Video: SAT Physics Exam 1 Q68

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.

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Video Transcript

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 is negative.

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