Cambridge Physics Forums
Let me reiterate that questions over here are not homework questions - I am merely doing them for sport. The questions are based on the circa-1991 A Level syllabus. They are for students preparing for Special Papers and are of 1st-year undergraduate difficulty.
Believe me I tried very hard to answer the questions but I really ran out of ideas to solve them. I would genuinely appreciate if anyone could provide the workings to solve the questions.
There will be more than one question posted in this thread. I do not intend to post a new thread for every single question as I do not want to make TSR look messy. If anyone begs to differ, please let me know and I will post new questions in new threads. I will start off with the easier questions; the more difficult, calculus-based questions will be posted later on. I would appreciate if you could write down what question(s) you asked yourself when you're attempting the questions.
Here goes:
A bag containing a mass M of sand is suspended from a hook on the arm of a balance at a height h above the balance pan. At a time t=0, the sand starts to pour from a hole at the bottom of the bagm falling onto the pan beneath, and continues at a constant rate r (mass/unit time) until the bag is empty.
i. Find the mass required on the other pan to maintain balance under steady conditions when a continuous stream of sand is falling from the bag (a simple balance with equal arms is envisaged).
ii. Show graphically the variation with time of the mass required to maintain balance throughout the experiment, indicating by suitable labelling the quantities involved. Assume ideal conditions, under which air resistance, balance inertian and damping effects may be ignored.
My working:
When we consider the pouring of sand from a height h above the pan, there exists 3 phases: the falling sand yet to reach the pan; the steady stream of sand whose momentum is destroyed upon reaching the pan and the final stage when the steady stream of sand reduces to zero when all of the sand in the bag is depleted.
i. So for the "mass required on the other pan to maintain balance under steady conditions when a continuous stream of sand is falling from the bag", I tried starting from the conservation of energy gh = 0.5 v^2. Since the height is yet to be found, I couldn't use the impulsive force on destroying of momentum too.
I apologise for not having mathematical workings to present, for I am uncertain of what terms - and how many of them - should I introduce into the working.
Please help. Thank you.
Believe me I tried very hard to answer the questions but I really ran out of ideas to solve them. I would genuinely appreciate if anyone could provide the workings to solve the questions.
There will be more than one question posted in this thread. I do not intend to post a new thread for every single question as I do not want to make TSR look messy. If anyone begs to differ, please let me know and I will post new questions in new threads. I will start off with the easier questions; the more difficult, calculus-based questions will be posted later on. I would appreciate if you could write down what question(s) you asked yourself when you're attempting the questions.
Here goes:
A bag containing a mass M of sand is suspended from a hook on the arm of a balance at a height h above the balance pan. At a time t=0, the sand starts to pour from a hole at the bottom of the bagm falling onto the pan beneath, and continues at a constant rate r (mass/unit time) until the bag is empty.
i. Find the mass required on the other pan to maintain balance under steady conditions when a continuous stream of sand is falling from the bag (a simple balance with equal arms is envisaged).
ii. Show graphically the variation with time of the mass required to maintain balance throughout the experiment, indicating by suitable labelling the quantities involved. Assume ideal conditions, under which air resistance, balance inertian and damping effects may be ignored.
My working:
When we consider the pouring of sand from a height h above the pan, there exists 3 phases: the falling sand yet to reach the pan; the steady stream of sand whose momentum is destroyed upon reaching the pan and the final stage when the steady stream of sand reduces to zero when all of the sand in the bag is depleted.
i. So for the "mass required on the other pan to maintain balance under steady conditions when a continuous stream of sand is falling from the bag", I tried starting from the conservation of energy gh = 0.5 v^2. Since the height is yet to be found, I couldn't use the impulsive force on destroying of momentum too.
I apologise for not having mathematical workings to present, for I am uncertain of what terms - and how many of them - should I introduce into the working.
Please help. Thank you.
I strongly suggest a separate thread for each question, otherwise this thread will become long and muddled.
For this question, it is solved using momentum change per second for the sand striking the pan.
The velocity of the mass striking the pan is as you say, from
The mass striking the pan per second is r
Assuming the mass is brought to rest, the change in momentum per second is
mass per second x velocity of sand
You have all the terms to put into this equation.
The mass on the other pan is the mass whose weight is equal to the force found from this equation.
For this question, it is solved using momentum change per second for the sand striking the pan.
The velocity of the mass striking the pan is as you say, from
The mass striking the pan per second is r
Assuming the mass is brought to rest, the change in momentum per second is
mass per second x velocity of sand
You have all the terms to put into this equation.
The mass on the other pan is the mass whose weight is equal to the force found from this equation.
(edited 12 years ago)
You also have to bear in mind the sand that is in midair and is thus not exerting any force on the pan while it falls.
This used to a favourite discussion question when I used to coach Oxbridge candidates.
This used to a favourite discussion question when I used to coach Oxbridge candidates.
You also have to bear in mind the sand that is in midair and is thus not exerting any force on the pan while it falls.
Indeed. That is in fact an interesting one.
Original post by teachercol
This used to a favourite discussion question when I used to coach Oxbridge candidates.
This used to a favourite discussion question when I used to coach Oxbridge candidates.
Are you implying that you've done this question before? Does this mean you know which book am I using?
Thank you for your help, teachercol. And many thanks to Stonebridge too.
Not sure which book you're using but yes, I have seen it before.
I also have a file with old special papers and my answers!
I also have a file with old special papers and my answers!
Assuming the mass is brought to rest, the change in momentum per second is
mass per second x velocity of sand
(...)
The mass on the other pan is the mass whose weight is equal to the force found from this equation.
mass per second x velocity of sand
(...)
The mass on the other pan is the mass whose weight is equal to the force found from this equation.
Consider this diagram:
Since the rate of pouring of sand is r, and that the velocity attained on hitting the pan (thereby the change in velocity) is v = (2gh)^0.5, then how do I show that r.(2gh)^0.5 is M?
And since m2 doesn't represent the whole of M, and that the mass of the whole left arm is already M, how do I ensure that
(impulsive force by m2) + m1 + m3 equals to M (which is the answer for the mass of the object on the right arm)?
Could you please show me the mathematical solution for this question (at least by pm)? It's been bugging me for far too long now and I'm this close to losing hope to solve it already.
You might like to take a look at this thread I started a while back which has an interesting problem based on exactly the same principle.
The solution is given towards the end.
http://www.thestudentroom.co.uk/showthread.php?p=31430619
I may have been slightly misleading in my first answer above as I hadn't noticed that the bag of sand is also suspended from the arm of the balance.
This doesn't alter the maths of the momentum change of the sand on hitting the pan, but it does make a difference to the other mass required to balance the system in the steady state.
The solution is given towards the end.
http://www.thestudentroom.co.uk/showthread.php?p=31430619
I may have been slightly misleading in my first answer above as I hadn't noticed that the bag of sand is also suspended from the arm of the balance.
This doesn't alter the maths of the momentum change of the sand on hitting the pan, but it does make a difference to the other mass required to balance the system in the steady state.
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