1. Feb 22, 2015

### Faris Shajahan

Last edited by a moderator: May 7, 2017
2. Feb 22, 2015

### Staff: Mentor

We don't work that way here. We can't do your homework for you. We can only provide hints once we see some work from you.

3. Feb 22, 2015

### Faris Shajahan

Well its not my homework......its a doubt.....
I know how to solve the problem completely but there is a doubt I have which is killing me!

4. Feb 22, 2015

### Staff: Mentor

Then why did you post it needing an immediate answer?

5. Feb 22, 2015

### Staff: Mentor

What is the doubt? Perhaps that can lead to a better understanding.

6. Feb 23, 2015

### Faris Shajahan

The doubt goes like while solving problems like the one shown in the figure, we take component of v1 along the string i.e. v1cos$\theta$ and write v1cos$\theta$=v2.....................but why don't we write v2cos$\theta$=v1???

7. Feb 23, 2015

### Staff: Mentor

It looks like in this example the string can't stretch and the pulley has no mass (hence no torque).

The v1cos(theta) is the projection of the speed along the string caused by the ball moving in the v1 direction. Since the string is inextensible (i.e. no stretching) then the v1cos(theta) speed is directly related to the speed of the block moving upward.

I can see the confusion you're having here If you were to do it starting with v2 then v2 would be along the string and hence the projection of v2 on the plane where the ball is moving is v2cos(theta).

I just can't see the reason of why one is chosen over the other. I still need to ponder this.

Someone else here at PF may have a better explanation.

8. Feb 23, 2015

### Staff: Mentor

Because the angle $\theta$ is between the direction of v1 and the direction of the string at m1. It is not an angle associated with v2 or m2. The direction of v2 is the same as the direction of the string at m2, so there's no angle involved there.

9. Feb 24, 2015

### Faris Shajahan

Could you please explain a little more......because I did not get what you exactly meant!
Thanks though, it cleared "some" parts of my doubt........

10. Feb 24, 2015

### jbriggs444

Saying it in slightly different words...

The motion of m1 can be seen has having two components. One in the direction of the string (i.e. at angle $\theta$ below the horizontal) and one at right angles to that (i.e at angle $\theta$ left of vertical).

Since component in the left-of-vertical direction is at right angles to the string, it has no effect on the motion of m2. It acts to "unwind" some string from the pulley without actually causing the pulley to rotate. The component in the below-horizontal direction does have a direct effect on the motion of m2.

So we separate the leftward motion of m2 into those two components. The left-of-vertical component is equal to v1 sin $\theta$ and is discarded. The below-horizontal component is equal to v1 cos $\theta$ and is the important piece.

11. Feb 24, 2015

### Staff: Mentor

Let h be the distance of the pulley above the table. In terms of h and θ, what is the horizontal component of distance of the ball from the pulley? What is the distance of the ball from the pulley along the hypotenuse? What is the time derivative of the distance along the hypotenuse, given that h is constant? What is the time derivative of the horizontal component of distance of the ball from the pulley, again given that h is constant? What is the relationship between these two time derivatives?

Chet

12. Feb 24, 2015

### Staff: Mentor

Another approach:
$v_1 \cos \theta = v_2 \cos \phi$ where $\phi$ is the angle between the direction of motion of object 2 and the string. This angle happens to be zero here, so the second cosine term is 1 and can be ignored. $v_2 \cos 0 = v_2 \cdot 1 = v_2$.

13. Feb 24, 2015

### Staff: Mentor

In answer to the very first post: the component of v1 in the direction of the string is the method to use.

The right-angled triangle you see outlined by the string and the horizontal is not a velocity vector triangle, neither is it a distance vector triangle. While its base and hypotenuse can change, this triangle's height remains fixed by the location of the pulley, so it doesn't represent any vector triangle. You'll have to construct your own vector triangle for the analysis, and it won't look like the right-triangular shape formed by the string around the pulley. The latter is just a clever distraction to trip up the unwary.

14. Feb 24, 2015

### jeffery_winkle

This is a related question. I have heard it said that with a system of pulleys, it reduces the force you need to lift a given weight, so someone could lift something heavier than they could normally lift. Well, if that were true, by simply adding more pulleys, you could lift an unlimited amount of weight. With enough pulleys, one person could lift ten tons. Well, that can't possibly be true. What's the flaw in the reasoning?

15. Feb 24, 2015

### Staff: Mentor

You can.
But the distance you have to pull the rope to lift the weight increases with the same factor as the weight you lift, so you need the same energy.
Please start a new topic if you want to discuss that in more detail.

16. Feb 24, 2015

### Staff: Mentor

Are you saying that the method that I outlined in post #11 will give the wrong answer? Because I don't think so.

H = hypotenuse
L = horizontal length

$$H = \frac{h}{sinθ}$$
$$\frac{dH}{dt}=v_2=-\frac{h}{sin^2θ}cosθ\frac{dθ}{dt}$$

$$L=\frac{h}{tanθ}$$
$$\frac{dL}{dt}=v_1=-\frac{h}{tan^2θ}sec^2θ\frac{dθ}{dt}=-\frac{h}{sin^2θ}\frac{dθ}{dt}$$

So, $v_2 = v_1cosθ$

Regarding the loss of the triangular shape formed by the string around the pulley, I can make the pulley as small as I wish relative the H and L.

Chet

17. Feb 24, 2015

### Staff: Mentor

Sorry. I started my response when there were only 2 other replies, and I was addressing post #1. Only after I'd posted did I become aware of still more replies, but I haven't studied them.

Keep up your good work, Chet!

18. Feb 25, 2015

### Faris Shajahan

I know what you mean and this method, I believe, uses constraints.
It goes like.........

Let the length of the string between $m_1$ and the pulley at any time t be $L$.
And let the distance between the pulley and the table be $y$. Let the horizontal distance between $m_1$ and the pulley be $x$.
Then as $m_1$ moves $L$ changes, $x$ changes, but $y$ remains constant.

Considering the right angled triangle,
$L^2=x^2+y^2$
Differentiating,
$2L\frac{dL}{dt}=2x\frac{dx}{dt}+2y\frac{dy}{dt}$

From the figure, $\frac{dL}{dt}=v_2$ (as with whatever velocity the string moves on the right side of the pulley, the same on the left) $;\frac{dx}{dt}=v_1; \frac{dy}{dt}=0$
Also $x=Lcos\theta$

Hence substituting,
$Lv_2=Lcos\theta v_1$
$=> v_2=v_1cos\theta$

Clear.!
But any other way than going for all this or in other words, how do we find this directly?

19. Feb 25, 2015

### Staff: Mentor

Sure. See post #16.

Chet

20. Feb 25, 2015

### Faris Shajahan

Clearly possible.
An example may be as shown,
Let T be the tension in the string joining m and the closest pulley.

Then in this figure we require a force>=T......where T=mg to lift the mass.
But here,

though T=mg, we only require force>=T/2 to lift the mass m!!!!!