What is the tension in a rope when a monkey accelerates up?

[Lnewqban]

I just want to be verified that I understand it right. All rope does when you say “The tension force pulls outward along the two ends of a rope” is that it is pulling inwards what is attached to it on both ends when the rope is under tension?

I believe that I did not say that, it is a statement in that linked website.

The internal forces in the rope ("molecular tension") prevent the rope from being ripped apart by the external forces acting at both ends.
The internal forces are resistive, are a natural reaction to external energy.
If the rope breaks, the energy of the external forces did the destructive work.

 

[rudransh verma]

I believe that I did not say that.

I mean the link says that.

 

[Lnewqban]

I mean the link says that.

As the magnitudes of external forces and internal forces are similar, they use tension and force alternately.
Again, I may be wrong.

 

[kuruman]

To @rudransh verma :
The important thing to remember about tension in a rope is that a force is associated with it. To find its direction, first you need a system on which this force acts. Once you have defined the system, the force due to tension acts along the rope or string in a direction away from the system. In short, you cannot push with a rope.

For example, in the climbing monkey problem we can choose to define the monkey as the system (see post #48.) The force due to the rope tension is up and away from the monkey. We can also choose as our system a 10 cm piece of rope above the point of contact with the monkey's hand. In that case, there is a down force on the 10 cm piece of rope due to the monkey's hand (equal and opposite to the one exerted by the rope on the monkey and away from the system), and an "up" force, also away from the system exerted by the rope that is above the 10 cm system. We can also choose a piece of rope of any length as the system. There will be two forces at each end of the system, both pointing away from the system.

At this stage, I think that's all you need to know to figure out the direction of the force due to tension in order to use it in free body diagrams. Knowing the details of the molecular forces that give rise to this force is not necessary for drawing and interpreting FBDs just like it is not necessary to know exactly how a car engine works in order to drive one to a friend's house. Defining the system before you attempt to draw the force due to tension is as important as knowing what a car looks like before you get in it.

 

[rudransh verma]

The attachment points pull outward on the rope. The rope pulls inward on the attachment points. For an ideal (massless) rope, the magnitudes of all four of those forces are identical.

One can see that they are identical by applying Newton's third law at the two attachment points ends and Newton's second and third laws in a daisy-chain all the way along the rope.

Can you clarify my post#50

 

[jbriggs444]

This casual explanation is making my life very difficult from morning (and now its night time) because what I knew is Tension T's direction when the rope is under stress is upwards and the force applied by monkey is downwards. So $T=F_{MR}=-mg$

You said you do not want casual. Ask and you shall receive.

For Tension as a stress in a rope, it has neither direction "up" nor direction "down". It has only direction "vertical". In three dimensions, tension has a coordinate representation as a 3 by 3 matrix. If you choose a coordinate system in which the first coordinate is vertically aligned with the vertical rope, the stress is uniform and the cross-sectional area of the rope is $a$ then this tensor will be $$
\begin{bmatrix}
-T/a & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{bmatrix}$$More generally, if one chooses a poorly aligned coordinate system, the tension will manifest as two or three negative terms on the main diagonal that sum (via the Pythagorean theorem) to the magnitude of the tension (per unit area) in the rope.

If one wants to recover the force transmitted by a uniform tension through an imaginary dividing line in the rope, one multiplies the stress tensor by the directed area of a cross-section through the rope.

A "directed area" is a planar area represented as a vector. The area is represented by the magnitude of this vector. The direction is represented by the direction of the vector (normal to the plane). If one multiplies a 3x3 stress tensor by a 1x3 directed area, one recovers a 3x1 force vector. That vector is the force resulting from the tension acting through the area.

Even less casually, tension need not be uniform across the rope. One could instead take a surface integral, adding up local stress times incremental directed area across a surface that bisects the rope.

What this means for a rope attached to the monkey is that you multiply the stress tensor in the body of the rope by a directed area pointing outward at the monkey and recover a vector force that points inward toward the rope. This is the force of rope on monkey.

More casually, we have a useful invariant. No matter where we choose to slice an ideal massless rope with a directed surface between the potion exerting a force and the portion on which a force is being exerted, the resulting vector force will be aligned with the rope and will have a direction opposite to the direction of the slice. [It does not matter whether you slice neatly at right angles, diagonally or along a jagged curve. The result is always the same -- aligned with the rope, not with the cut].

So tension is basically a force that the rope applies back when it is under stress. It is an inward force. Tension T's direction at end points of rope where its attached to the body and ceiling is inwards. Tension is what we pull something with not push.

Casually speaking, yes.

Someone said that tension is pair of opposite forces acting on small rope elements all along the rope cancelling each other. At the ends there is just one force acting inwards. Am I right?

Yes, there is one force from rope on attached object. Though, of course, Newton's third law still applies. There is an equal and opposite force from attached object on rope.

 

[rudransh verma]

You said you do not want casual. Ask and you shall receive.

Ok! Very clever. Thanks man!