Previous in Forum: Why is the Sky Blue?   Next in Forum: How I Was Almost Killed
Close
Close
Close
65 comments
Rate Comments: Nested
Guru
United Kingdom - Member - Not a New Member Hobbies - Musician - New Member Hobbies - Fishing - New Member

Join Date: May 2006
Location: Reading, Berkshire, UK. Going under cover.
Posts: 9684
Good Answers: 468

Maths on Friday

07/06/2012 11:50 AM

A rectangular steel bar is pivoted as shown below. It is released from a position 1° away from vertical. How long does it take to reach the horizontal position?

Assume a 1G gravitational field. Neglect friction. State any other assumptions you need to make.

P.S. This was inspired by a near miss with a toilet seat .

__________________
"Love justice, you who rule the world" - Dante Alighieri
Register to Reply
Interested in this topic? By joining CR4 you can "subscribe" to
this discussion and receive notification when new comments are added.

"Almost" Good Answers:

Check out these comments that don't yet have enough votes to be "official" good answers and, if you agree with them, vote them!
Guru
Hobbies - CNC - New Member Hobbies - DIY Welding - New Member Engineering Fields - Electromechanical Engineering - New Member

Join Date: Aug 2007
Posts: 23647
Good Answers: 420
#1

Re: Maths on Friday

07/06/2012 12:05 PM

P.S. This was inspired by a near miss with a toilet seat

was anybody hurt?

Is one degree enough to have the moment of enertia offset enough it to make it fall?

I say no!

Now if we were talking about toilet seat instead of a steel bar........

__________________
“ When people get what they want, they are often surprised when they get what they deserve " - James Wood
Register to Reply
Guru
United Kingdom - Member - Not a New Member Hobbies - Musician - New Member Hobbies - Fishing - New Member

Join Date: May 2006
Location: Reading, Berkshire, UK. Going under cover.
Posts: 9684
Good Answers: 468
#8
In reply to #1

Re: Maths on Friday

07/06/2012 1:31 PM

I'd say that 1° would be plenty to get it moving (remember we're neglecting friction). In fact in an ideal situation it would move if the rod were anything other than vertical.

__________________
"Love justice, you who rule the world" - Dante Alighieri
Register to Reply
Guru
Hobbies - Fishing - Old Salt Hobbies - CNC - New Member United States - US - Statue of Liberty - New Member

Join Date: Mar 2007
Location: Rosedale, Maryland USA
Posts: 5197
Good Answers: 266
#2

Re: Maths on Friday

07/06/2012 12:08 PM

http://www.youtube.com/watch?v=c86QENS2Uq4

Can take longer then you think with no more near misses.

__________________
Life is not a journey to the grave with the intention of arriving in a pretty, pristine body but rather to come sliding in sideways, all used up and exclaiming, "Wow, what a ride!"
Register to Reply
Guru
Hobbies - CNC - New Member Hobbies - DIY Welding - New Member Engineering Fields - Electromechanical Engineering - New Member

Join Date: Aug 2007
Posts: 23647
Good Answers: 420
#3
In reply to #2

Re: Maths on Friday

07/06/2012 12:13 PM

Can take longer then you think with no more near misses.

But will I have time to finish the job?

__________________
“ When people get what they want, they are often surprised when they get what they deserve " - James Wood
Register to Reply
Guru
United Kingdom - Member - Not a New Member Hobbies - Musician - New Member Hobbies - Fishing - New Member

Join Date: May 2006
Location: Reading, Berkshire, UK. Going under cover.
Posts: 9684
Good Answers: 468
#7
In reply to #2

Re: Maths on Friday

07/06/2012 1:24 PM

I have a "slow closer" at home - tho' it moves pretty swiftly for the first 60° or so. The "near miss" was at work.

__________________
"Love justice, you who rule the world" - Dante Alighieri
Register to Reply
Guru

Join Date: Mar 2007
Location: by the beach in Florida
Posts: 33392
Good Answers: 1817
#4

Re: Maths on Friday

07/06/2012 1:00 PM

You need one of these....

Automatic toilet seat.... one wave = lid opens....another wave = seat opens...auto close in 15 sec...

__________________
All living things seek to control their own destiny....this is the purpose of life
Register to Reply Score 1 for Good Answer
Guru
Hobbies - CNC - New Member Hobbies - DIY Welding - New Member Engineering Fields - Electromechanical Engineering - New Member

Join Date: Aug 2007
Posts: 23647
Good Answers: 420
#6
In reply to #4

Re: Maths on Friday

07/06/2012 1:07 PM

You need a secondary one that lifts the seat and TO PUT IT DOWN AFTER YOUR DONE USING IT.

