 Follow
 4
 01102016 14:29
A balloon contains a bag of sand of initial mass , and the combined mass of balloon+sand is . The balloon experiences a constant upthrust R and is initially at rest and in equilibrium, with the upthrust compensating exactly for the gravitational force. At time t = 0 the sand is released at a constant rate dm/dt. It is fully disposed of after a time T.
(i) Determine the velocity of the balloon v(t) at any time between t = 0 and T. (Hint: Derive your expression in terms of )
(ii) Find the height gained by the balloon as a function of time.
(iii) For small values of express your results for the velocity and height gain at time T to first order in .Post rating:1 
 Follow
 5
 01102016 17:32
(Original post by langlitz)
A balloon contains a bag of sand of initial mass , and the combined mass of balloon+sand is . The balloon experiences a constant upthrust R and is initially at rest and in equilibrium, with the upthrust compensating exactly for the gravitational force. At time t = 0 the sand is released at a constant rate dm/dt. It is fully disposed of after a time T.
(i) Determine the velocity of the balloon v(t) at any time between t = 0 and T. (Hint: Derive your expression in terms of )
(ii) Find the height gained by the balloon as a function of time.
(iii) For small values of express your results for the velocity and height gain at time T to first order in . 
 Follow
 6
 01102016 17:37
(Original post by tangotangopapa2)
Really tough question. My previous solution for (i) was: but now I know that I missed out something. I used F = M(x) dv/dt which is wrong. Need to do it again writing F as F = M(x) dv/dt + dM/dt times v and solve the differential equation again. I am not sure I will be able to solve the differential equation. 
 Follow
 7
 01102016 18:22
(Original post by tangotangopapa2)
Really tough question. My previous solution for (i) was: but now I know that I missed out something. I used F = M(x) dv/dt which is wrong. Need to do it again writing F as F = M(x) dv/dt + dM/dt times v and solve the differential equation again. I am not sure I will be able to solve the differential equation.(Original post by langlitz)
Yes you're right there, it's an easy mistake to make as normally the mass isn't changing in mechanics questions. The integration is not too bad actually, I have faith in you
I am sorry but I just looked at the solution in the following sheet, last question: http://www2.ph.ed.ac.uk/~egardi/MfP3..._Workshop3.pdfLast edited by tangotangopapa2; 01102016 at 20:03. 
 Follow
 8
 01102016 21:00
One end of a thin inexstendable, but perfectly flexible, length of string 'L' with uniform mass per unit length is held at a point on a smooth table a distance 'd' (< L) away from a small vertical hole in the surface of the table. The string passes through the hole so that a length L  d of the string hangs vertically. The string is released from rest. Assuming the height of the table is greater than L, find the time taken for the end of the string to reach the top of the hole.
tangotangopapa2 this is a bit easy for youTagged: 
 Follow
 9
 01102016 22:01
(Original post by Ipsooo)
One end of a thin inexstendable, but perfectly flexible, length of string 'L' with uniform mass per unit length is held at a point on a smooth table a distance 'd' (< L) away from a small vertical hole in the surface of the table. The string passes through the hole so that a length L  d of the string hangs vertically. The string is released from rest. Assuming the height of the table is greater than L, find the time taken for the end of the string to reach the top of the hole.
tangotangopapa2 this is a bit easy for youSpoiler:Showsqrt (L/g) arc cosh (L/ (LD)) 
 Follow
 10
 01102016 22:04
Let me find some harder ones for you hmmPost rating:1 
 Follow
 11
 01102016 22:06

 Follow
 12
 02102016 09:32
(Original post by tangotangopapa2)
Wait a moment, If m_0 is very small compared to M_0. and dm/dt is fairly small, M(x)dv/dt could be reasonable approximation to F. In that case, rewriting above equation using R = M_0 g, we get: and then after integrating this (had to use integration of parts to solve integration of ln(1kt) by writing it as 1 times ln (1kt)) we can arrive at the following equation:
I am sorry but I just looked at the solution in the following sheet, last question: http://www2.ph.ed.ac.uk/~egardi/MfP3..._Workshop3.pdf 
 Follow
 13
 02102016 09:47
(Original post by langlitz)
Well done, that question is damned brutal (even when you cheat ). I have a similar and possibly harder one which has no answer online if you'd like to try it? 
 Follow
 14
 02102016 11:36
(Original post by tangotangopapa2)
Sure!!!
