How does force get distributed in a body?

Say we hold a rope in our hand at arms length. If we move our arm towards our body slowly the whole rope moves such that there is no bending and it moves as if a straight rod. But if we move the same rope towards us quickly the lowermost part remains where it is for a split second and then begins to move with the rest of the body. Thus there is a delay in the movement of the lowermost part. The only force acting on the rope was the one that our hand applied on the upper end of the rope.

I know that this is because of inertia of the lower most part (that is the high-school physics textbook answer). What I want to know is that why is there a difference in the way force got distributed across the rope, depending on whether our force was impulsive or not. My book said that "the force doesn't have enough time to spread across the body". But that statement needs a bit more explanation, such as a mathematical one.

The force got distributed in the same way. You just chose two extreme examples that look like they are different.

When you pull the rope towards you, you increase the tension on the part of the rope nearest you. Now, to be completely accurate we would have to consider how fast this change in tension is distributed. It gets distributed at the speed of sound, which is really fast. Faster than anything in your experiments. So we'll ignore it... but the effect is indeed there.

So when you pull on the rope you increase the tension force. Now that the tension force is higher than the force of gravity pulling on the rope, the rope actually gets pulled upwards. You can sometimes see this at the bottom of the rope. The key here is that you build up a gravitational potential. The system wants to let the rope extend back down fully, so that it is at the lowest height possible.

So now you have two forces. You have the force of tension from your hand, which pulls the top of the rope towards your body, and pulls the rest of the rope along the length of the rope, and the force of gravity, pulling the rope down. Intuitively, the force of tension on the rope is going to cause the rope to curve. You're pulling it horizontally at the top of the rope, but the rope can only pull along its length so the force is all vertical at the bottom.

Gravity is going to be trying to undo this curve. The curve means the rope is slightly higher on average than it could be without a curve. So, over time, gravity is going to straighten the rope.

What you have here is a balance. If you move quickly, you apply a lot of force with your hand, and its effects dominate. The result is that the rope sort-of follows your hand, being pulled along a smooth curve. You apply this force quickly, so there is not much time for gravity to act. On the other hand, if you move slowly, with less hand force over a longer period of time, gravity has more time to act. Thus the rope ends up straighter.

The rope is never straight, no matter how slowly you move your hand. There's always some curvature caused by this balance. However, you likely cannot see it with the naked eye. Its effects are dwarfed by all sorts of other effects, like the fact that your hand is never moving perfectly horizontally. It bobs up and down as a natural effect of how our brain controls our muscles. But the curving is there. It's not like there's a sudden point where the rope goes from moving as a straight rod to moving on a curve.

Incidentally, even if the rope was replaced with a straight rod, this effect would occur. However, rods tend to be stiffer, so they deflect a lot less.

• Does this tension originate due to deformation (elongation of the rope)? – Mayank Kashyap May 10 at 3:30
• @MayankKashyap The deformation is certainly related to it. Its one of those circular-logic things where I'm a little wary of saying what causes the other. At the deepest level, you have some atoms in your hand, and you apply a force that starts accelerating them. This acceleration leads to stretching of bonds that, in the end, we bundle up and call "tension" – Cort Ammon May 10 at 4:42
• And by f=mdv/dt, smaller time interval would mean larger force and thus larger deformation and thus larger tension. That's cool, thank you – Mayank Kashyap May 10 at 14:03

If I understood correctly, you have a rope (which is not massless) suspended vertically with the lower end free.

This system is somewhat difficult to treat because the tension on the rope is not constant across its length (and therefore it is more complicated to treat this analytically).

You can somewhat interpret the experiment in your post as moving one end of the rope and observing the vibration propagate across the length of the rope.

If you ignore gravity, the velocity at which the vibration propagates is given by $$$$v = \sqrt{\frac{T}{\mu}},$$$$ where $$\mu=M/L$$ is the linear density and $$T$$ is the tension on the rope (suppose that you have tension because you are stretching the rope). You can find a nice derivation of this formula on the Wikipedia page "string vibration".

You can think intuitively that the force of gravity acts as a damping mechanism for these vibration waves and tends to straighten the rope.

Depending on the speed you are performing the movement you can either be in the regime where gravity dominates and the rope just lays vertically (if you move your hand slowly) or, if you move your hand fast, where the wave propagation dominates. In this latter case, you will notice the delay you are mentioning before the other side of the rope will start moving (this is due to the previous formula for the velocity at which vibration moves across the rope).

• You mean like in harmonic motion with one end free, with gravity as damping force? Cool. – Mayank Kashyap May 10 at 3:28
• @MayankKashyap Gravity tends to make the oscillation smaller because any bending of the rope makes the center of gravity rise, increasing the gravitational potential energy. In this sense you can see it as a damping effect. Probably modeling it is not so easy, though. – Davide Dal Bosco May 10 at 13:04