A child stands at rest on a plank
of wood that is also at rest. The plank is on a flat surface of
thick ice on a frozen river. The friction between the wood and
the ice is negligible. The child starts to walk along the
plank towards the river bank shown in the diagram. Which of the following most
correctly describes the displacement of the child and of the plank as the child
walks? Consider positive displacement as
toward the river bank.
Okay, before we get to these
descriptions of the child’s and the plank’s displacement, let’s consider this
diagram. In it, we see a child walking along
this wooden plank, which is initially at rest on the frozen ice of a river. We’re told that the friction
between the wood plank and the ice is negligibly small. So we can assume there is no
friction between these two surfaces. The child walks along the plank
toward this bank. And we’re told that displacement in
this direction is positive.
Okay, now, let’s consider a set of
descriptions of the displacement of the child and of the plank as the child
walks. Description (a) displacement of the
child: zero, displacement of the plank: negative. (b) Displacement of the child:
negative, displacement of the plank: negative. (c) Displacement of the child:
positive, displacement of the plank: positive. (d) Displacement of the child:
positive, displacement of the plank: zero. And lastly (e) displacement of the
child: positive, displacement of the plank: negative.
So all of these are descriptions of
how the child and the plank may be displaced as the child walks along the plank. We want to figure out which one is
most correct. Let’s start by thinking about the
motion of the child and the plank too.
From our diagram, we can see that
the child is walking forward along this plank. If we were to take a close-up view
of the child’s foot as it presses on the plank, then as far as forces go, there
would be a downward force that the child’s foot exerts on the plank. This force is due to the child’s
weight. And there would also be a force
acting to the left because the child is pushing on the plank to move forward, which
is to the right.
If we focus on this force that’s
acting to the left exerted by the child’s foot on the plank, we know that so long as
there is friction between the foot and the plank. The plank will respond with its own
frictional force pushing forward on the child’s foot. If this force was absent, if there
was no friction between the plank and the foot, then the child’s foot would just
slide backward and the child wouldn’t move forward at all.
But as far as we know from our
problem statement, there is a frictional force between these two surfaces. And so, the plank resists this
backward-acting force from the child’s foot pushing the foot forward. And therefore, the child gets
traction on the plank and is able to move forward relative to it.
Now that statement “relative to it”
is important. Because recall that we’re told that
there is negligible friction between the plank and the ice underneath it. So even though there is friction
between the child’s foot and the plank, there is none between the plank and the
ice. And this means that as the child
starts to move ahead, the plank will start to move relative to the ice. It will start to slide backward to
the left on it. This is due to the lack of friction
between these surfaces.
Now if that was the whole story, we
might expect something like this to happen. The child walks ahead on the
plank. But because there’s no friction
between the plank and the ice, the plank just slides backward to the left. And the child’s displacement ends
up being zero. That is, the child effectively
walks in place while the plank underneath them moves to the left. If this was what took place, we
would choose option (a) as our answer. This says that the displacement of
the child is zero — they’re effectively walking in place — while the plank’s
displacement is negative to the left.
But in our analysis so far, we’ve
left out something important. The plank of wood itself that the
child is walking on has some amount of mass. And we know that the plank, like
the child, starts out at rest. The plank, therefore, has
inertia. And it resists being put into
motion. With a plank as small as the one we
have here, this is a bit harder to see.
So let’s imagine that, instead of a
small wooden plank, the child is walking on a gigantic wooden beam that weighs
several hundred kilograms. In this case, it’s easier to see
that even if there’s still no friction between this wooden beam and the ice beneath
it. Thanks simply to the tendency of
the beam to remain as it is, stationary, the child will be able to walk to the right
on the beam and actually move forward, actually have a nonzero displacement. The beam would gradually start to
move to the left, but not so fast as the child moves to the right.
Now if we go back from our gigantic
beam to our original small wooden plank, the same thing will take place but just in
a less dramatic way. As the child walks to the right on
the plank, the plank will move to the left since there’s no friction between it and
the ice below it. But thanks to the inertia of the
plank, its tendency to stay at rest, its displacement to the left along the ice will
be less than the child’s displacement to the right along the plank. And therefore, relative to a fixed
point, say a point on the bank of this river, the child really will move to the
right with a nonzero displacement.
Our convention in this exercise is
that displacement to the right, towards the river bank, is considered positive and
displacement to the left is considered negative. So, we can say that the
displacement of the plank is negative, while the displacement of the child is
positive. And this description matches answer
option (e). Thanks to the fact that there’s no
friction between the plank and the ice and that the plank has inertia, the child’s
displacement in walking along the plank is positive, while the plank’s displacement