Question regarding kinematics

[sankalpmittal]

Homework Statement

There are two rods with one having ring "P" and the other having ring "Q". The rings are connected by a massless and inextensible string which is then connected from "Q" to ceiling. If ring "Q" moves down with velocity "v", what is the velocity of ring "P" ? The string connecting "P" and "Q" makes angle "θ" with the horizontal.

See image : http://postimg.org/image/ljczx6iqv/

Homework Equations

----------

The Attempt at a Solution

If ring Q moves down with velocity v, then the ring "P" must move up with velocity vsinθ. But it is not the answer. How should I proceed. I am out of ideas.

Please help !

Thanks in advance...

 

[PeterO]

Homework Statement

There are two rods with one having ring "P" and the other having ring "Q". The rings are connected by a massless and inextensible string which is then connected from "Q" to ceiling. If ring "Q" moves down with velocity "v", what is the velocity of ring "P" ? The string connecting "P" and "Q" makes angle "θ" with the horizontal.

See image : http://postimg.org/image/ljczx6iqv/

Homework Equations

----------

The Attempt at a Solution

If ring Q moves down with velocity v, then the ring "P" must move up with velocity vsinθ. But it is not the answer. How should I proceed. I am out of ideas.

Please help !

Thanks in advance...

re-draw (or imagine) the situation if ring Q was attached to the ceiling with a string 10cm longer.

Now imagine that Q had moved from the position originally shown, to the new position, at constant speed "v".

What will P have been doing?

 

[lewando]

Am I to understand from the drawing that there is really only one string connected to P which passes through Q (not attached to Q) and is then secured at the ceiling?

If ring Q moves down with velocity v, then the ring "P" must move up with velocity vsinθ.

How did you arrive at this conclusion?

 

[PeterO]

Am I to understand from the drawing that there is really only one string connected to P which passes through Q (not attached to Q) and is then secured at the ceiling?



How did you arrive at this conclusion?

I see your interpretation, and understand that possibility. When the original description said "The rings are connected by a massless and inextensible string..." I assumed the string was attached to each ring.

You may be right.

To get a feel for the situation, I would take some possible numerical values and investigate.

eg: assume the rods are 20cm apart, and the string is 40 cm long.

If the ring Q is against the ceiling, you can easily work out where P will be.
Then consider if Q is 5 cm down - where is P
Then consider if Q is 10 cm down - where is P
etc.

If Q was traveling at 5 cm/s, this would leave to a position time graph for P, from which you may be able to see its velocity.

Then try to relate to the original, general case (or perhaps some values were given?)

 

[sankalpmittal]

Can I also do this question by making a constraint equation of motion ? I do not see how. Can anyone please give me hints for that ?

 

[sankalpmittal]

Anyone ?

Just preventing my thread to be oblivion. :(

Any hints will surely do. Can I not do this question by making constraint equations ? Yes ?

 

[PeterO]

Anyone ?

Just preventing my thread to be oblivion. :(

Any hints will surely do. Can I not do this question by making constraint equations ? Yes ?

Who's diagram is it? supplied with the question or your interpretation?

 

[voko]

Can I also do this question by making a constraint equation of motion ?

That is the way to go.

I do not see how. Can anyone please give me hints for that ?

You could start by writing down what constraints you think this motion has.

 

[sankalpmittal]

Who's diagram is it? supplied with the question or your interpretation?

My classmate gave me this question along with the diagram and everything.

That is the way to go.
You could start by writing down what constraints you think this motion has.

By constraint equation, we have,

x²+y² = l²

Since x, the distance between two rods remain constant, we have on differentiating this equation with respect to time :

2y(dy/dt) = 2l(dl/dt)

dy/dt = l/y(dl/dt)

where l is length of string...

dy/dt = sec(θ) dl/dt

Now what shall I do ahead ?

Note: θ is angle the string makes with horizontal.

Thanks...

More hints please...

 

[voko]

What is y?

You have two rings, I would expect you need two symbols to denote their vertical positions.

 

[sankalpmittal]

What is y?

You have two rings, I would expect you need two symbols to denote their vertical positions.

Let position of ring Q be y_Q and ring P be y_P. These two variables denote the Y coordinates of the two rings. It is already given that dy_Q/dt = -v.. See question:image.

I am assigning downward negative and upward positive.

Now, we have the constraint,

y_Q² + x² = l_Q²

And , y_P² + x² = l_P²

I have 2 equations. We know that l_P+l_Q=constant, as length of string is same.

l_P is length of string shortened by Ring P and l_Q, that length of string shortened by Ring Q.

Am I on the right track ?

 

[voko]

You are on the right track, but you you need to reconsider your geometry.

I understand $l_Q$ is the length of the string from the Q ring to the top. Then I simply do not see why there is x in the equation for $y_Q$ and $l_Q$. There is no triangle involved.

The other equation makes more sense, because there is a triangle, but does it contain the entire side $y_P$ ?

 

[sankalpmittal]

You are on the right track, but you you need to reconsider your geometry.

I understand $l_Q$ is the length of the string from the Q ring to the top. Then I simply do not see why there is x in the equation for $y_Q$ and $l_Q$. There is no triangle involved.

The other equation makes more sense, because there is a triangle, but does it contain the entire side $y_P$ ?

It does not contain the entire side $y_P$. Correct ?

 

[voko]

It does not contain the entire side $y_P$. Correct ?

Given my question, I can't say otherwise :)

 

[sankalpmittal]

Given my question, I can't say otherwise :)

What ? =.= ...

-.-... Hmm just answering yes isn't against the rules of forums, right ? :(


Shall I differentiate this constraint with respect to time ? y_P² + x² = l_P²

Throw me the rope !

And its also not obvious to me that why there is not a triangle involved in constraint equation for ring Q, while there is doubtfully a triangle involved for constraint equation of ring P. I cannot fathom.

Please, can I get more hints ?... :)

 

[voko]

I would say you need to get the constraints right to begin with.

$l_Q$ is just a vertical line - where is a triangle there?

 

[dreamLord]

Draw the figure at an arbitrary time with Q at a distance y_q and P at a distance y_p from the support. What is the length of the string connecting the 2 rings? It is simple geometry.

 

[sankalpmittal]

Ok thanks to Voko for the help till now.

My both the equations were wrong then ? :(

So in none of the constraint equations there is a triangle involved. :(

Pythagoras theorem fails then. I would have to try something else, I don't know what. Please suggest something.

dreamLord, is it easier to use geometry than by constraint equation ?

 

[voko]

I really am unsure why you are having trouble with this.

Even in the drawing in #1, it is quite obvious that $y_Q$, which is the distance from the top bar to Q, is just a vertical line. It will always be a vertical line, because Q is constrained to move along the rod. It is also clear that $y_P = l_P$, which is the length of the string from the top to Q.

The relationship between $y_P$ and $l_P$, where the latter the length of the string from Q to P, is more complicated. If you draw a line parallel to the top bar and passing though Q, the line will also pass through the rod on the left; let's label that point R. PR and $l_P$ are two sides of triangle PQR, so they are related. But how is PR related to $y_P$?