__________________
“ When people get what they want, they are often surprised when they get what they deserve " - James Wood
Register to Reply
Guru

Join Date: Mar 2012
Posts: 2189
Good Answers: 84
#48
In reply to #4

Re: Maths on Friday

07/10/2012 3:52 AM

This is going to save a lot of marriages.

Register to Reply
Guru

Join Date: Jun 2011
Location: Phnom Penh
Posts: 4019
Good Answers: 102
#51
In reply to #4

Re: Maths on Friday

07/10/2012 4:23 AM

Toilet automation can be uncomfortable.

eg the toilets at Changi airport autoflush when a client gets up off the seat however they do have a tendency to false trigger and initiate a flush when one leans forward just a bit.....not fun. You can all picture it for yourselves.

Anyhow, default position for a toilet seat is "raised" and should be held there by a detent. No more problem.

__________________
Difficulty is not an obstacle it is merely an attribute.
Register to Reply
Guru
United States - Member - USA! Hobbies - Musician - Sound Man Engineering Fields - Mechanical Engineering - More than a Hobby Technical Fields - Technical Writing - New Member

Join Date: Oct 2008
Location: City of Roses.
Posts: 2056
Good Answers: 101
#5

Re: Maths on Friday

07/06/2012 1:06 PM

42 henways

__________________
Don't believe everything you read on the Internet!
Register to Reply
Guru

Join Date: Feb 2011
Location: Atlanta, GA
Posts: 1053
Good Answers: 110
#9

Re: Maths on Friday

07/06/2012 3:37 PM

Calculating this seems like too much work.

However, it stimulated some thought and a search. Chimneys break around the middle when they fall, as demonstrated in this video: http://myweb.lmu.edu/gvarieschi/chimney/toy2m-forweb.mov

Here's why that is.

So your bar will flex in the middle as it falls (imperceptibly of course, if shaped as you've drawn it.)

Imagine what would happen if you had a four foot long piece of uncooked spaghetti.

Had you asked for the rotational velocity of your steel bar, I could have stolen this solution (suitably modified).

Perhaps extra credit could be given for times starting from .25 degrees, 1 degree, 10 degrees and 45 degrees from vertical.

__________________
Think big. Drive small.
Register to Reply
Guru
Hobbies - CNC - New Member Hobbies - DIY Welding - New Member Engineering Fields - Electromechanical Engineering - New Member

Join Date: Aug 2007
Posts: 23647
Good Answers: 420
#10
In reply to #9

Re: Maths on Friday

07/06/2012 3:39 PM

yeah.... do your own homework....

__________________
“ When people get what they want, they are often surprised when they get what they deserve " - James Wood
Register to Reply
Guru
Popular Science - Evolution - New Member Popular Science - Weaponology - New Member

Join Date: May 2006
Location: The 'Space Coast', USA
Posts: 11119
Good Answers: 918
#11

Re: Maths on Friday

07/06/2012 6:30 PM

I think the mass is not important, but we do need to know the length.

There are two forces at play here, one is gravity pulling down and another is the force of the pivot point acting on the lower end of the rod. Since the rod rotates, a portion of the downward gravitational force is converted into rotational torque, effectively slowing down the rod's fall time. However, the top of the rod end will actually be traveling faster (angular velocity) than a normal free falling object.

The angular acceleration, a = (3/2)(g/L) cos(rho). This assumes that the initial value of rho is 89° before release.

However, we are solving for the point in time when the rod is horizontal or rho = 0.

If we have the acceleration we can rearrange for t from v = a * t = {(3/2)(g/L)cos(rho)} t, v is the angular velocity.

So, the initial angle is 1.571 radians. the time to fall should be equal to a final angle of 0° with the total arc = 1.571 radians. so when rho final = 0 = (1.571) - (1/2)(angular acceleration) t^2

Rearrange the equation and solve for t:

-1.571 = (1/2)(angular acceleration) t^2

t = sqrt[(1.571*2)/(angular acceleration/2)]

t = sqrt[3.142((2/3)(g/L)]

Probably screwed up something...

The 89° arc makes this a little awkward.

Register to Reply
Guru
Engineering Fields - Optical Engineering - Member Engineering Fields - Engineering Physics - Member Engineering Fields - Systems Engineering - Member

Join Date: Apr 2010
Location: Trantor
Posts: 5363
Good Answers: 647
#12
In reply to #11

Re: Maths on Friday

07/06/2012 8:48 PM

I used a stick about 40 cm (.4 m) long, and it took about 1 sec to fall from nearly vertical. The calculation using your formula yields 7 sec.

So there must be an error there, but I'm on vacation and have my brain mostly turned off. Too dumb and lazy right now to review your work and offer any helpful suggestions.

/Good try, though.

