# Hard Mechanics Discussion Thread

Watch
Announcements

Inspired by the

Topics of discussion : Hard questions from the A level Physics/ Mathematics (Mechanics) and the first-year 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

(Note : Binary stars orbits around the combined centre of mass.)

If you have some interesting questions please post them.

**hard integral thread !!!**Topics of discussion : Hard questions from the A level Physics/ Mathematics (Mechanics) and the first-year 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.

2

reply

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.

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.

0

reply

Report

#3

(Original post by

Inspired by the

Topics of discussion : Hard questions from the A level Physics/ Mathematics (Mechanics) and the first-year 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

(Note : Binary stars orbits around the combined centre of mass.)

If you have some interesting questions please post them.

**tangotangopapa2**)Inspired by the

**hard integral thread !!!**Topics of discussion : Hard questions from the A level Physics/ Mathematics (Mechanics) and the first-year 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.

1

reply

Report

#4

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 .

(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 .

1

reply

(Original post by

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 .

**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 .

0

reply

Report

#6

(Original post by

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.

**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.

0

reply

**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

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

**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.pdf

0

reply

Report

#8

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

tangotangopapa2 this is a bit easy for you

0

reply

(Original post by

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

**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

Spoiler:

Show

sqrt (L/g) arc cosh (L/ (L-D))

0

reply

0

reply

Report

#12

(Original post by

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(1-kt) by writing it as 1 times ln (1-kt)) 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

**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(1-kt) by writing it as 1 times ln (1-kt)) 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

0

reply

(Original post by

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?

**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?

0

reply

Report

#14

1

reply

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.

If you have something please post it.

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.

0

reply

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) M

**v**+ delta m

**u**= 0 Differentiating with respect to time and writing, dM/dt = 0 and d

**u**/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:

0

reply

Report

#17

**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

0

reply

Report

#18

That's a neat one. ^^

Spoiler:

You can treat the string as a particle falling through space, but with variable mass.

Show

You can treat the string as a particle falling through space, but with variable mass.

Spoiler:

Show

This 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.

You should get a 2nd order differential equation in time that should be quite easy to solve.

1

reply

(Original post by

Hint pls

**NatoHeadshot**)Hint pls

0

reply

Report

#20

(Original post by

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) M

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,

**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) M

**v**+ delta m**u**= 0 Differentiating with respect to time and writing, dM/dt = 0 and d**u**/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

0

reply

X

### Quick Reply

Back

to top

to top