Calculating Angular Velocity and Time for Insect on Rotating Rod

[Shahid0072]

A rod AB having mass M ,pivoted at mid point at 0 is in horizontal..An insect having same mass M Fall vertically on rod at mid point C between 0 and B with speed V..It starts moving towards end B such that angular velocity of rod remains constant ..If the insect reached at B when rod has rotated to an angle of 90degree..Calculate V...How can we calculate time insect take to cover from C to B?? How moving of insect towards end B made angular velocity of rod constant?

 

[tiny-tim]

Welcome to PF!

Hi Shahid0072! Welcome to PF!

Once the insect lands on the rod, the total angular momentum will be constant (because then there is no external torque).

if the insect moves towards B, that will increase the moment of inertia (which will tend to slow the rod, exactly as a skater slows when he extends his arms).

So calculate the moment of inertia when the insect is at distance r from O.

 

[Shahid0072]

Thank you tim..Yeah..Great..When insect will move from C to B,moment of inertia of system will increase which will tend to descrease their angular velocity..But what abt the torque due to weight of insect Of mass m..Won't it be increasing the angular speed of rod..But torque will also be changing as insect will move to B...Will both effects cancel each other,thus maintaining angular velocity of rod constant

 

[tiny-tim]

Yup! That's why we have to have specially trained insects!

You need to teach yours the following formulas:

τ = d/dt(Iω) = Idω/dt + ωdI/dt

dω/dt = 0

 

[Shahid0072]

Sir..In this question we have to calcute V..Velocity with wh4ch insect falls on rod...For that we ha ve to calculate angular velocity using the fact the insect reached the mid way between 0(pivot point)to end B..During the time rod rotated by pie/2 radians...So if we calculate the time insect takes to reach from C to end B..That time is equal to time rod takes to rotate by constant angular velocity...So dividing the time by pie/2 will give angular velocity and eventually speed by which insect falls on rod..But how to calculate time??

 

[tiny-tim]

(don't call me sir! … I'm only a little goldfish! )

No, you calculate the intial angular velocity by conservation of angular momentum …

exactly the same way as you'd use conservation of ordinary momentum for an ordinary collision.

 

[Shahid0072]

Alryt goldfish..I know how to apply conservation of angular momentum in this case..Problem is that we are given velocity of insect with which it falls as V..We have to calculate V IN Numerals..We are given length of rod(L) =1.8 m...And mass M SAME as of insect..Read the question again...After applying coam..We get omega in terms of V and L..But we just know that time taken by insect to reach end b is equal to time taken by rod to rotate by 90 degree..Can u calculate the time,tim?

 

[tiny-tim]

Can u calculate the time,tim?

No, that's your job …

start writing out all the equations, and see where you get.

 

[Ketman]

I arrived on this forum as a questioner, not a helper, but even I know that there is an external torque acting on this system tending to increase the velocity while the action of the insect tend to decrease it. It isn't a simple problem by any means.

 

[tiny-tim]

Welcome to PF!

Hi Ketman! Welcome to PF!

It isn't a simple problem by any means.

And that's exactly why it's best to write out all the equations first, and not try to work out some energy-saving short-cut.

 

[Ketman]

Thanks for the welcome.

Returning to the problem, is there an external torque or not? I don't think the questioner can begin writing out his equations until he knows the answer to that.

 

[tiny-tim]

Hi Ketman!

Yes, that's the "τ" in "τ = d/dt(Iω) = Idω/dt + ωdI/dt"

 

[Ketman]

Yep. It's just that it conflicts with your first reply to him, which says no external torque.

I hope my posts don't seem too much like butting in, but I'm trying to get back into practice with theoretical mechanics after many years away from the subject. And one way is to look at other people's problems and have a go at solving them, which is why I'm interested in this one. In the meantime I hope someone will have a go at the medieval catapult problem I posted, which is certainly foxing me.

Shahid - why not start with an equation for how much angular momentum there is after the insect lands, even if it's only variables to begin with? At least post one equation. You know, one step at a time...

 

[Shahid0072]

Thanks for helping you two.. Tim,don't get me wrong ..When i asked you to calculate time,i meant the approach to calculate it.Ketman,u r ryt..It ain't easy..Before attempting ny question, i try to solve it in mind first..First we apply angular momentum conservation to get angular velocity in terms of v and l..Now we need time taken by insect to reach end B..how can we calculate that..Tim,your chain rule formula is ryt but torque is variable in this case ,its increasing as insect is going towards end B,simultneously its decreasing as it rod is rotating..First the rod is horizontal it makes angle0..When its vertical it makes pie/2..Torque varies as Mgx cos§ where x is distance from rod's centre 0..And § is angle rod makes with horizontal..Tell me please...Angular velocity of rod is constant but angular momentum is changing..As insect is moving away,MOI changes...

 

[Shahid0072]

Alryt first we as insect land at C( mid point between 0(centre ) and end B) with velocity V..We apply conservation of angular momentum we get omega=12 V /7L where L=1.8 metres given..Now we have to find V in nos...First let's denote omega (angular velocity) by¥..We have V=7L¥/12 ..We need to calculate ¥ using the fact time taken by rod to rotate 90(from horizontal position to vertical) is equal to time taken by insect to reach from C TO B...If we could calculate time,we can divide it by pie/2 to get omega ..

 

[Doc Al]

Perhaps I'm missing something, but I don't see how the angular velocity of the rod can remain constant. There's a torque on the system due to the insect's weight, thus the system will have an angular acceleration.

