KK2.22) A uniform rope of weight W hangs between two trees. The ends of the rope are the same height, & each of them make equal angle θ (acute) with the trees. Find the tension at the middle of the rope.
I did in the following way: considering equilibrium of the left/right portion of the rope, horizontal component of tension at the end, which is the horizontal component of reaction of the tree on the rope, must be equal in magnitude to the horizontal tension in the middle. Therefore, T(end) sinθ=T(mid); it appears to be correct & matches the criterion of the solution.
But normally we do equate forces considering a single point of action, such as C.G. of a body, in Newtonian mechanics. Am I justified in carrying the problem out in this way? Am I effectively equating horizontal forces acting at a single point?

Off the topic

I was thinking about adding one more Law to Newtons's law of motion, which is All motion proceeds in a forward motion. It seems to me that its impossibly to move backwards. Even if time travelling was designed it would only be possible to travel forwards. Time dilation also travels forwards. In fact you have to move forwards before you can move backwards, which is not exactly backwards but a subset or variation of its continual forwards motion.

*A ball at rest(0 value) will always remain at rest unless an object acts on it, which sets it in motion, and any motion and direction is a positive magnitude (more values than zero), setting it forward.

Of course it has a positive MAGNITUDE. Magnitude is always greater than or equal to zero by definition. However, actual motion, displacement, velocity, acceleration, etc. is neither forward or backward, it is simply a vector in some n-space. This vector could have positive compoents, negative components, or zero components in any of the n-dimensions of the space. So while it may FEEL right to say forward. It has no real maning to say that.

-AntonK

Of course it has a positive MAGNITUDE. Magnitude is always greater than or equal to zero by definition. However, actual motion, displacement, velocity, acceleration, etc. is neither forward or backward, it is simply a vector in some n-space. This vector could have positive compoents, negative components, or zero components in any of the n-dimensions of the space. So while it may FEEL right to say forward. It has no real maning to say that.

-AntonK
That is where time dilation comes in. It emphasis my theory by clearifying that it is at least impossible to travel back in time. If time does does permit backwards travel, so does space. Since matter occupies space matter has no impetus and mechanics to travel backwards, at least on the quantum level. Also note that things must travel forwards to be ABLE to travel backwards, which in retrospect means that all initial motion must be forwards. The law should better be revised as all initial motion is a forwards motion

Again, you are using a term that has no meaning within physics. Motion is a vector. If I am moving at +1.0 m/s on the X axis and I turn completely around and move the opposite direction at the same speed, I am now moving -1.0m/s on the X axis. This is NOT backwards, nor was the first direction forwards. All I would have to do is reverse my coordinate system and at first I was moving -1.0 m/s on the X axis and after I turn around I am moving +1.0 m/s on the X axis. This is why we use vector notation rather than imprecise words like forwards and backwards.

Likewise with time, if we treat time as a fourth dimension then we are simply moving on the T axis at some rate. The fact that time dilation is only capable of reducing that rate to a limit of 0 does not imply that the direction of time cannot be negative. Or in fact we could reverse our coordinate system again and say that we are ALREADY moving negative through the T axis and that time dilation is incapable of resulting in a POSITIVE rate. Do you see now how words like forward and backward have no meaning and would never be incorporated into a set of laws like that.

-AntonK

Yes but there are some realities. One reality is that it is impossible for the universe to reverse back exactly to its initial position, just as its impossibly for me to return to my initial position at birth. Let me explain. If you kick a ball, it moves in the magnitude and direction of the resultant vector, this direction as you say is inprecise, but it is nonetheless a direction relative to the origin. Lets say as you kicked the ball I travelled around the universe and back in the speed of light at the same magnitude and direction as the ball, I would have gained a lot more time and space relative to the ball's. I would have travelled into the future relative to the ball. Do we agree on this? Yes. Here is the kicker, the ball does not travel backwards...or any direction for that matter, yet it manages to remain in the past(initial position). So you see you don't have to travel in any direction to remain in the past, all you need is any other object faster than you.

Location and time
Location is dependent of time but time is not dependent of location. Location needs space, and space warps time. Time is a measurement of Scalar quantities, totaly independent. Location is also dependent on the observer, Time is independednt of the observer. You cannot have location without space-time but you can have initial location and final location; and time is used to measure the rate of change from the initial to the final. The time at the initial is always before the time at the end, hence forwards/backwards is merely used to identify the initial from the end. Unless you can turn the universe the other way around as you say...it is in motion, and it is in a forwards motion relative to its initial.

neeklash, I am also going through K & K (as well as several other physics texs) so maybe we can talk. Off the top of my head, 2Tsin theta = mg, where T is the vector describing the tension force at one end of the hanging rope. From there you can find the middle rope by using T*cos (theta) = T,mid. Wherere T,mid is the 'tension' acting in one direction on the middle of the rope.

You are in effect treating the rope as a point mass. Of course, it helps that we can say mg/2 is the weight of half of the rope (because we can translate this result to our particle without the morbid integration used in solving gravitation problems).

