Direction of motion of points on a rope as a wave travels

[vcsharp2003]

Homework Statement

A rope RS has a pulse at $t=0$ traveling towards the right at $1 m/s$ as shown in the first diagram below.
What will be the direction of motion of points A, B and C at $t=0$?

Relevant Equations

None

The second diagram is my attempt at the solution, in which the dotted part is the pulse in the rope a very small interval of time after $t=0$.

Point A should be at rest since we know wave is moving towards right and point A on the rope becomes a part of initial horizontal part of the string a moment after $t=0$.

I am still trying to figure out the reasoning behind the direction of motion of points B and C.

 

[haruspex]

Your diagram is good, making it quite easy to see where B and C have moved to on the dotted line. Try marking them as B', C'.

 

[vcsharp2003]

Your diagram is good, making it quite easy to see where B and C have moved to on the dotted line. Try marking them as B', C'.

I am assuming that the length of rope from point $R$ upto points $A, B, C$ will be a constant for each point in question i.e. lengths $RA, RB, RC$ are going to be constant. We can also say that $RA=RA'$,
$RB=RB'$ and $RC=RC'$.

With above assumption, I came up with following diagram for new positions $A', B', C'$. It seems as the interval of time for dotted wave approaches $0$, $B'$ approaches a position vertically under $B$; similar reasoning will apply to $C'$.

So, points $A, B$ will be moving vertically down at $t=0$.

 

[vcsharp2003]

Your diagram is good, making it quite easy to see where B and C have moved to on the dotted line. Try marking them as B', C'.

If my reasoning in post#3 is correct, then is there
an alternate easier/shorter reasoning?

 

[haruspex]

I am assuming that the length of rope from point $R$ upto points $A, B, C$ will be a constant for each point in question i.e. lengths $RA, RB, RC$ are going to be constant. We can also say that $RA=RA'$,
$RB=RB'$ and $RC=RC'$.

With above assumption, I came up with following diagram for new positions $A', B', C'$. It seems as the interval of time for dotted wave approaches $0$, $B'$ approaches a position vertically under $B$; similar reasoning will apply to $C'$.

So, points $A, B$ will be moving vertically down at $t=0$.

View attachment 297221

Again, your diagram is good, though in the usual idealisation of a transverse wave the elements do not undergo any longitudinal movement. That also leads to the answer you arrived at.

 

[vcsharp2003]

In the usual idealisation of a transverse wave the elements do not undergo any longitudinal movement.

Can we conclude based on this fact the directions (either up or down) of the points? For a shorter reasoning, we can say that a rope always has a transverse wave and therefore points on it will move perpendicular to the direction of travel of the wave. But, how do we now say that the correct direction is either up or down?

 

[haruspex]

For a shorter reasoning, we can say that a rope always has a transverse wave and therefore points on it will move perpendicular to the direction of travel of the wave. But, how do we now say that the correct direction is either up or down?

That won't do it because the direction of the wave matters.

 

[vcsharp2003]

So, points A,B will be moving vertically down at t=0.

Sorry, the above statement should be as below.

So, points $B, C$ will be moving vertically down at $t=0$.

Note: I could not edit my post#3 and therefore, I am posting a correction here for others.