Conditions for pulleys and strings

[andyrk]

Homework Statement

Why are pulleys frictionless and massless in the pulley questions? What would happen if they are not frictionless and not massless?
Why are the strings connecting the blocks in pulley questions massless and inextensible? What would happen if they are not massless an not inextensible?

 

[voko]

Consider each part of your question separately: e.g., what would happen if a pulley were massive and a force were applied to it?

 

[andyrk]

I don't know ...I am asking that..sorry but could you please explain a bit more?

 

[voko]

Take a simple case: a rope goes over a massive pulley, with two unequal masses at its ends. Clearly the masses will be accelerating. And the pulley will have to accelerate with them. But it is massive, i.e., it has inertia. How do you think this will affect its acceleration, and the acceleration of the masses?

 

[andyrk]

What I think of it

Intuitively I think that it would be difficult to rotate the pulley as it would we wanting to stay in its position of rest so it would apply friction to the tension to the string going over it, so it would hinder the actual acceleration of the blocks due to friction and we'll get a lesser value..is that it?
But this involves friction also...what would happen when the pulley is friction less but has mass and therefore has inertia? Also I think that the strings are mass less so that tension is same throughout the string. if the string wouldn't be mass less then different part of the string would have different tension..What about it being in-extensible?

 

[voko]

Massiveness and friction are independent. They have separate effects.

Massiveness of strings is similar to massiveness of pulleys. You would need to take into account their own motion.

Finally, an extensible string is also known as "spring".

 

[andyrk]

Inability

Ok..thanks for that but I am unable to conclude the effect(s) that would arise out of massiveness and friction..could you please tell them with the explanation? What I thought I wrote above..rest I don't know whether they are correct or not..

 

[andyrk]

I think that an extensible string would mean that the object connected with it would be having different accelerations...

 

[voko]

Massiveness = inertia.

Friction = resistance to motion.

 

[andyrk]

So what would inertia do? The pulley would spin slowly but will that have any effect on the values of tension and acceleration in the string and blocks respectively?

 

[voko]

Think again on #4: how would that be different if the pulley was massless.

 

[andyrk]

The string would slide/skid over the pulley rather than moving with it..is that correct? I need this for an assignment please if you know tell..its urgent! :)

 

[voko]

The string would slide/skid over the pulley rather than moving with it..is that correct?

That is a possible outcome if the pulley is very massive, and the static friction between the string is pulley is not very great, and the tensile strength of the string is high. Do you see how many complications you may have if your pulleys and strings are not ideal?

 

[andyrk]

So if the string DOES skid what are the effects? I don't see any..

 

[voko]

And if it does NOT?

 

[andyrk]

Even then nothing happens..

 

[voko]

So mass is a meaningless concept for you, then.

 

[andyrk]

Had I known the answer to that I wouldn't have started the thread in the first place! If you know it please tell it! nothing more that I can say!

 

[andyrk]

So mass is a meaningless concept for you, then.

I think that the acceleration of the pulley would also have to be added to the masses connected via the string since tension in the rope acts as torque to the pulley..so the pulley rotates and would have an angular acceleration..as a result it would be having a tangential acceleration which won't play any role for the pulley but would impart that tangential acceleration to the string which in turn would impart the extra acceleration to the bodies connected by the string..Is that correct?

 

[voko]

I think that the acceleration of the pulley would also have to be added to the masses connected via the string since tension in the rope acts as torque to the pulley..so the pulley rotates and would have an angular acceleration..as a result it would be having a tangential acceleration which won't play any role for the pulley but would impart that tangential acceleration to the string which in turn would impart the extra acceleration to the bodies connected by the string..Is that correct?

This is not exactly correct, but you are going in the right direction. It is true that a massive pulley will have to accelerate, and its acceleration depends on its mass. Because the pulley and the masses are connected with a rope, their accelerations must be equal (at least when the rope is not skidding) - and what does that mean?

 

[andyrk]

According to me, that means that the pulley and the string as a result the block would have the same acceleration. But that would also be the case when the pulley is mass less..I am stuck here..

 

[voko]

Imagine there are two stones. One is very light - you could say massless - and another is very heavy.

You have to lift one of those.

There is some amount of force your muscles can exert on either one.

Naturally, when you lift a stone, your hands (at least) have to move with the same acceleration as the stone does.

According to you, the motion of your hands would be the same with either stone.

 

[andyrk]

Hmmm..so what would be the condition required for the pulleys tangential acceleration and the strings and blocks acceleration to be the same?
Let T be the tension in the string, then Td=ζ (Torque); ---(1)where 'd' is the radius of the pulley.
Also, dα=a_t---(2); where 'α' is the angular acceleration of the pulley imparted by the tension T.

Also, ζ=Iα---(3) ; where 'α' is the angular acceleration of the pulley imparted by the tension T and 'I' is the Moment of Inertia of the pulley about its own axis of rotation. So, If the pulley is considered a cylinder (since we don't consider it of negligible thickness otherwise it would have been a disc). Let the mass of the pulley be m_pulley. Also this is a solid cylinder.
Therefore I= (1/2)m_pulleyd²---(4)
So, Td=(1/2)m_pulleyd²(a_t/d)---(5)
=>a_t=2T/m_pulley---(6)
So if masses of blocks are m₁ and m₂; m₂>m₁, then their acceleration if the pulley is massless is:
a_blocks=((m₂-m₁)/(m₂+m₁))g---(7); where g is acceleration due to gravity.
a_t=a_blocks, RHS of (6) and (7) should be equal,
2T/m_pulley=((m₂-m₁)/(m₂+m₁))g---(8)
Also, T=((2m₁m₂)/(m₁+m₂))g;---(9)
So,
((4m₁m₂)/(m₁+m₂)m_pulley)g=((m₂-m₁)/(m₂+m₁))g

=>((4m₁m₂)/(m₂-m₁))=m_pulley

Is that the condition required? For the acceleration of the blocks to remain the same even though the pulley has mass? If this condition is not met, then the acceleration of the blocks would be the same as the acceleration of the pulley provided that the blocks don't skid. Is that correct? Phew!

 

[voko]

Not exactly. There is no physical reason for equating (6) and (7).

It is true that while the rope does not skid, the acceleration of the masses and what you call the tangential acceleration of the pulley must be equal (note this assumes the rope is inextensible). But that does not mean they must be equal to the acceleration of the masses when the pulley is massless!

What you should do instead is consider the FBD for each object, including the pulley.

 

[andyrk]

Free Body Diagrams

Here are the FBDs:

 

[voko]

If the tension at the left-hand mass is equal with the tension at the right-hand mass, then the net torque on the pulley is zero, so it won't accelerate at all.

 

[andyrk]

So you mean to say that tensions should be different in the string on either side of the pulley?

 

[voko]

If they are the same, what is the net torque on the pulley?

 

[andyrk]

then the torque would be zero...so the pulley won't rotate so the string would skid..is that it? But why would the tensions be different?

 

[voko]

And why should they be equal?