Does Newton's third law apply to Atwood's machine?

[Syrus]

Homework Statement

Consider Atwood's machine with two masses, m₁ and m₂ with m₁ less than m₂. Now, according to application of Newton's third law, the system accelerates in the direction of the heavier mass (here, m₂). This, however, seems to contradict Newton's third law, which implies that the interaction between the block and the rope are equal and opposite. That is, the tension of the string should counteract the force of gravity on each block. Obviously, though, there is a fallacy someplace in my argument. Does anyone mind pointing it out?

Homework Equations

The Attempt at a Solution

 

[rollcast]

Going by my applied maths textbook and having a quick look at a diagram for an attwood machine. It appears that the force acting on the pulley is eqivilent to 2 times the force of tension in the string over it.

I'll do a quick diagram to show why.

 

[PhanthomJay]

Homework Statement

Consider Atwood's machine with two masses, m₁ and m₂ with m₁ less than m₂. Now, according to application of Newton's [strike]third[/strike] second[/color] law, the system accelerates in the direction of the heavier mass (here, m₂).

I think you mean that the center of mass of the system accelerates, with respect to the ground, in the direction of the net force, which is down

This, however, seems to contradict Newton's third law, which implies that the interaction between the block and the rope are equal and opposite.

this is correct regarding Newton 3, but there is no contradiction.

That is, the tension of the string should counteract the force of gravity on each block. Obviously, though, there is a fallacy someplace in my argument. Does anyone mind pointing it out?

the string tension does not counteract the force of gravity. That would be true only if the system is not accelerating. Newton 3 says that if the block exerts a force on the string equal to 'A', then the string exerts an equal and opposite force on the block with a magnitude of 'A'. 'A' is generally not equal to the block's weight.

Homework Equations

The Attempt at a Solution

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[rollcast]

Heres my quick diagram.

Then consider the pulley and only the forces acting directly on it.

 

[Syrus]

rollcast, it sounds like you're atttesting to Newton's third law holding for the pulley itself and the rope. I perhaps should have made it more clear that i was interested in the third law interaction between the rope and each mass (i.e. that the interactions between the rope and each mass would cancel out and hence the net acceleration should be zero).

 

[rollcast]

rollcast, it sounds like you're atttesting to Newton's third law holding for the pulley itself and the rope. I perhaps should have made it more clear that i was interested in the third law interaction between the rope and each mass (i.e. that the interactions between the rope and each mass would cancel out and hence the net acceleration should be zero).

I thought you meant the pulley when you said block as in a pulley block.

 

[SHISHKABOB]

rollcast, it sounds like you're atttesting to Newton's third law holding for the pulley itself and the rope. I perhaps should have made it more clear that i was interested in the third law interaction between the rope and each mass (i.e. that the interactions between the rope and each mass would cancel out and hence the net acceleration should be zero).

well Syrus, you can still see by his diagram that all of the tension in that set-up cancels out, thus we are left with only the weights, and they are unbalanced

 

[Syrus]

Shishkabob,

I guess where my confusion arises is in figuring why the tensions are equal when (it seems) they should not be (due to the fact that the blocks are not identical in mass and hence, by N3, the tension force applied to each mass should counteract the gravitational force- whence rendering no acceleration).

A previous remark addressed this issue by stating that the downward force of the mass was not, in fact, equal to the gravitational force acting on it, but i am still failing to grasp why this is so.

 

[SHISHKABOB]

I'm pretty sure it's the weight of the heavier block minus the weight of the less massive block.

Imagine the blocks laid out flat on the ground with the ropes connecting them. Block one has a force acting on it equal to its weight, and the other block has a force acting on it equal to its weight

or even imagine it like two guys pulling on it in opposite directions. One guy pulls with a constant force that is larger than the other guy, which direction does the whole rope-two guy system move?

 

[PhanthomJay]

A previous remark addressed this issue by stating that the downward force of the mass was not, in fact, equal to the gravitational force acting on it, but i am still failing to grasp why this is so.

You should apply Newton's 2nd Law to the masses. Let's look at the heavier one, in a free body diagram. Let's say it has a mass of 20 kg, or a weight of 200 N. Newton 2 says that

F_{net} = ma, and since the net force is 200 N down and T Newtons up, then

200 - T = ma

Now since the system is accelerating and not at rest, then the acceleration is greater than 0, and thus, T must be less than the gravitational force of 200 N.

The tension of a string wrapped around a frictionless and massless ideal pulley is always the same on both sides of the pulley.

Newton 3 holds for all cases of accelerating or non accelerating bodies. In this case, the rope exerts a force of T upward on the 20 kg mass, so the mass exerts an opposite force of T downward on the rope. T is not equal to 200 N. Newton 3 refers to equal and opposite forces on different objects. Newton 2 applies to forces on the same object.