Problem 5 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A \(1.60-\mathrm{m}\) tall person lifts a 1.95 -kg book off the ground so it is 2.20 \(\mathrm{m}\) above the ground. What is the potential energy of the book relative to \((a)\) the ground, and \((b)\) the top of the person's head? (c) How is the work done by the person related to the answers in parts \((a)\) and \((b) ?\)

Step-by-Step Solution

Answer

Potential energy relative to the ground is 42.007 J; relative to the top of the person's head is 11.469 J.

Step 1: Define the Potential Energy Formula

The potential energy (PE) of an object at height h is given by the formula: \[ PE = mgh \]where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity (approximately \( 9.81\, \text{m/s}^2 \) on Earth's surface), and \( h \) is the height above the reference point.

Step 2: Calculate Potential Energy Relative to the Ground

We are given: - Mass \( m = 1.95 \, \text{kg} \)- Height relative to the ground \( h = 2.20 \, \text{m} \)- Plug these into the potential energy formula: \[ PE_{ ext{ground}} = 1.95 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 2.20 \, \text{m} \]Calculate the result.

Step 3: Calculation for Height Relative to Top of the Head

The height relative to the top of the person's head is the difference between the book's height and the person's height:\[ h' = 2.20 \, \text{m} - 1.60 \, \text{m} = 0.60 \, \text{m} \]Plug this into the potential energy formula:\[ PE_{ ext{top of head}} = 1.95 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 0.60 \, \text{m} \]Calculate the result.

Step 4: Relate Work Done to Potential Energy

The work done by the person in lifting the book is equal to the change in the book's potential energy with respect to different reference points. Therefore, the work done \( W \) is equivalent to the potential energy calculated in parts (a) and (b), since work is the energy transferred to the book as it is lifted.

Key Concepts

Work Done by Force

When calculating the work done by a force, we are essentially determining how much energy is transferred by the force to move an object over a distance. In the case of lifting a book, the force at play is the force exerted by a person to overcome gravity. Gravity constantly pulls the book downward, and the person applies an upward force to lift it.

The work done can be measured using the formula:

• \( W = Fd \cos(\theta) \)

Here, \( F \) is the force applied, \( d \) is the distance through which the force acts, and \( \theta \) is the angle between the force and the direction of motion. When lifting directly upward, this angle is 0, making \( \cos(0) = 1 \). Hence, the work done simplifies to \( W = Fd \).

For this specific example, the force applied is equal to the weight of the book \( (mg) \), and the distance \( d \) is the height the book is lifted. Therefore, the work done by the person equals the gravitational force times the height lifted, connecting directly to the change in potential energy.

Energy Transfer

Energy transfer occurs when energy is shifted from one system or object to another. In this context, lifting the book involves transferring energy from the person to the book. The person uses muscular energy to perform work on the book, transferring energy into the book in the form of gravitational potential energy.

This transfer is vital to understanding many mechanical processes and systems. The key point to remember is that during the lifting process, all the energy transferred to the book is stored as potential energy. If the book were to fall, this stored energy would convert back into kinetic energy as it approaches the ground, illustrating the energy conservation principle.

• Energy is not lost; it merely changes forms, such as from muscular energy to potential energy.
• This process upholds the law of conservation of energy, one of the fundamental principles in physics.

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object due to its position above a reference point, often the ground. This energy is given by the formula \( PE = mgh \), where:

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

For example, calculating the potential energy of a book relative to the ground involves using these variables to find out how much energy is stored because the book is 2.20 meters above the ground.

If calculating for a different reference point, such as the top of the person's head, adjust the height accordingly, using the difference between the holding level and the top of the head. This situation shows that potential energy is relative to the chosen reference point. The higher the reference point, the less potential energy an object has relative to that point.

Changing the height or the reference point demonstrates how this energy varies with position, emphasizing the potential in gravitational interactions.