__________________
Whiskey, women -- and astrophysics. Because sometimes a problem can't be solved with just whiskey and women.
Register to Reply
Guru
United Kingdom - Member - Not a New Member Hobbies - Musician - New Member Hobbies - Fishing - New Member

Join Date: May 2006
Location: Reading, Berkshire, UK. Going under cover.
Posts: 9684
Good Answers: 468
#13
In reply to #11

Re: Maths on Friday

07/06/2012 9:13 PM

"The 89° arc makes this a little awkward."

That's cool - call it 90° - just imagine someone gave it a miniscule shove (call it a perturbation) to get it started.

Confess I haven't studied your answer (will try tomorrow) - but I suspect the solution invovles some fancy calculus (as you've indicated there is a component of "G" acting, depending on the (varying) angle).

A' first-order' approach could ignore the moment of inertia of the bar (among other things). Think it's a d3θ/dθ3 problem (AKA a 3-pipe job).

__________________
"Love justice, you who rule the world" - Dante Alighieri
Register to Reply
Guru
United Kingdom - Member - Not a New Member Hobbies - Musician - New Member Hobbies - Fishing - New Member

Join Date: May 2006
Location: Reading, Berkshire, UK. Going under cover.
Posts: 9684
Good Answers: 468
#22
In reply to #13

Re: Maths on Friday

07/07/2012 10:32 AM

Ewe silly bugger JohnDG - don't think you can do it at all without considering the moment of inertia somewhere along - tho' the mass does cancel out before you have to do any arithmetic.

__________________
"Love justice, you who rule the world" - Dante Alighieri
Register to Reply
Guru
Panama - Member - New Member Hobbies - CNC - New Member Engineering Fields - Marine Engineering - New Member Engineering Fields - Retired Engineers / Mentors - New Member

Join Date: Dec 2006
Location: Panama
Posts: 4273
Good Answers: 213
#15
In reply to #11

Re: Maths on Friday

07/06/2012 11:16 PM

If we model this as the gravitational force acting on the mass at the centroid of the bar or toilet seat or whatever, then the moment induced at the pivot point is going to increase with time- in other words, I don't think you can consider angular acceleration as a constant in this case. The force due to gravity would be constant, but the moment arm would be increasing as the angle decreased from 89º to 0º.

It is too late on Friday night to work through the dynamics of this...and the wife awaits...

Register to Reply
Guru

Join Date: Jan 2011
Posts: 668
Good Answers: 3
#34
In reply to #11

Re: Maths on Friday

07/08/2012 5:17 AM

a clock maker from the 17th century could answer this question without your confusing numberic jibberish.

save it for nasa. sorry, i just remembered, nasa is'nt interested either.

Register to Reply
Guru
Engineering Fields - Optical Engineering - Member Engineering Fields - Engineering Physics - Member Engineering Fields - Systems Engineering - Member

Join Date: Apr 2010
Location: Trantor
Posts: 5363
Good Answers: 647
#14

Re: Maths on Friday

07/06/2012 11:11 PM

Ok, I'm looking at the initial and final potential and rotational kinetic energies (PE and RKE).

The initial PE is m g h, where h=half the length of the bar. Call the length of the bar L.

Then PEi = m g (L/2).

The final PE is zero.

The initial RKE is zero. The final RKEf = 1/2 (m (L2/3) ωf2)

Equating RKEf to PEi and cancelling terms I get:

1/6 L2 ωf2 = g L/2; so ωf = √ (3 g/L)

Then assuming the average ω is half the final ωf, and since the rotation is through θ=1/2 pi radians, and further assuming that L=0.4 m.

t = θ⁄ω

I get t = (pi/2)/[0.5(√(3×9.8⁄0.4)], or t = 0.366 sec.

Of course, I could be wrong, too. Likely at this late at night (~ 11 PM) I've made a mistake somewhere.

__________________
Whiskey, women -- and astrophysics. Because sometimes a problem can't be solved with just whiskey and women.
Register to Reply
Guru
Popular Science - Evolution - New Member Popular Science - Weaponology - New Member

Join Date: May 2006
Location: The 'Space Coast', USA
Posts: 11119
Good Answers: 918
#23
In reply to #14

Re: Maths on Friday

07/07/2012 10:39 AM

A quick check with a .6m bar shows it drops in about 1.2 second ±.20 seconds.

The same calculation with your formula is .45 seconds. While much closer than mine, it seems that it still is wrong.

If the arc is 90° from vertical I still get t = SQRT[ {(pi)(2/3)(L/g)} ], which is .55s seconds for my bar.

Someone will spank us with the right answer.

Register to Reply
Guru
United Kingdom - Member - Not a New Member Hobbies - Musician - New Member Hobbies - Fishing - New Member

Join Date: May 2006
Location: Reading, Berkshire, UK. Going under cover.
Posts: 9684
Good Answers: 468
#25
In reply to #23

Re: Maths on Friday

07/07/2012 11:04 AM

My Excel model comes up with 0.81 second for a 0.6m bar .

