Problem 27 - University Physics with Modern Physics

Question

What is the gravitational potential energy of a \(2.0-\mathrm{kg}\) book \(1.5 \mathrm{~m}\) above the floor?

Step-by-Step Solution

Answer

Answer: The gravitational potential energy of the book is 29.43 joules.

Step 1: Identify the given values

In this problem, we are given the mass (m) of the book as 2.0 kg, the height (h) above the floor as 1.5 m, and the acceleration due to gravity (g) as approximately 9.81 m/s².

Step 2: Write down the formula for gravitational potential energy

The formula to calculate gravitational potential energy is GPE = m * g * h

Step 3: Plug in the given values into the formula

Now, we can substitute the values of m, g, and h into the formula: GPE = (2.0 kg) * (9.81 m/s²) * (1.5 m)

Step 4: Multiply the values

Multiply the values together: GPE = 2.0 kg * 9.81 m/s² * 1.5 m = 29.43 kg m²/s²

Step 5: Convert the gravitational potential energy to joules

Since 1 kg m²/s² is equivalent to 1 joule (J), the gravitational potential energy of the book is equal to 29.43 J. So, the gravitational potential energy of the 2.0 kg book 1.5 meters above the floor is 29.43 joules.

Key Concepts

Mass and its role in gravitational potential energy

Mass is a measure of the amount of matter in an object. In physics, mass is a crucial factor when dealing with forces and energy. It is usually measured in kilograms (kg) in the International System of Units (SI units). When it comes to calculating gravitational potential energy, mass represents how much material is involved.

You can think of mass as the "weight" of the object - although, scientifically, weight is the force due to gravity pulling the mass downwards. In our example of a book, the mass of the book, which is given as 2.0 kg, directly influences the potential energy calculation.

• Heavier objects have more mass, leading to increased potential energy for the same height.
• The formula for gravitational potential energy involves multiplying the mass with gravity and height: \( GPE = m \cdot g \cdot h \).

Understanding mass helps us estimate potential energy accurately by knowing how "heavy" an object is.

Understanding acceleration due to gravity

Acceleration due to gravity is the pull that the Earth exerts on objects. This force gives us the sensation of weight. On Earth, this acceleration is approximately constant at 9.81 meters per second squared (m/s²). This value might vary slightly depending on geographical location, but it is fairly consistent for most calculations.

Gravity is a key component in determining potential energy because it affects how strongly an object is pulled downwards. In our scenario with the book, we used the standard gravitational acceleration value of 9.81 m/s².

• It ensures that the potential energy calculation reflects true physical conditions.
• It is a multiplier that amplifies the effect of mass and height in the energy formula: \( GPE = m \cdot g \cdot h \).

Familiarity with this value is essential for anyone performing physics calculations related to energy and motion.

Breaking down the energy formula

The energy formula used here is specific to gravitational potential energy (GPE). It is represented as \( GPE = m \cdot g \cdot h \), where:\

• \( m \) is the mass of the object.
• \( g \) is the acceleration due to gravity.
• \( h \) is the height above the reference point, usually the ground or floor.

This formula tells us that gravitational potential energy is directly proportional to each of these three variables. Higher mass, gravity, or height will increase potential energy.

In our problem, the book’s potential energy was computed as \(29.43\) Joules by inserting \(2.0\, \text{kg}\) for mass, \(9.81\, \text{m/s}^2\) for gravity, and \(1.5\, \text{m}\) for height into the formula.

Understanding this formula is crucial as it forms a foundational concept for analyzing energy in physics, showing how various factors combine to create potential energy.

SI units in physics

SI units, or the International System of Units, are the standard units of measurement used in science and engineering. They ensure that calculations and interpretations of measurements are consistent worldwide. The major units in our scenario include:

• Mass in kilograms (kg).
• Distance and height in meters (m).
• Time in seconds (s), used in composite units such as m/s² for acceleration.
• Energy in joules (J), where \( 1\, \text{J} = 1\, \text{kg} \cdot \text{m}^2/\text{s}^2 \).

Using SI units provides clarity and uniformity, especially when dealing with international collaborations or studies.

For our gravitational potential energy problem, utilizing SI units allows us to directly compare and convert measured or calculated values reliably.