A spring is hanging from the ceiling. Attaching a 500 g physics book to the spring causes it to stretch 20 cm in order to come to equilibrium. What is the book's maximum speed?

Verified step by step guidance

1. Determine the spring constant (k) using Hooke's Law: \( F = kx \). Here, \( F \) is the force exerted by the book due to gravity, \( F = mg \), where \( m = 0.5 \; \text{kg} \) and \( g = 9.8 \; \text{m/s}^2 \). The displacement \( x \) is given as \( 0.2 \; \text{m} \). Solve for \( k \) using \( k = \frac{F}{x} \).
2. Recognize that the maximum speed of the book occurs when all the potential energy stored in the spring is converted into kinetic energy. The potential energy stored in the spring is given by \( U = \frac{1}{2}kx^2 \).
3. Set the spring's potential energy equal to the book's kinetic energy at maximum speed: \( \frac{1}{2}kx^2 = \frac{1}{2}mv^2 \), where \( v \) is the maximum speed. Cancel out the \( \frac{1}{2} \) terms on both sides.
4. Rearrange the equation to solve for \( v \): \( v = \sqrt{\frac{kx^2}{m}} \). Substitute the values of \( k \), \( x \), and \( m \) into this equation.
5. Simplify the expression to find the maximum speed \( v \). Ensure all units are consistent (e.g., meters, kilograms, seconds) before performing the calculation.

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to the amount it is stretched or compressed, represented mathematically as F = kx, where F is the force, k is the spring constant, and x is the displacement from the equilibrium position. This principle is essential for understanding how the spring behaves when a mass is attached.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In the context of the spring and the attached book, the potential energy stored in the spring when stretched is converted into kinetic energy as the book moves, allowing us to calculate the maximum speed of the book.

Maximum Speed in Simple Harmonic Motion

In simple harmonic motion, the maximum speed of an object occurs as it passes through the equilibrium position. This speed can be calculated using the formula v_max = ωA, where ω is the angular frequency and A is the amplitude of the motion. Understanding this concept is crucial for determining the book's maximum speed as it oscillates around the equilibrium point.