Stop press! The starting angle is very important: I get 1.13 seconds (0.6m bar) if I start from 90° (and give it a shove to start it moving ).

__________________
"Love justice, you who rule the world" - Dante Alighieri
Register to Reply
Guru
Popular Science - Evolution - New Member Popular Science - Weaponology - New Member

Join Date: May 2006
Location: The 'Space Coast', USA
Posts: 11119
Good Answers: 918
#26
In reply to #25

Re: Maths on Friday

07/07/2012 12:14 PM

That sounds good!

Yes, starting angle makes a huge difference.

Register to Reply
Guru

Join Date: Aug 2007
Location: London England.
Posts: 583
Good Answers: 10
#16

Re: Maths on Friday

07/07/2012 5:05 AM

Get a toilet seat made of wood or plastic,missing the bowl could start rusting in steel.

Bazzer

__________________
When I die I'd like to go peacefully in my sleep like my dad,not shouting and screaming like his passengers.
Register to Reply
Guru

Join Date: Jan 2006
Location: Tamworth, UK.
Posts: 1782
Good Answers: 45
#17

Re: Maths on Friday

07/07/2012 6:32 AM

I guess you need to take moments of the force of gravity acting on the center of gravity about the fulcrum. This will give the force of the turning moment which will increase in proportion to the sine of the angle from the vertical. From which the incremental acceleration and velocity can be calculated. and hence the time per degree

A bit of integral calculus will given the total time. But I'm pretty sure the air pressure under the seat will have a deceleration effect that increases with angle (otherwise parachutes would not work) so we will need to factor in the area of the seat.

My maths is too rusty to do the sums myself.

__________________
When arguing, remember mud-slinging = lost ground.
Register to Reply
Guru

Join Date: Jul 2005
Location: Stoke-on-Trent, UK
Posts: 4496
Good Answers: 137
#18

Re: Maths on Friday

07/07/2012 7:35 AM

Assuming length 0.4m (about right for a toilet seat) I make it 0.865 sec. Had to use numerical integration.

__________________
Give masochists a fair crack of the whip
Register to Reply
Guru
United Kingdom - Member - Not a New Member Hobbies - Musician - New Member Hobbies - Fishing - New Member

Join Date: May 2006
Location: Reading, Berkshire, UK. Going under cover.
Posts: 9684
Good Answers: 468
#21
In reply to #18

Re: Maths on Friday

07/07/2012 10:20 AM

I did an Excel model with 0.5° steps, and got 0.382 seconds for a 0.4m bar. Fully prepared to be wrong! (Tho' it's pretty close to USBport's answer).

__________________
"Love justice, you who rule the world" - Dante Alighieri
Register to Reply
Guru
United Kingdom - Member - Not a New Member Hobbies - Musician - New Member Hobbies - Fishing - New Member

Join Date: May 2006
Location: Reading, Berkshire, UK. Going under cover.
Posts: 9684
Good Answers: 468
#24
In reply to #21

Re: Maths on Friday

07/07/2012 10:53 AM

New result: 0.540 seconds (forgot that the force is acting at (L/2), not L). So I'm somewhere between Usbport's and Codemaster's results.

Never mind - nearly time to go to the pub! .

__________________
"Love justice, you who rule the world" - Dante Alighieri
Register to Reply
Guru

Join Date: Jul 2005
Location: Stoke-on-Trent, UK
Posts: 4496
Good Answers: 137
#37
In reply to #24

Re: Maths on Friday

07/08/2012 10:36 AM

I'd like to post my workings, but I can't get the hang of copying images, it seems overly complicated. But it's not hard to write the equation of motion. The neat bit is when you get dω/dt as a function of θ, put dω/dt = dθ/dt*dω/dθ = ω*dω/dθ. That integrates OK to give ω2 = 3g/L*(cos 1° - cos θ) but the next step to get t as a function of θ needs numerical integration.

For various initial angles, I (with help of Mathcad) get time, sec

1° - 0.865

0.1° - 1.25

0.01° - 1.62

0.001° - 2.00

0.0001° - 2.38

0.00001° - 2.76

0.000001° - 3.06

For smaller angles, Mathcad overloads. But until that point, the increase in time with decreasing angle is surprisingly slow, so if anybody can fault the maths.......? BTW, for different lengths, time varies as √L.

On the subject of toilet seats, the ones that irritate me are when the seat hits the cistern before going overcentre, so it doesn't stay up at all. It's quite acrobatic holding it up with one foot while doing the biz!

