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
𝐹 is applied to the end of the second rope, both objects uniformly accelerate in
the direction of 𝐹 across a smooth surface. Tension 𝑇 one is produced in the
rope that the force is applied to, and tension 𝑇 two is produced in the rope that
connects the objects. Which of the following statements
correctly represents the relationship between 𝑇 one and 𝑇 two?
Before we consider these
statements, let’s take a look at our diagram. We see here two masses, which we’re
told are identical, connected by this rope. Then there’s another rope here,
which has a constant force 𝐹 applied to the end of it, pointing to the right. Under this influence, we’re told
that both objects accelerate uniformly in that direction. And note that there’s no friction
force opposing this acceleration because we’re told that the movement is across a
smooth surface. So we have two ropes, one under
tension 𝑇 one and the other under tension 𝑇 two. And we want to pick out what
relationship correctly describes them.
Here are our options. (A) 𝑇 one equals 𝑇 two, (B) 𝑇
one equals 𝑇 two divided by two, (C) 𝑇 one plus 𝑇 two equals zero, (D) 𝑇 one
equals two times 𝑇 two.
Now, here’s 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 𝐹 being
applied to the system and causing it to accelerate. This can remind us of Newton’s
second law of motion, which tells us that the net force acting on an object of mass
𝑚 is equal to that mass multiplied by the object’s acceleration. Now, in our case, if we think only
of forces acting in a horizontal direction, we can say that there’s one external
force acting on our system. That’s the force 𝐹. 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, 𝐹 is equal to our
system’s mass times its acceleration. Now, we’re told that the two ropes
in our scenario are massless, but we’re not told the masses of these two
objects. We do know, though, that they’re
identical. So, just to give them a name, let’s
say that they each have a mass 𝑚. This means that the total mass of
our system is two times 𝑚. Again, the mass of our ropes is
considered zero. So, if we call the acceleration of
our two objects 𝑎, then we can say that 𝐹 is equal to two times 𝑚 times 𝑎.
But then, looking at our diagram,
we see that this force 𝐹 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 𝑚 times 𝑎 is also equal to 𝑇 one. Because we don’t know 𝑚 or 𝑎, we
can’t go about calculating a numerical value for 𝑇 one. But all we want to do is compare it
to the other tension force 𝑇 two to arrive at an expression for that variable. Instead of considering our two
masses and the two ropes, let’s just consider the second mass and the rope under
tension 𝑇 two. Focusing in here, we can say that
𝑇 two is the only horizontal force acting on this second mass. And therefore, by Newton’s second
law, it’s equal to the mass of this object, which is 𝑚, times its acceleration
𝑎.
And note that this object’s
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
𝑇 two is equal to this unknown quantity 𝑚 times 𝑎 and that the tension force 𝑇
one is equal to two times that same quantity. So, if we replace 𝑚 times 𝑎 here
with 𝑇 two, which is equal to that product, then we find that two times 𝑇 two is
equal to 𝑇 one. And we see that that corresponds to
answer choice (D). The correct relationship between
these two tension forces is that 𝑇 one is equal to two times 𝑇 two.
