Question Video: Comparing the Changes in Gravitational Potential Energy of Objects in Different Positions Physics • 9th Grade

Video Transcript

A room contains books, a shelf, and an open hatch that leads to a basement, as shown in the diagram. The books, all of which have identical masses, are placed in various positions in the room. Books 1 and 2 are at the same height, with book 2 over the open hatch. Similarly, book 3 is also over the open hatch. The books are released from the points shown in the diagram. Their gravitational potential energies change because of a difference in height between their starting positions and the positions where they make contact with a supporting surface. The table below orders the books by the size of this change in gravitational potential energy. Which row of the table shows the correct order? (A) Row A, (B) row B, (C) row C, (D) row D.

In the diagram, we have been shown the positions from which five books are released. As they fall toward the ground, they lose gravitational potential energy. And we have to order the books in terms of how much gravitational potential energy each of them loses from least to greatest. Let’s start by recalling that the gravitational potential energy of an object, or GPE for short, is equal to the mass of the object 𝑚 multiplied by the gravitational field strength on Earth 𝑔 multiplied by the height of the object above the ground ℎ. In this question though, we are not asked to find the gravitational potential energy of each book. Instead, we are interested in the change in gravitational potential energy of each book as it falls down.

Let’s see what this change would be for one of the books, say, book 2. Let’s say it starts at a height ℎ sub 𝑖 and ends up at a height ℎ sub 𝑓. In this case, we can see that ℎ sub 𝑖 is the height of the floor level in the room, while ℎ sub 𝑓 is the height of the floor in the basement. The difference in gravitational potential energy for this book after it falls can be found by calculating the final gravitational potential energy, 𝑚𝑔ℎ sub 𝑓, minus the initial gravitational potential energy, 𝑚𝑔ℎ sub 𝑖. We have common factors 𝑚 and 𝑔 in both terms in this expression. So we can factorize it to find that the change in gravitational potential energy is equal to 𝑚 times 𝑔 times ℎ sub 𝑓 minus ℎ sub 𝑖.

We could equally write this difference of heights as Δℎ. Since this is the change in height that book 2 undergoes, let’s call this term Δℎ two. So, the change in gravitational potential energy, ΔGPE, for any of these books will be equal to 𝑚 times 𝑔 times Δℎ for that book. Labeling the height changes as we have done in the diagram, we can write an expression for the change in gravitational potential energy for each book. Notice that each of these expressions uses the same mass 𝑚. That’s because we’ve been told that all of these books have the same mass. The only thing that’s different between these five expressions is the values of Δℎ. So, if we want to compare all the gravitational potential energy changes, all we really need to do is compare the values of Δℎ for each book, that is, how far each book falls.

In other words, the book that loses the most gravitational potential energy is the book that falls the largest distance. And the book that loses the least gravitational potential energy is the one that falls the smallest distance. Looking at our diagram, we can see that book 1 is already on a flat surface so will not fall any distance. That is, Δℎ one is zero. Since all the other books fall some distance, we can confidently say that the smallest gravitational potential energy change is experienced by book 1.

Looking at the distances the other books will fall, we can see that book 4 falls the next smallest distance, then book 2, then 5, and then finally book 3 falls the greatest distance. This order of heights tells us how the gravitational potential energy changes are ordered. That is, book 1’s gravitational potential energy changes the least, then book 4’s, then book 2’s, then book 5’s, then book 3’s. This matches the order shown in row B of the table. Our answer then is option (B). Row B correctly shows the change in gravitational potential energy of each book from least to greatest.