__________________
Give masochists a fair crack of the whip
Register to Reply
Guru
United Kingdom - Member - Not a New Member Hobbies - Musician - New Member Hobbies - Fishing - New Member

Join Date: May 2006
Location: Reading, Berkshire, UK. Going under cover.
Posts: 9684
Good Answers: 468
#38
In reply to #37

Re: Maths on Friday

07/08/2012 11:49 AM

Agree up to 0.00001°. VB6 claps out if the counter in a for...next loop is greater than 16,777,215 (= 224-1) so I can't go any further (without pulling some tricks) - and anyway the run time is getting excessive.

May see if VB10 is any better tomorrow (don't have it loaded on my home pc).

__________________
"Love justice, you who rule the world" - Dante Alighieri
Register to Reply
Guru

Join Date: Jul 2005
Location: Stoke-on-Trent, UK
Posts: 4496
Good Answers: 137
#39
In reply to #38

Re: Maths on Friday

07/08/2012 12:48 PM

OK, good to see 2 different approaches give similar results

__________________
Give masochists a fair crack of the whip
Register to Reply
Guru

Join Date: Feb 2011
Location: Atlanta, GA
Posts: 1053
Good Answers: 110
#44
In reply to #37

Re: Maths on Friday

07/09/2012 11:35 AM

0.00001° - 2.76

0.000001° - 3.06

For smaller angles, Mathcad overloads. But until that point, the increase in time with decreasing angle is surprisingly slow, so if anybody can fault the maths.......?

I can't fault the maths. I agree that the increase in time with decreasing angle is slower than I would have guessed. However, I tried balancing a broom handle quite carefully in my shop, and even given its longer length, it was still hard to get it to "just stand there" for more than 2 seconds before getting into the "obviously falling" stage.

__________________
Think big. Drive small.
Register to Reply
Power-User

Join Date: May 2008
Posts: 380
#60
In reply to #37

Re: Maths on Friday

07/10/2012 12:48 PM

What if you do cos1º≈1,cosΘ≈1-Θ²/2.then for so small angles log(Θ1/Θ2)appears..

Register to Reply
Guru

Join Date: Jul 2005
Location: Stoke-on-Trent, UK
Posts: 4496
Good Answers: 137
#62
In reply to #60

Re: Maths on Friday

07/10/2012 3:15 PM

Good idea, I'll have a look at that when I have a minute.

__________________
Give masochists a fair crack of the whip
Register to Reply
Guru
United Kingdom - Member - Not a New Member Hobbies - Musician - New Member Hobbies - Fishing - New Member

Join Date: May 2006
Location: Reading, Berkshire, UK. Going under cover.
Posts: 9684
Good Answers: 468
#36
In reply to #18

Re: Maths on Friday

07/08/2012 10:31 AM

I've re-worked my Excel model in VB (and corrected an error in the calculations). For a 0.4m bar falling from 89°, with 10000 (or more) increments I get 0.865 sec. I reckon you're right.

__________________
"Love justice, you who rule the world" - Dante Alighieri
Register to Reply
Power-User

Join Date: May 2008
Posts: 380
#19

Re: Maths on Friday

07/07/2012 9:22 AM

1)mg½lsinΘ=Idω/dt , ω=dΘ/dt I≈(ml²)/3; 2)mg½lωsinΘ=Iωdω/dt ;3)mgcosΘ+I½ω²=const ; 4)√((k-mgcosΘ)/I)=dΘ/dt;5)Δt=∫(dt/dΘ)dΘ=∫1/(√....)dΘ;this last eq.must be calc.by numerical methods..no for saturday..

Register to Reply
Guru

Join Date: Jun 2011
Location: Phnom Penh
Posts: 4019
Good Answers: 102
#20

Re: Maths on Friday

07/07/2012 9:42 AM

The time taken to fall will be dependent on the length ie (height ) of the arm.

It will also fall faster if you've had a bevvy or 6.

If you are holding a can of beer in your, err, other hand, then it will be faster than you really want and will lead to a panicked solo sword fight.

__________________
Difficulty is not an obstacle it is merely an attribute.
Register to Reply
Guru

Join Date: Aug 2007
Location: London England.
Posts: 583
Good Answers: 10
#33
In reply to #20

Re: Maths on Friday

07/08/2012 4:27 AM

Also,get your willy out of the way before it slams shut,Owwww.

Bazzer.

__________________
When I die I'd like to go peacefully in my sleep like my dad,not shouting and screaming like his passengers.
Register to Reply
Guru

Join Date: Mar 2012
Posts: 2189
Good Answers: 84
#50
In reply to #33

Re: Maths on Friday

07/10/2012 4:11 AM

[SPLARFFFF!!!]

D*mn you! There's coffee everywhere!

ROFLMAO

Register to Reply Off Topic (Score 5)
Guru

Join Date: Mar 2011
Location: Sebastopol, California
Posts: 1205
Good Answers: 54
#27

Re: Maths on Friday

07/07/2012 12:28 PM

Heck this is easy!