 

[Ketman]

Perhaps I'm missing something, but I don't see how the angular velocity of the rod can remain constant. There's a torque on the system due to the insect's weight, thus the system will have an angular acceleration.

It'll have angular acceleration due to gravity, but angular deceleration due to the increased moment of inertia as the insect travels away from the axis. If they cancel each other out, the angular velocity will stay constant.

 

[Ketman]

Alryt first we as insect land at C( mid point between 0(centre ) and end B) with velocity V..We apply conservation of angular momentum we get omega=12 V /7L where L=1.8 metres given..

12?... 7?... Where did those numbers come from?

Edit: okay I get the 12, but still don't know about 7.

 

[Doc Al]

It'll have angular acceleration due to gravity,

Right.

but angular deceleration due to the increased moment of inertia as the insect travels away from the axis.

I don't know what you mean. The rotational inertia will increase as the insect moves along, which (by itself) acts to reduce the angular acceleration.

 

[Shahid0072]

Ketman..Applying conservation of angular momentum about rod centre..We have(i am using LL INSTEAD OF L SQUARE
MVL/4=(MLL/12 +MLL/16)¥

 

[Shahid0072]

Again..M just remindin..Angular momentum does not remain constant ,only angular velocity does!

 

[Doc Al]

Again..M just remindin..Angular momentum does not remain constant ,only angular velocity does!

Yes, my bad. Tiny-tim stated that in post #4.

 

[Shahid0072]

Let me summarize..Torque is there due to insects weight..And torque varies as Mgx cos§..Where x is distance from centre and § is angle made by rod with horizontal..Torque is equal to ¥d(I)/dt..Where ¥ is constant angular velocity and I is moment of inertia of rod and insect,which is also varying

 

[Shahid0072]

I guess it requires the use of calculus..

 

[Shahid0072]

If we somehow calculate ¥(angular velocity)..We can calculate speed V with which insect falls..Don't u think we have to calculate time taken by using the fact that time taken by rod to rotate 90 degree is the same time taken by insect to reach ground!

 

[tiny-tim]

Hi Shahid0072!

Ketman..Applying conservation of angular momentum about rod centre..We have(i am using LL INSTEAD OF L SQUARE
MVL/4=(MLL/12 +MLL/16)¥

(have a theta: θ and an omega: ω and try using the X² tag just above the Reply box )

Again..M just remindin..Angular momentum does not remain constant ,only angular velocity does!

Angular momentum is conserved at the instant of impact, but not afterwards.

Let me summarize..Torque is there due to insects weight..And torque varies as Mgx cos§..Where x is distance from centre and § is angle made by rod with horizontal..Torque is equal to ¥d(I)/dt..Where ¥ is constant angular velocity and I is moment of inertia of rod and insect,which is also varying

Yes, that's all correct.

Now … show us your equations!

 

[Doc Al]

Nice problem. Easier than I expected!

 

[Doc Al]

I guess it requires the use of calculus..

A trivial bit.

 

[Shahid0072]

We obviously know change in angular momentum..

 

[Shahid0072]

Tiny tim..I don't have a compatible handset..

 

[Shahid0072]

Tiny tim...Which equations?

 

[tiny-tim]

Do you mean you can't read the symbols below?

(if you can, just copy one and paste it )

 

[Doc Al]

Moderator's note: Moving to Intro Physics, since this is more about solving a particular problem than about broader conceptual issues.

 

[Shahid0072]

M writing thru my cell..Its not allowing me to click on those letters..I can just c them

 

[tiny-tim]

Tiny tim...Which equations?

The equation which relates the speed V to the angular velocity ω,

and the τ = ωdI/dt equation

(with all the numbers, please ).

 

[Shahid0072]

I wrote it..We are just given Length of rod (L)in numbers which is 1.8 m..And i wrote already that ¥(angular velocity)=12V/7L..We have to calculate V in nos..

 

[Shahid0072]

Tim..Torque is variable..We know change in angular momentum ..What can we do?

 

[tiny-tim]

Tim..Torque is variable..We know change in angular momentum ..What can we do?

Call the position of the insect x, and the angle θ, and find the moment of inertia as a function of x, and the torque as a function of x and θ.

 

[Shahid0072]

I guess u haven't read my earliear post..Torque varies as Mgx cos§ where x is insect's distance from centre and § is angle rotated by rod..Initial moment of inertia of rod +insect is 7MLL/12 AND when rod is vertical is MLL/3..How to calculate V in numbers?

 

[tiny-tim]

I guess u haven't read my earliear post..Torque varies as Mgx cos§ where x is insect's distance from centre and § is angle rotated by rod..Initial moment of inertia of rod +insect is 7MLL/12 AND when rod is vertical is MLL/3..How to calculate V in numbers?

hmm … that can't be right … 1/3 < 7/12, but the moment of inertia should be increasing.

If L is the length AB of the whole rod, then I for the rod is 1/12 ML², to which you have to add I for the insect for L/4 ≤ x ≤ L/2.

 

[Shahid0072]

Oh..Pardon..Its 7MLL/48 Instead of 7MLL/12..Sorry..But what can we do to calculate V?

 

[tiny-tim]

Oh..Pardon..Its 7MLL/48

That's better!

ok, then, as you said …

¥(angular velocity)=12V/7L

Now use τ = 12V/7L dI/dt.

 

[Shahid0072]

But torque is variable??

 

[Shahid0072]

Its a function of both x and § as stated earlier..How can we find V in numerals by using that!