Unit 2.8: Spring Forces

Introduction

Springs and other elastic materials obey **Hooke’s Law**, which describes the force exerted by a spring when it is stretched or compressed. The force is proportional to the displacement from equilibrium and acts in the direction opposite to the deformation.

Key Concepts

Hooke’s Law: The restoring force of a spring is given by:
\( F_s = -k x \) where **\( k \)** is the spring constant (N/m) and **\( x \)** is the displacement from equilibrium.
Elastic Potential Energy: The energy stored in a compressed or stretched spring:
\( U_s = \frac{1}{2} k x^2 \).
Spring Constant \( k \): A measure of the stiffness of the spring. Higher values indicate stiffer springs.
Equilibrium Position: The natural position of a spring when no external forces act on it.

Tip: The negative sign in Hooke’s Law indicates that the force exerted by the spring is always opposite to the direction of displacement.

Graphical Representations

Spring force and energy can be visualized with graphs:

Force vs. Displacement Graph: A straight-line graph with slope **\( k \)**, showing that force increases linearly with displacement.
Energy vs. Displacement Graph: A parabolic curve representing the quadratic relationship of elastic potential energy.

Exam Strategy: The **area under a force vs. displacement graph** represents work done on or by the spring.

Mathematical Routines

Solving spring problems involves:

Using Hooke’s Law to calculate the force exerted by a spring.
Applying **Newton’s Second Law** to analyze forces in spring-mass systems.
Using conservation of energy to solve motion problems involving springs.

Important: If a spring is in equilibrium with other forces acting on it, use **\( F_s = kx \)** to determine equilibrium position.

Practice Activities

Activity 1: Finding Spring Force

A spring with a constant of **200 N/m** is stretched **0.1 m** from equilibrium. What force does it exert?

Activity 2: Elastic Potential Energy

A **50 N/m** spring is compressed **0.2 m**. How much potential energy is stored in the spring?

Activity 3: Equilibrium Condition

A **3 kg** block hangs from a vertical spring with a spring constant of **100 N/m**. Find the equilibrium displacement of the block.

Summary & Exam Preparation Tips

Understanding spring forces is critical for AP Physics. Key takeaways:

Hooke’s Law governs how springs exert force: \( F_s = -k x \).
Elastic potential energy follows the equation \( U_s = \frac{1}{2} k x^2 \).
The force exerted by a spring always acts opposite to displacement.
Springs obey conservation of energy, transitioning between kinetic and potential energy.

Practicing problems with spring forces and energy will reinforce these concepts for the AP Physics exam.