Time to fall (not looking) = 0

Time to fall (watching) = about one second

Time to fall (watching with baited breath) = Forever + one day

See how easy that was?

__________________
Most people are mostly good most of the time.
Register to Reply
Guru

Join Date: Mar 2007
Location: Etherville
Posts: 12362
Good Answers: 115
#28

Re: Maths on Friday

07/07/2012 12:40 PM

This is horribly complex for a weekend problem. I'm just going to print the thread off, and show it to Mrs K as justification for leaving the toilet seat up .

__________________
For sale - Signature space. Apply on self addressed postcard..
Register to Reply
Power-User

Join Date: May 2008
Posts: 380
#29

Re: Maths on Friday

07/07/2012 5:56 PM

I think with help of GRAPHCALC i got the results:Δt(10º-90º)=1.839√(l/g), that means .455 sec for l=.6m, and .372sec for l=.4m.That's using: Δt=(√(l/3g))∫dΘ/√(1-cosΘ), cosΘo≈1. For case 0‹Θ‹10º: Δt=.816•(√(l/g))•ln(Θ1/Θ2), so Δt(5º-10º)=.18√l=.11sec for.4m and .14sec for .6 m.Anyway Δt is infinite if you start in 0º not only improper integral and is natural because it could stay in that pos.for ever (therically).-

Register to Reply
Guru
Panama - Member - New Member Hobbies - CNC - New Member Engineering Fields - Marine Engineering - New Member Engineering Fields - Retired Engineers / Mentors - New Member

Join Date: Dec 2006
Location: Panama
Posts: 4273
Good Answers: 213
#30

Re: Maths on Friday

07/07/2012 9:35 PM

This whole problem goes away if we could just convince our ladies not to install fluffies on the toilet seat when there is a resident male...

Register to Reply
Guru
Engineering Fields - Optical Engineering - Member Engineering Fields - Engineering Physics - Member Engineering Fields - Systems Engineering - Member

Join Date: Apr 2010
Location: Trantor
Posts: 5363
Good Answers: 647
#31
In reply to #30

Re: Maths on Friday

07/07/2012 9:53 PM

Hey! That's just a shameless attempt to get a GA!

__________________
Whiskey, women -- and astrophysics. Because sometimes a problem can't be solved with just whiskey and women.
Register to Reply
Guru

Join Date: Mar 2007
Location: Etherville
Posts: 12362
Good Answers: 115
#32
In reply to #31

Re: Maths on Friday

07/08/2012 2:19 AM

I'd give him one, except the problem goes away if the seat is held back by elastic bands.

__________________
For sale - Signature space. Apply on self addressed postcard..
Register to Reply
Guru

Join Date: Jan 2011
Posts: 668
Good Answers: 3
#35

Re: Maths on Friday

07/08/2012 6:40 AM

about .5 sec before you realize you may never have kids again.

Register to Reply
Guru

Join Date: Mar 2007
Location: City of Light
Posts: 3943
Good Answers: 183
#40

Re: Maths on Friday

07/09/2012 7:33 AM

The numerical integration gives following results:

start angle 1° time to reach 90° 1.06 s end rotation speed 6.89 rad/s

start angle 2° 0.92 s and final velocity 6.88 rad/s

Time is only function of gravity and bar length (assuming that mass is uniform over the length).

Register to Reply
Guru

Join Date: Jun 2011
Location: Phnom Penh
Posts: 4019
Good Answers: 102
#41
In reply to #40

Re: Maths on Friday

07/09/2012 7:43 AM

Bar length is the CG distance so uniform mass distribution isn't an issue. If it is bottom heavy then the resultant arm length is shorter and vice versa,

__________________
Difficulty is not an obstacle it is merely an attribute.
Register to Reply
Guru

Join Date: Jul 2005
Location: Stoke-on-Trent, UK
Posts: 4496
Good Answers: 137
#42
In reply to #40

Re: Maths on Friday

07/09/2012 8:20 AM

What bar length? Back-calculating I make it ~ 0.6m.

__________________
Give masochists a fair crack of the whip
Register to Reply
Guru

Join Date: Jun 2011
Location: Phnom Penh
Posts: 4019
Good Answers: 102
#43
In reply to #42

Re: Maths on Friday

07/09/2012 10:33 AM

I think the bar length is supposed to model a hypothetical dunny lid.

__________________
Difficulty is not an obstacle it is merely an attribute.
Register to Reply
Guru

Join Date: Jul 2005
Location: Stoke-on-Trent, UK
Posts: 4496
Good Answers: 137
#46
In reply to #43

Re: Maths on Friday

07/09/2012 5:39 PM

Original post said a rectangular bar, inspired by a toilet seat. Time of fall doesn't mean much without giving the length it's based on.

