# Question regarding kinematics

1. Jun 18, 2013

### sankalpmittal

1. The problem statement, all variables and given/known data

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/ [Broken]

2. Relevant equations

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3. 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.

Last edited by a moderator: May 6, 2017
2. Jun 18, 2013

### PeterO

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?

Last edited by a moderator: May 6, 2017
3. Jun 18, 2013

### 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?

How did you arrive at this conclusion?

4. Jun 18, 2013

### PeterO

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 travelling 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?)

5. Jun 19, 2013

### 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 ?

6. Jun 20, 2013

### 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 ?

7. Jun 20, 2013

### PeterO

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

8. Jun 20, 2013

### voko

That is the way to go.

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

9. Jun 20, 2013

### sankalpmittal

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

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...

10. Jun 20, 2013

### voko

What is y?

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

11. Jun 20, 2013

### sankalpmittal

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

I am assigning downward negative and upward positive.

Now, we have the constraint,

yQ2 + x2 = lQ2

And , yP2 + x2 = lP2

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

lP is length of string shortened by Ring P and lQ, that length of string shortened by Ring Q.

Am I on the right track ?

12. Jun 20, 2013

### 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 $y_P$ ?

13. Jun 22, 2013

### sankalpmittal

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

14. Jun 22, 2013

### voko

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

15. Jun 22, 2013

### sankalpmittal

What ? =.= .....

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

Shall I differentiate this constraint with respect to time ? yP2 + x2 = lP2

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 ?... :)

16. Jun 22, 2013

### 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?

17. Jun 22, 2013

### dreamLord

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

18. Jun 24, 2013

### 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 ?

19. Jun 24, 2013

### 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 $y_P$?