Problem 19 - University Physics with Modern Physics

Question

A spring of negligible mass has force constant \(k=\) 1600 \(\mathrm{N} / \mathrm{m}\) . (a) How far must the spring be compressed for 3.20 \(\mathrm{J}\) of potential energy to be stored in it? (b) You place the spring vertically with one end on the floor. You then drop a \(1.20-\mathrm{kg}\) book onto it from a height of 0.80 \(\mathrm{m}\) above the top of the spring. Find the maximum distance the spring will be compressed.

Step-by-Step Solution

Answer

(a) The spring must be compressed by 0.0633 m. (b) The maximum compression of the spring is 0.1086 m.

Step 1: Understanding Spring Potential Energy Formula

The potential energy stored in a compressed or stretched spring is given by the formula \( PE = \frac{1}{2} k x^2 \), where \( PE \) is the potential energy, \( k \) is the spring constant, and \( x \) is the compression distance. In part (a), we need to find \( x \) when \( PE = 3.20 \) J.

Step 2: Solving for Compression Distance

Rearrange the spring potential energy formula to solve for \( x \): \( x = \sqrt{\frac{2 \cdot PE}{k}} \). Substituting \( PE = 3.20 \) J and \( k = 1600 \) N/m, we get: \( x = \sqrt{\frac{2 \cdot 3.20}{1600}} \approx 0.0633 \) m. The spring must be compressed by approximately 0.0633 meters.

Step 3: Understanding Energy Conversion for Part (b)

When the book is dropped, its gravitational potential energy is converted into spring potential energy at the maximum compression point of the spring. We equate the gravitational potential energy (GPE) with spring potential energy (SPE). The GPE is given by \( mgh \), where \( m \) is mass, \( g \) is gravity (9.81 m/s²), and \( h \) is height.

Step 4: Calculating Initial Gravitational Potential Energy

Compute the gravitational potential energy when the book is dropped from \( h = 0.80 \) m: \( GPE = mgh = 1.20 \times 9.81 \times 0.80 = 9.408 \) Joules.

Step 5: Solving for Maximum Compression with Energy Conservation

Using energy conservation, set \( GPE = SPE \): \( \frac{1}{2} k x^2 = 9.408 \). Solve for \( x \), \( x = \sqrt{\frac{2 \cdot 9.408}{1600}} \approx 0.1086 \) m. The maximum compression of the spring is approximately 0.1086 meters.

Key Concepts

Energy Conservation

Energy conservation is an important concept in physics that implies energy cannot be created or destroyed but can only change forms. When dealing with springs and gravity, this principle becomes useful in predicting how energy moves between objects. For example, when you compress a spring by dropping a book onto it, the gravitational potential energy (GPE) of the book transforms into the spring's potential energy (SPE).
This means the energy the book has due to its height is transferred into the spring, causing it to compress. The key is to remember that the initial energy from the book converts into the energy stored in the spring, showing perfect energy transformation from gravitational to elastic potential energy.

Spring Constant

The spring constant, symbolized as k, is a measure of a spring's stiffness. A higher spring constant indicates a stiffer spring which requires more force to compress it a certain distance.
This constant is fundamental in calculating the potential energy stored in a spring. For instance, the spring in the exercise has a constant of 1600 N/m, suggesting it’s quite stiff. In mathematical terms, this constant is used in the spring potential energy formula:

• Potential Energy (PE) = \( \frac{1}{2} k x^2 \)

where \( x \) is the displacement from equilibrium. Thus, understanding the spring constant helps predict how much energy it can store when compressed or stretched.

Gravitational Potential Energy

Gravitational Potential Energy (GPE) depends on three factors: the mass of the object, the height from which it falls, and the force of gravity. Mathematically, we express it as:

• GPE = \( mgh \)

where \( m \) stands for mass, \( g \) is the gravitational acceleration (approximately 9.81 m/s² on Earth), and \( h \) is the height.
In our exercise, when the 1.20 kg book is dropped from a height of 0.8 meters, it possesses gravitational potential energy. Once the book lands on the spring, this energy is converted into the spring's potential energy, allowing us to calculate the maximum compression of the spring using energy conservation principles. Understanding GPE helps in predicting how much spring compression will occur, given the height and mass of the falling object.