__________________
Give masochists a fair crack of the whip
Register to Reply
Guru
United Kingdom - Member - Not a New Member Hobbies - Musician - New Member Hobbies - Fishing - New Member

Join Date: May 2006
Location: Reading, Berkshire, UK. Going under cover.
Posts: 9684
Good Answers: 468
#47
In reply to #46

Re: Maths on Friday

07/09/2012 7:05 PM

When I first posted, I hadn't done any "back of envelopes" or even thought much about it. I simplified it to a steel bar, to try to avoid distractions relating to density, air resistance and whatever else could have been thrown in (despite my direction to neglect friction).

As I hadn't thought about it, I didn't know (at that stage) whether or how the length, mass or anything else would affect the time to fall. Having scratched out some sums, I now have a reasonable idea of what's going on (which probably should've been obvious, but wasn't initially - I claim a "senior moment" never blame the beer ).

I did ask for assumptions made to be declared (this would clearly include the length).

We've had 400mm suggested as a reasonable length (relative to the toilet seat which really did trigger the thoughts), and I think 600mm came in from AH's experiments.

Maybe I should've dropped the toilet seat reference. Just thought some context would make it a bit more of a "real world" problem.

__________________
"Love justice, you who rule the world" - Dante Alighieri
Register to Reply
Guru

Join Date: Mar 2007
Location: Etherville
Posts: 12362
Good Answers: 115
#49
In reply to #47

Re: Maths on Friday

07/10/2012 3:56 AM

Don't drop the toilet seat (!), the 'added value' of humerous situations has held my attention - not that it's helped me come up with a solution, but I'm having a fun time pondering all this. For reasons best unexplained, I've found myself distracted by roller coasters (with people falling out), pendulums, and assorted futile paths.

If AH doesn't mind, could we stick with the 40cm (ie mass at a radius of 20 cm) so as to avoid confusion ?

__________________
For sale - Signature space. Apply on self addressed postcard..
Register to Reply
Guru

Join Date: Jul 2005
Location: Stoke-on-Trent, UK
Posts: 4496
Good Answers: 137
#52
In reply to #49

Re: Maths on Friday

07/10/2012 4:26 AM

Problem with that is it's not equivalent to a bar. For mass m and length L, MoI = m*(L/2)2 = m*L2/4, but for a bar MoI = m*L2/3.

__________________
Give masochists a fair crack of the whip
Register to Reply
Guru

Join Date: Mar 2007
Location: Etherville
Posts: 12362
Good Answers: 115
#53
In reply to #52

Re: Maths on Friday

07/10/2012 5:10 AM

Well now you've scrambled my eggs !

There's a nice simplicity to the question, which is much more intriguing than first inspection would suggest. The type of thing that I'd have expected to be in standard texts - but it isn't (not directly as John posted).

__________________
For sale - Signature space. Apply on self addressed postcard..
Register to Reply
Guru

Join Date: Jun 2011
Location: Phnom Penh
Posts: 4019
Good Answers: 102
#54
In reply to #52

Re: Maths on Friday

07/10/2012 5:10 AM

Really?

__________________
Difficulty is not an obstacle it is merely an attribute.
Register to Reply
Guru

Join Date: Jul 2005
Location: Stoke-on-Trent, UK
Posts: 4496
Good Answers: 137
#57
In reply to #54

Re: Maths on Friday

07/10/2012 8:46 AM

Yes, really Not sure what point you're making

__________________
Give masochists a fair crack of the whip
Register to Reply
Guru

Join Date: Jun 2011
Location: Phnom Penh
Posts: 4019
Good Answers: 102
#58
In reply to #57

Re: Maths on Friday

07/10/2012 9:29 AM

My ignorance perhaps. I don't get the division by 3 instead of 2 bit.

Does that follow from MoI = m*(L/√3)2 perhaps?

__________________
Difficulty is not an obstacle it is merely an attribute.
Register to Reply
Guru

Join Date: Jul 2005
Location: Stoke-on-Trent, UK
Posts: 4496
Good Answers: 137
#61
In reply to #58

Re: Maths on Friday

07/10/2012 2:29 PM

That's right, MoI of a rod about one end m*(L/√3)2 = m*L2/3. (don't usually bother with the (√3)2 = 3 part ). Not hard to work out if you have a smattering of calculus. For a point mass at distance L/2 it's m*(L/2)2 = m*L2/4.

__________________
Give masochists a fair crack of the whip
Register to Reply
Commentator
United States - Member - New Member

Join Date: May 2009
Location: Minnesota, USA
Posts: 89
Good Answers: 2
#45

Re: Maths on Friday

07/09/2012 11:56 AM

Why do they call them "near misses" and not "near hits?"

