Hooke's Law and Spring Constants

Hooke's Law

• A stretched spring stores energy, called elastic potential energy.
• A spring's ability to do work when returning to its original position indicates stored energy.
• Springs have varying stiffness, indicated by the spring constant (k).

Hooke's Law Lab

• A force-vs-stretch graph results in a straight line.
• The slope of the graph equals the spring constant (k).
• The spring constant (k) conversion from the example given (from data provided) is 10 N/m.
• The spring constant (k) conversion from the example given (from the data provided is also 4 N/0.4m which is 10 N/m)

Hooke's Law - Restoring Force

• A stretched spring exerts a restoring force to return to its equilibrium position (Newton's Third Law).
• Hooke's Law: F = -kx
  • F = restoring force
  • k = spring constant (N/m)
  • x = displacement (m)
• Excessive force can permanently deform or break a spring.

Example Calculations - Spring Constant

• Example 1: Given a spring constant of 175 N/m and a stretch of 30cm, the elastic force is calculated at 52.5 N.
• The calculation of spring constant when given a 2.0 kg mass hanging on a spring stretched 4.0cm from its rest position equals 490 N/m.

Energy in a Spring

• Stored energy in a spring is elastic potential energy (Ep).
• Formula for elastic potential energy: Ep = ½kx²
  • Ep = elastic potential energy (joules)
  • k = spring constant (N/m)
  • x = displacement (m)
• The work done on a spring equals the area under a force-vs-stretch graph

Example Calculations - Energy in a Spring

• Example (page 6): A spring with a constant of 240 N/m and a 0.80 kg mass suspended from it has an extension of 0.0327 m and elastic potential energy of 0.13 J
• Example (page 7): The work done to compress a spring 4.0 cm with a 55N/m constant is 0.044J.
• Example (page 7): A toy gun with a 410 N/m spring compressed 3.0cm and 0.05kg projectile has launch velocity of 2.7m/s.
• Example (page 10): A 9.0N/m spring with a .0100kg mass displaced 5.0cm to the right of its rest position shows a -45 m/s² acceleration and a velocity of 1.5 m/s when at rest.
• Example (page 10): 0.0189=0.010V² (calculation) with velocity of 1.4m/s, when at the stated displacement.
• Example (page 11): A 0.020kg toy with spring constant of 220N/m and 0.030 m compression reaches maximum height of 0.50m.
• Example (page 11): A 0.50 Kg ball attached to a horizontal spring and compressed 0.25m from the equilibrium position, is released. A maximum speed of 1.5m/s is calculated and the spring constant is 18 N/m. The speed when 0.125m from its equilibrium position is 1.3 m/s.

Simple Harmonic Motion

• Repetitive back-and-forth motion around an equilibrium position.
• The force for the motion is directly proportional to the displacement from equilibrium.
• Examples include a mass on a horizontal spring.
• Example (page 8) and (page 9) demonstrates this motion through calculations, showing the relationship between the displacement and acceleration.

Types of Collisions

• Elastic Collisions: Kinetic energy is conserved. Objects bounce apart.
• Inelastic Collisions: Kinetic energy is not conserved. Objects usually stick together.
• Example (page 13): Elastic vs Inelastic Collision.

Example Calculations - Collisions (Elastic/Inelastic)

• Example (page 14): A 1200 kg car traveling East at 25m/s collides with a 1300kg car traveling West at 19m/s and then rebounds at 27m/s. The collision here is Inelastic.
• Example (page 14): A 95 kg hockey player traveling East at 2.3m/s collides with a 104kg player traveling West at 1.2 m/s. The final speed of the entangled players is 0.47m/s East. The collision here is also Inelastic .