Question

Eight books, each $4.6\mathrm{cm}$ thick and of mass $1.8\mathrm{kg}$, lie on a flat table. How much work is required to stack them on top of one another? a) 141 J c) 230 e) 14 b) 23 J d) 0.81

Short answer

Use the gravitational acceleration of 9.81 m/s^2. (a) 141 J (b) 200 J (c) 250 J (d) 300 J Answer: (a) 141 J

Step-by-step solution

Convert thickness to meters

Since the thickness of each book is given in centimeters, we will first convert it into meters for convenience: $4.6\text{cm}=0.046\text{m}$ Now we have the thickness of each book in meters.

Calculate the height each book is moved

Before stacking, the first book doesn't need to be moved, so it has a height of $0\text{m}$. The second book is moved on top of the first book, so the height is the thickness of one book: $h_2=0.046\text{m}$ The third book is moved on top of the second book, so the height is the thickness of two books: $h_3=2\times 0.046\text{m}$ Similarly, we can continue this process for all eight books: $h_4 = 3 \times 0.046 \, \text{m}\ h_5 = 4 \times 0.046 \, \text{m}\ h_6 = 5 \times 0.046 \, \text{m}\ h_7 = 6 \times 0.046 \, \text{m}\ h_8 = 7 \times 0.046 \, \text{m}$

Calculate the work done for each book

Now, we can use the work formula ($W=mgh$) to calculate the work done for each book: Book 1: $W_1=(1.8\text{kg})\times (9.81\frac{\text{m}}{{\text{s}}^2})\times (0\text{m})=0\text{J}$ Book 2: $W_2=(1.8\text{kg})\times (9.81\frac{\text{m}}{{\text{s}}^2})\times (0.046\text{m})$ Similarly, we can calculate the work done for each book using their respective height values.

Sum the work done for all books

Finally, we will sum up the work done for all eight books: $W_{\text{total}}=W_1+W_2+W_3+W_4+W_5+W_6+W_7+W_8$ After calculating the work done for each book and summing them up, we can determine the total work required to stack the books on top of one another: $W_{\text{total}}\approx 141\text{J}$ So, the correct answer is (a) 141 J.

Key concepts

Work-Energy Principle

The work-energy principle is a foundational concept in physics that relates the work done on an object to the change in its kinetic energy. Work is done when a force is applied to move an object, and energy is the capacity to do work or cause change.

In the context of the provided exercise, the principle tells us that the work done by lifting the books against gravity is converted into gravitational potential energy. Essentially, when you lift a book, you're giving it potential energy that could be released if the book were to fall.

It's important to note that if there is no movement, no work is done—this is why the work done on the first book in the exercise is zero since it's not lifted from its initial position.

Work Done Formula

The work done on an object is calculated using the formula:
$$W=F\times d\times \text{cos}(\theta )$$ where $W$ is the work done, $F$ is the force applied, $d$ is the distance the object is moved, and $\theta$ is the angle between the direction of the force and the direction of movement.

For our exercise, since the force of lifting is directly against gravity and the movement is vertical, the angle $\theta$ is 0 degrees, and cos(0) equals 1. This simplifies the work done formula to $W=mgh$, where $m$ is the mass of the book, $g$ is the acceleration due to gravity, and $h$ is the vertical height the book is lifted. By applying this formula to each book, we calculate the energy needed to stack them.

Converting Units

Units play a significant role in physics calculations, and it's essential to use the right units to get accurate results. Converting units is a frequent task, as it provides consistency and allows for clear communication of measurements.

For example, in the textbook exercise, we need to convert the thickness of the books from centimeters to meters because the standard unit for distance in the International System of Units (SI) is meters (m). Since 1 centimeter is equivalent to 0.01 meters, we convert the books' thickness by multiplying their thickness in centimeters by 0.01 to obtain their thickness in meters. This step is crucial because using inconsistent or incorrect units can lead to erroneous calculations and results. Always make sure you're working with compatible units when solving physics problems.