Problem 21 - University Physics with Modern Physics

Question

A spring of negligible mass has force constant \(k =\) 1600 N/m. (a) How far must the spring be compressed for 3.20 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-kg book onto it from a height of 0.800 m above the top of the spring. Find the maximum distance the spring will be compressed.

Step-by-Step Solution

Answer

(a) 0.0632 m; (b) 0.1084 m.

Step 1: Understanding Spring Potential Energy

The potential energy stored in a compressed spring is given by the formula \( PE = \frac{1}{2}kx^2 \). We need to find the compression distance \( x \) for which the spring stores \( PE = 3.20 \) J.

Step 2: Solving for Compression (a)

We have the equation \( 3.20 = \frac{1}{2} \times 1600 \times x^2 \). Solving for \( x \), we first multiply both sides by 2 to get \( 6.4 = 1600x^2 \). Then, divide both sides by 1600: \( x^2 = \frac{6.4}{1600} = 0.004 \). Take the square root: \( x = \sqrt{0.004} = 0.0632 \) meters.

Step 3: Calculating Initial Energy of Dropped Book (b)

The gravitational potential energy (GPE) of the book before it's dropped can be calculated using \( GPE = mgh \), where \( m = 1.20 \) kg, \( g = 9.8 \text{ m/s}^2 \), and \( h = 0.8 \) m. Therefore, \( GPE = 1.20 \times 9.8 \times 0.8 = 9.408 \text{ J} \).

Step 4: Equating Energies for Maximum Compression

When the book compresses the spring to its maximum extent, all its initial GPE is converted into the spring's potential energy. Therefore, set \( PE = 9.408 \) J equal to the spring potential energy formula: \( \frac{1}{2} \times 1600 \times x^2 = 9.408 \).

Step 5: Solving for Maximum Compression (b)

Multiply both sides by 2: \( 1600x^2 = 18.816 \). Divide by 1600: \( x^2 = \frac{18.816}{1600} = 0.01176 \). Take the square root: \( x = \sqrt{0.01176} = 0.1084 \) meters.

Key Concepts

Hooke's Law

Hooke's Law describes the relationship between the force exerted by a spring and its displacement from its equilibrium position. This law is fundamental in understanding how springs behave. The equation for Hooke's Law is given by

• \( F = -kx \)

where:

• \( F \) is the force exerted by the spring (in newtons),
• \( k \) is the spring constant (in newtons per meter), and
• \( x \) is the displacement from the equilibrium position (in meters).

The negative sign in the formula indicates that the force exerted by the spring is always opposite to the direction of displacement. If you compress the spring, it pushes back; if you stretch it, it pulls back. Understanding Hooke's Law helps us determine how a spring will respond under various forces, which is crucial for calculating potential energy stored in the spring.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy that an object possesses because of its position relative to the Earth. It depends on three factors:

• Mass of the object \((m)\)
• Gravitational acceleration \((g)\), which is approximately \(9.8 \text{ m/s}^2\) on Earth's surface
• Height \((h)\) of the object above the reference point

The formula for calculating gravitational potential energy is:

• \( GPE = mgh \)

For example, if you have a 1.20 kg book 0.800 m above the spring, its gravitational potential energy is \(1.20 \times 9.8 \times 0.8 = 9.408 \text{ J}\). This energy will be transformed into spring potential energy when the book is dropped onto the spring. Understanding GPE is important in physics because it allows us to calculate how energy is stored due to an object's position and how it might be transformed into other forms of energy.

Energy Conservation in Physics

Energy conservation is a fundamental principle in physics stating that energy cannot be created or destroyed, only transformed from one form to another. In this exercise, we see this principle applied when a book falls onto a spring, converting gravitational potential energy (GPE) into spring potential energy. Initially, the book has a certain amount of GPE due to its height above the spring. When it is released and falls:

• This GPE decreases as the book loses height.
• Simultaneously, this lost GPE is converted into kinetic energy.
• As the book hits the spring, the kinetic energy is transformed into potential energy stored in the spring.

At the maximum compression of the spring, all the gravitational potential energy initially possessed by the book is stored in the spring. This process exemplifies energy conservation: the total energy before the book is dropped equals the total energy after it compresses the spring. Understanding this concept is critical, as it allows us to predict and calculate the outcome of various physical processes accurately.