### Video Transcript

Two identical objects are connected
to each other by a rope as shown in the diagram. A second rope is connected to one
of the objects. The masses of the ropes are
negligible. A short time after a constant force
๐น is applied to the end of the second rope, both objects uniformly accelerate in
the direction of ๐น across a smooth surface. Tension ๐ one is produced in the
rope that the force is applied to, and tension ๐ two is produced in the rope that
connects the objects. Which of the following statements
correctly represents the relationship between ๐ one and ๐ two?

Before we consider these
statements, letโs take a look at our diagram. We see here two masses, which weโre
told are identical, connected by this rope. Then thereโs another rope here,
which has a constant force ๐น applied to the end of it, pointing to the right. Under this influence, weโre told
that both objects accelerate uniformly in that direction. And note that thereโs no friction
force opposing this acceleration because weโre told that the movement is across a
smooth surface. So we have two ropes, one under
tension ๐ one and the other under tension ๐ two. And we want to pick out what
relationship correctly describes them.

Here are our options. (A) ๐ one equals ๐ two, (B) ๐
one equals ๐ two divided by two, (C) ๐ one plus ๐ two equals zero, (D) ๐ one
equals two times ๐ two.

Now, hereโs one way we can think
about this scenario. We essentially have a system where
that system consists of these two identical masses in the two ropes. We have an external force ๐น being
applied to the system and causing it to accelerate. This can remind us of Newtonโs
second law of motion, which tells us that the net force acting on an object of mass
๐ is equal to that mass multiplied by the objectโs acceleration. Now, in our case, if we think only
of forces acting in a horizontal direction, we can say that thereโs one external
force acting on our system. Thatโs the force ๐น. That force is transmitted through
the first rope and then pulls on the first mass, then transmitted through the second
and pulls on the second mass.

Effectively then, this force is
pulling our whole system, both masses and both ropes. Therefore, ๐น is equal to our
systemโs mass times its acceleration. Now, weโre told that the two ropes
in our scenario are massless, but weโre not told the masses of these two
objects. We do know, though, that theyโre
identical. So, just to give them a name, letโs
say that they each have a mass ๐. This means that the total mass of
our system is two times ๐. Again, the mass of our ropes is
considered zero. So, if we call the acceleration of
our two objects ๐, then we can say that ๐น is equal to two times ๐ times ๐.

But then, looking at our diagram,
we see that this force ๐น is being applied to the end of our first rope. And therefore, the tension in this
rope is equal to that applied force. This means we can write that two
times ๐ times ๐ is also equal to ๐ one. Because we donโt know ๐ or ๐, we
canโt go about calculating a numerical value for ๐ one. But all we want to do is compare it
to the other tension force ๐ two to arrive at an expression for that variable. Instead of considering our two
masses and the two ropes, letโs just consider the second mass and the rope under
tension ๐ two. Focusing in here, we can say that
๐ two is the only horizontal force acting on this second mass. And therefore, by Newtonโs second
law, itโs equal to the mass of this object, which is ๐, times its acceleration
๐.

And note that this objectโs
acceleration is equal to the acceleration of the system overall. This is because both of our masses
move together and accelerate equally. We see then that the tension force
๐ two is equal to this unknown quantity ๐ times ๐ and that the tension force ๐
one is equal to two times that same quantity. So, if we replace ๐ times ๐ here
with ๐ two, which is equal to that product, then we find that two times ๐ two is
equal to ๐ one. And we see that that corresponds to
answer choice (D). The correct relationship between
these two tension forces is that ๐ one is equal to two times ๐ two.