__________________
"Obviously you're not a golfer." - The Dude
Register to Reply Off Topic (Score 5)
Guru

Join Date: Jun 2011
Location: Phnom Penh
Posts: 4019
Good Answers: 102
#55

Re: Maths on Friday

07/10/2012 6:52 AM

This is a pendulum on its side.

The travel of the falling pendulum is arrested abruptly approx quarter way through its travel if starting from nearly vertical.

The time to reach that point would be quarter of the natural pendulum period (loss-less pendulum)

pendulum period

L=0.4m, g=9.8ms-2

gives 1.269 seconds

Quarter of that is 0.32secs. for L =0.6 we get 0.39secs

Notes:

  • 89°/358° is little less (not much) than 1/4 for 1° start offset, (good enough for this illustration)
  • if we started at say 5° then the quotient would be 85°/350°, 10° would be 80°/340°.....
  • If you want to go with minute start offsets of fractional degrees then the quotient just asymptotes to 1/4 any way.
  • if you dropped the lid from say 1° above horizontal then the quotient would be 1°/182° and this extreme asymptotes to zero.
__________________
Difficulty is not an obstacle it is merely an attribute.
Register to Reply
Guru
United Kingdom - Member - Not a New Member Hobbies - Musician - New Member Hobbies - Fishing - New Member

Join Date: May 2006
Location: Reading, Berkshire, UK. Going under cover.
Posts: 9684
Good Answers: 468
#56
In reply to #55

Re: Maths on Friday

07/10/2012 7:49 AM
__________________
"Love justice, you who rule the world" - Dante Alighieri
Register to Reply
Guru

Join Date: Mar 2007
Location: Etherville
Posts: 12362
Good Answers: 115
#59
In reply to #56

Re: Maths on Friday

07/10/2012 11:40 AM

It was that (roughly 5o) limitation the left me headed for the roller coasters - have the thing enter a (circular, not clothoid) loop with just enough velocity to reach the top.

__________________
For sale - Signature space. Apply on self addressed postcard..
Register to Reply
Guru

Join Date: Mar 2007
Location: City of Light
Posts: 3943
Good Answers: 183
#63

Re: Maths on Friday

07/16/2012 9:09 AM

Starting angle 2° from vertical.Equations are: J= M*l^2/3 valid which ever comments are ONLY for an uniform mass distribution which gives dM=M/l*dx and J= ∫dM*x^² = M/l*∫x²*dx. If not J= ∫dM*x² and has another equation.The second equation d²φ/dt²=1.5*g/l*sin(φ) can be for easier integration written asdω/dt=1.5*g/l*sin(φ) and dφ/dt=ω.

Angle in radian versus time in s for start between 0.005 and 0.035 rad. Graph is limited at π/2 which corresponds to 90°. In this graph the angle is measured with respect to vertical. The 1st graph had a shift of 90°.
Evolution of rotation speed for same start values in rad/s.

Register to Reply
Guru
United Kingdom - Member - Not a New Member Hobbies - Musician - New Member Hobbies - Fishing - New Member

Join Date: May 2006
Location: Reading, Berkshire, UK. Going under cover.
Posts: 9684
Good Answers: 468
#64
In reply to #63

Re: Maths on Friday

07/16/2012 9:33 AM

Nice - was it a maths package? Too hot and too far away from my desk to verify your calcs - on holiday in Montpellier, France.

I take it from what everyone's come back with that there's no analytical solution.

__________________
"Love justice, you who rule the world" - Dante Alighieri
Register to Reply
Guru

Join Date: Mar 2007
Location: Etherville
Posts: 12362
Good Answers: 115
#65
In reply to #64

Re: Maths on Friday

07/16/2012 10:09 AM

I'll desist from calling you a sun-lounging ***. No matter which way I slice this, I can't see an easy way to solution. As good as the input from everybody is, I have a nagging feeling that we miss something. The complexity from such a seemingly straightforward question is amazing.

It's a good one for a school teacher to use. The students will not get it right, but wow will they learn a lot along the way. Thanks, John, it's a superbly provoking question.

May we move on to some further ananytics - towels on sun loungers at 7 a.m, .

__________________
For sale - Signature space. Apply on self addressed postcard..
Register to Reply
Register to Reply 65 comments

"Almost" Good Answers:

Check out these comments that don't yet have enough votes to be "official" good answers and, if you agree with them, vote them!
Copy to Clipboard

Users who posted comments:

Anonymous Hero (3); Bazzer Englander (2); Codemaster (9); cwarner7_11 (2); Deefburger (1); durtieduck (2); europium (2); ferquiza (3); horace40 (1); JohnDG (12); K_Fry (2); Kris (6); nick name (2); ozzb (1); pauls_14 (1); phoenix911 (4); RVZ717 (1); SolarEagle (1); Usbport (3); Wal (7)

Previous in Forum: Why is the Sky Blue?   Next in Forum: How I Was Almost Killed

Advertisement