Post rating:1 
 Follow
 15
 03102016 11:48
A stick of mass density per unit length ρ rests on a circle of radius R.The stick makes an angle θ with the horizontal and is tangent to the circle at its upper end. Friction exists at all points of contact and it is large enough to keep the system at rest. Find the friction force between the ground and the circle.
Spoiler:Show(Yes, the information given is complete. You do not need coefficient of friction or the frictional force between stick and circle/ground). Good Luck haha.
If you have something please post it. 
 Follow
 16
 05102016 22:04
a) Where (v)s are vectors, don't know how to write vectors in LaTex.
b) Momentum is only conserved if there is no external force, as there is external drag momentum is not conserved. The change in energy accounts for the work done against the resistive force.
c) (By conservation of momentum) Mv + delta mu = 0 Differentiating with respect to time and writing, dM/dt = 0 and du/dt = 0, we get the given equation. (I have taken reference frame to be the frame stationary with respect to rocket just before the propulsion.) (This is the famous rocket equation where F = 0.)
d) Integrating the equation given in c) with limits M= M_o + m_o to M = M_o + m_o  m_1 we get the equation in v.
e) As, v_o and u are parallel, we could just write their magnitudes in the equation. Now we simply have to plug in values to obtain:

 Follow
 17
 06102016 19:10
(Original post by Ipsooo)
One end of a thin inexstendable, but perfectly flexible, length of string 'L' with uniform mass per unit length is held at a point on a smooth table a distance 'd' (< L) away from a small vertical hole in the surface of the table. The string passes through the hole so that a length L  d of the string hangs vertically. The string is released from rest. Assuming the height of the table is greater than L, find the time taken for the end of the string to reach the top of the hole.
tangotangopapa2 this is a bit easy for you 
 Follow
 18
 06102016 20:47
That's a neat one. ^^
Spoiler:Show
You can treat the string as a particle falling through space, but with variable mass.Spoiler:ShowThis is because the mass of the horizontal part of the string does not cause acceleration (balanced by table reaction force), but the mass of the vertical part of the string does. So the mass of the object is effectively the mass per unit length times the length of string hanging vertically, which is a function of time. Set up the equations allowing mass to be a function of this length, and this length to be a function of the displacement of the end of the string, which (via F=ma integrated) is a function of the mass.
You should get a 2nd order differential equation in time that should be quite easy to solve.Last edited by mik1a; 06102016 at 20:49.Post rating:1 
 Follow
 19
 06102016 21:34
(Original post by NatoHeadshot)
Hint plsLast edited by tangotangopapa2; 06102016 at 21:49. 
 Follow
 20
 06102016 22:12
(Original post by tangotangopapa2)
Are my answers correct?
a) Where (v)s are vectors, don't know how to write vectors in LaTex.
b) Momentum is only conserved if there is no external force, as there is external drag momentum is not conserved. The change in energy accounts for the work done against the resistive force.
c) (By conservation of momentum) Mv + delta mu = 0 Differentiating with respect to time and writing, dM/dt = 0 and du/dt = 0, we get the given equation. (I have taken reference frame to be the frame stationary with respect to rocket just before the propulsion.) (This is the famous rocket equation where F = 0.)
d) Integrating the equation given in c) with limits M= M_o + m_o to M = M_o + m_o  m_1 we get the equation in v.
e) As, v_o and u are parallel, we could just write their magnitudes in the equation. Now we simply have to plug in values to obtain:
p.s. to write vectors in latex you do \vec{b} for example for a vector
Write a reply…
Reply
Submit reply
Register
Thanks for posting! You just need to create an account in order to submit the post Already a member? Sign in
Oops, something wasn't right
please check the following:
Sign in
Not got an account? Sign up now
Updated: October 8, 2016
Share this discussion:
Tweet
Related discussions:
 AQA AS Mathematics MM01 Mechanics 1 ? Friday 12th June ...
 AQA A2 Mathematics MM2B Mechanics 2  Monday 22nd ...
 AQA A2 MM2B Mechanics 2 ? 27th June 2016 [Exam ...
 AQA AS M1 Mechanics 1 21st June 2016 [Exam Discussion ...
 Additional Further Maths Discussion Thread
 AQA A2 MFP3 Further Pure 3 ? 18th May 2016 [Exam ...
 Aqa aslevel ict info 1 & ino 2 exam discussion thread! 1) (03 ...
 Edexcel FP1  May 14th, 2015 [Exam discussion thread]
 Edexcel M4 Exam Discussion Thread [June 5th]
 AQA A2 Mathematics MPC3 Core 3  Friday 5th June 2015 ...
TSR Support Team
We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.