KK2.22) A uniform rope of weight W hangs between two trees. The ends of the rope are the same height, & each of them make equal angle θ (acute) with the trees. Find the tension at the middle of the rope.

This is a difficult question. (What is K&K, by the way?)

Does the rope hang in a curve (catenary), or is it pictured as sagging in the middle with straight sections going to the trees? Because if it is a rope with mass, it will hang in a curve. Moreover, the tension will vary along the length of the rope.

Yes but there are some realities. One reality is that it is impossible for the universe to reverse back exactly to its initial position, just as its impossibly for me to return to my initial position at birth. Let me explain. If you kick a ball, it moves in the magnitude and direction of the resultant vector, this direction as you say is inprecise, but it is nonetheless a direction relative to the origin. Lets say as you kicked the ball I travelled around the universe and back in the speed of light at the same magnitude and direction as the ball, I would have gained a lot more time and space relative to the ball's. I would have travelled into the future relative to the ball. Do we agree on this? Yes. Here is the kicker, the ball does not travel backwards...or any direction for that matter, yet it manages to remain in the past(initial position). So you see you don't have to travel in any direction to remain in the past, all you need is any other object faster than you.

Location and time
Location is dependent of time but time is not dependent of location. Location needs space, and space warps time. Time is a measurement of Scalar quantities, totaly independent. Location is also dependent on the observer, Time is independednt of the observer. You cannot have location without space-time but you can have initial location and final location; and time is used to measure the rate of change from the initial to the final. The time at the initial is always before the time at the end, hence forwards/backwards is merely used to identify the initial from the end. Unless you can turn the universe the other way around as you say...it is in motion, and it is in a forwards motion relative to its initial.

Perhaps the best way to understand my observation is a circle. All radius of a circle(regardless of direction) goes towards and away from the center or origin. All movements away from the center are traveling in forwards motions because they are moving away from the center(origin). All movement towards the center are travelling backwards because that is a point of little or no space-time. Chords are not significant because they are not travelling from or towards the center of the circle. So its not so much about direction but about position relative to origin. Another important note is that cosmic background radiation now proves that the universe started with a big bang, and as long as the universe will continualy expand its applicable to say all initial motion is in a forwards motion. Do I make sense?

Fine, if you want to continue to argue that your statement:

"All initial motion is in a forward direction"

Then you must answer this. What precisely does this give us an addition to Newton's Laws, that Newton's Laws do not already give us? What predictions can be made with the addition of your statement that CANNOT be made without it? If the answer is none, it has no place.

-AntonK

I think you are being too conservative and obviously a newtonian. Listen, its very simple. "YOU CANNOT GO BACK IN TIME". If I travel the speed of light towards the universe and back I will arrive in the Future. Thus all speed faster than the slowest speed in the universe is into the future.

Yes, it may not apparently give us anything now but who knows in the future, and I am certain it will help in QM. Newtonian mechs is overated anyway. I have to confess I an not an expert in Physics but neither did I need anybody to tell me "all bodies remain at rest unless a foreign force acts on it". Besides its not about Newton, its a fundamental element of motion.Hey, just joking.

I am not a 'Newtonian'. I simply realize that under specific conditions, Newton provided an EXCELLENT predictive model. Relativity and QM improved on these given different specific conditions. In the future, someone will eventually construct a new model which will have even greater predictive abilities under even wider conditions.

In fact, at the time not everyone knew or agreed with Newton's Laws. For a very long time it was believed that the "natural" state of an object was to be still. That is, even given that you apply a force, the body will ALWAYS want to slow down til it has a velocity of 0. Newton showed this was not the case. In fact what he did was give us a ground work for PREDICTING motion. Any model that wants to contribute must give the ability to predict something that was not predicted before.

-AntonK

Good contribution. Thanks mate

Again, you are using a term that has no meaning within physics. Motion is a vector. If I am moving at +1.0 m/s on the X axis and I turn completely around and move the opposite direction at the same speed, I am now moving -1.0m/s on the X axis.

But if you turned around you are still going forward.. granted plots pertain a reversal to possible plots but the facts remains you have not stopped but rather changed trajectory in a forward momentum that is negative to your initial velocity.

I understand where the both of you are coming from, and indeed you are somewhat correct as is the other, but the above point is, you are still going forward, perhaps it would have been better to suggest a freeze and or an absolute Stop, and then a complete reversal of all steps and or potential energy negatively prior to the absolute Stop, which BTW is absolutely Impossible in the real world, or are you saying a wound up spring can actually wind its self up even further after our winding it up, rather than gaining a velocity back to a state of equilibrium when we release the spring. :D

>> All motion proceeds in a forward motion

or in other words

Force can only be transmitted via a 'push'.

There is no 'pull'

>> All motion proceeds in a forward motion

or in other words

Force can only be transmitted via a 'push'.

There is no 'pull'

And may I add push can only be possible if exerted upon an equal or greater opposing force, having said that I agree attraction is based on pseudo science.

Further more - Gravity is not via attraction either! Rather it is exerted!
And my money is on the 90~99% of mass that most individuals don't fully understand as yet, Err~that's is if we overlook the few that do understand the properties of near Vacuum.. :cool: