A student has six textbooks, each with a thickness of 4.8 cm and a weight of 26 N. What is the minimum work the student would have to do to place all the books in a single vertical stack, starting with all the books on the surface of the table?
To find the minimum work required to stack the textbooks, we need to calculate the change in potential energy as each book is lifted from the table to the top of the stack. The work done is equal to the change in potential energy.
Let's break down the problem step by step:
Step 1: Determine the total thickness of the textbooks.
Given that each textbook has a thickness of 4.8 cm, and the student has six textbooks, we can calculate the total thickness by multiplying the thickness of one textbook by the number of textbooks:
Total thickness = 4.8 cm * 6 = 28.8 cm
Step 2: Convert the total thickness to meters.
Since work is typically calculated using the metric system, we need to convert the total thickness from centimeters to meters. There are 100 centimeters in a meter, so the total thickness in meters is:
Total thickness = 28.8 cm / 100 = 0.288 meters
Step 3: Calculate the change in potential energy for each book.
The change in potential energy is equal to the weight of the book multiplied by the height to which it is lifted. In this case, the height will increase with each book added to the stack.
The height for each book is equal to the cumulative thickness of the textbooks below it. Therefore, the height for each book can be calculated as follows:
Height of book i = total thickness - thickness of books below i
For example, the height for the first book (i = 1) will be equal to the total thickness since there are no books below it.
Step 4: Calculate the work done for each book.
The work done for each book is given by the formula:
Work = force * distance
In this case, the force is equal to the weight of the book (26 N), and the distance is equal to the height to which the book is lifted.
Step 5: Calculate the total work done.
To find the minimum work required to stack all the textbooks, we need to sum up the work done for each book.
Let's calculate the work done for each book:
For the first book:
Height = total thickness = 0.288 meters
Work1 = force * distance = 26 N * 0.288 m
For the second book:
Height = total thickness - thickness of books below = 0.288 - 4.8 cm = (0.288 - 0.048) meters
Work2 = force * distance = 26 N * (0.288 - 0.048) m
Similarly, we can calculate the work done for the remaining books.
Step 6: Sum up the work done for each book to find the total work done.
Total work done = Work1 + Work2 + Work3 + Work4 + Work5 + Work6
By following these steps, you can calculate the minimum work required to stack all the textbooks.