Problem 1 - Chemistry: The Central Science

Question

Imagine a book that is falling from a shelf. At a particular moment during its fall, the book has a kinetic energy of \(24 \mathrm{~J}\) and a potential energy with respect to the floor of \(47 \mathrm{~J}\). (a) How do the book's kinetic energy and its potential energy change as it continues to fall? (b) What was the initial potential energy of the book, and what is its total kinetic energy at the instant just before it strikes the floor? (c) If a heavier book fell from the same shelf, would it have the same kinetic energy when it strikes the floor? [Section 5.1]

Step-by-Step Solution

Answer

(a) As the book continues to fall, its potential energy decreases and kinetic energy increases, while the total mechanical energy remains constant. (b) The initial potential energy of the book was 71 J, and its total kinetic energy just before striking the floor is 71 J. (c) No, a heavier book falling from the same shelf would have a greater kinetic energy when it strikes the floor, as it has a larger initial potential energy due to its larger mass.

Step 1: Describe the Energy Changes

As the book falls, it loses potential energy and gains kinetic energy. As the height decreases, the potential energy is converted into kinetic energy. While the book falls, the total energy (potential + kinetic) remains constant. (b) What was the initial potential energy of the book, and what is its total kinetic energy at the instant it strikes the floor?

Step 2: Applying the Principle of Conservation of Mechanical Energy

Let's consider EP_initial as the initial potential energy of the book. Given that at this particular moment, the book has EP1 = 47 Joule of potential energy and K1 = 24 Joule of kinetic energy. At this moment, the total energy (E_total) is the sum of its potential and kinetic energies: E_total = EP1 + K1 = 47 J + 24 J = 71 J This is the initial total mechanical energy, and it will remain constant during the fall.

Step 3: Calculate the Initial Potential Energy

Knowing that the total energy remains constant during the fall, we can now determine the initial potential energy (EP_initial): EP_initial = E_total - K_initial Since the book wasn't moving initially, K_initial is zero. Therefore, the initial potential energy of the book was: EP_initial = E_total - K_initial = 71 J - 0 J = 71 J

Step 4: Calculate its total kinetic energy just before it strikes the floor

Right before striking the floor, the potential energy of the book will be 0 J. At this moment, all the initial potential energy will have been converted into kinetic energy. Thus, the total kinetic energy right before striking the floor is equal to the initial total energy (E_total): K_before_striking_floor = E_total = 71 J (c) If a heavier book fell from the same shelf, would it have the same kinetic energy when it strikes the floor?

Step 5: Analyze the case if a heavier book fell from the same shelf

Since the heavier book has a larger mass, it will have a greater potential energy initially when compared to the lighter book. The potential energy at the same shelf height is given by the formula: EP = m * g * h As the mass (m) increases, so does the potential energy. Since the initial potential energies are different, the final kinetic energies of the two books at the moment they strike the floor will also be different. Since the heavier book has a greater initial potential energy, it will have a greater final kinetic energy.

Key Concepts

Potential Energy

Potential energy is the stored energy an object has due to its position or state.
For the falling book, its initial potential energy is because of its height above the floor. The formula to calculate potential energy is \( PE = m \cdot g \cdot h \), where:

• \( m \) is the mass of the object
• \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \mathrm{m/s^2} \))
• \( h \) is the height above the reference point, which is the floor here.

As the book falls, it descends and loses this potential energy as it gets closer to the floor. All this lost potential energy converts into kinetic energy as part of the energy transformation, keeping the total mechanical energy constant. Understanding potential energy helps us grasp how energy transitions from one form to another during motion.

Kinetic Energy

Kinetic energy is the energy of motion. For the book, its kinetic energy becomes evident as it accelerates downward during its fall.
The formula for kinetic energy is \( KE = \frac{1}{2}mv^2 \), where:

• \( m \) is the mass of the object
• \( v \) is the velocity of the object

As the book falls, its velocity increases since it is accelerating due to gravity, hence its kinetic energy increases too.
Initially, when the book starts to fall, its kinetic energy is zero, but as it approaches the floor, its kinetic energy becomes maximum, transforming all the initial potential energy into kinetic as evidenced when kinetic energy peaks.

Conservation of Energy

The conservation of energy principle says that energy cannot be created or destroyed; it can only transform from one form to another.
In the context of the falling book, the total mechanical energy, which is the sum of potential and kinetic energies, stays constant throughout the fall.

• Initially, all energy is potential as the book is at rest on the shelf.
• As it falls, this potential energy gets converted into kinetic energy.
• At the precise moment before it hits the floor, the energy has entirely shifted to kinetic.

This principle helps us predict how energy transformations occur in different moving and resting states of objects.
It is a fundamental rule of physics that applies to all energy transformations, ensuring total energy in a closed system remains unchanged.