Question

Trivia - Tension Trick

Answer 1

Let's say - we have to find a relation between the acceleration vectors of the blocks in the diagram (see attachment).

To keep it simple; don't consider torque (for m5). Although it won't be much of a trouble; but I think the system is already pretty complicated.

Note: All pulleys and strings are ideal.

Answer 2

To use the trick method; first draw all the tension vectors and acceleration vectors (corresponding to each mass).

Then according to Tension Trick -
\[\Sigma (T⋅a)=0\]

Answer 3

Now, try to find a proof for the above ^^ trick.

PS: You can use a single pulley system.

Answer 4

Can you choose a simple example to show us? I do not understand what a represents.

Answer 5

'T' represents Tension vector
'a' represent acceleration vector

For those who can't see the equation -
Σ(T⋅a)=0

^^ Its a dot product :)

Answer 6

Consider the simple pulley system -
According to Tension Trick -
(-T*A) + (T*a) = 0
implies A = a

Answer 7

^^ @Vincent-Lyon.Fr

Answer 8

But A = a is a simple consequence of kinematics, no forces need to be implied to prove it.

Sorry, I still do not understand. What if a pulley has its centre moving too? Does this trick help?

Answer 9

Yes, it is obvious here.

But what needs to be noted is - the trick will work irrespective of what arrangement of pulleys do we use.

Answer 10

Like in my last question - http://openstudy.com/users/lastdaywork#/updates/52dbfe05e4b003c643a019fb

I could easily say that A = a (and save a lot of lengthy calculation)
Was it obvious there too ??

Answer 11

So, if the trick is independent of the arrangement of pulleys; it must have a general proof, right?

Answer 12

who needs so many pulleys in the first place? :D :D..

Answer 13

"I could easily say that A = a (and save a lot of lengthy calculation) Was it obvious there too ??"

Of course it is obvious since the thread cannot be extended. Noneed of lengthy calculation to state that.

Answer 14

"who needs so many pulleys in the first place? :D :D.. "

It's only for the sake of making you mad, Mashy! ;-)

Answer 15

Conserving the length of thread might become complicated in some cases. I can upload some; if you are interested :)

@Mashy : It really is to make people mad :D

Answer 16

BTW, the answer is simply the fact that - in an ideal pulley system; net work done by tension (at any instance) is always zero.

You can involve torque and spring and all other stuff but the above fact survives :)

Answer 17

And yes - "What if a pulley has its centre moving too? Does this trick help? "
It will still work (as long as the pulley is mass-less)