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. From equilibrium, the book is pulled down 10 cm and released. What is the period of oscillation?

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's weight, \( F = mg \), where \( m = 0.5 \; \text{kg} \) and \( g = 9.8 \; \text{m/s}^2 \). The displacement \( x \) is 20 cm (converted to meters: \( x = 0.2 \; \text{m} \)). Rearrange Hooke's Law to solve for \( k \): \( k = \frac{F}{x} \).
2. Recall the formula for the period of oscillation of a mass-spring system: \( T = 2\pi \sqrt{\frac{m}{k}} \). Here, \( m \) is the mass of the book (0.5 kg), and \( k \) is the spring constant calculated in the previous step.
3. Substitute the value of \( m \) and the calculated \( k \) into the formula \( T = 2\pi \sqrt{\frac{m}{k}} \). Ensure that all units are consistent (mass in kilograms, spring constant in \( \text{N/m} \)).
4. Simplify the expression under the square root and calculate the value of \( T \) symbolically. This will give the period of oscillation in seconds.
5. Note that the amplitude of the oscillation (10 cm) does not affect the period, as the period of a mass-spring system depends only on the mass and the spring constant, not the amplitude.

Key Concepts

Hooke's Law

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

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In the case of the spring and the attached book, when the book is pulled down and released, it will oscillate back and forth about the equilibrium position, following a sinusoidal pattern characterized by a constant period and amplitude.

Period of Oscillation

The period of oscillation is the time it takes for one complete cycle of motion in a harmonic system. For a mass-spring system, the period T can be calculated using the formula T = 2π√(m/k), where m is the mass attached to the spring and k is the spring constant. This relationship allows us to determine how long it takes for the book to complete one full oscillation after being released.