Unit 6 Lesson 2 Mass on a Spring and Pendulums Mass Spring System - Energy Homework SERWAY Page 416: #’s 25, 29, 33 Page 416 #’s 23, 24, 26, 27, 28

Mass – Spring System Vertical  Spring Force vs. Gravity Horizontal  Spring Force vs. Friction Work = ½ kx 2 Net Work = Zero Kinetic Energy + Potential Energy = Total ½ mv 2 + ½ k A 2 = TE Velocity = √[{k/m}(A 2 - x 2 )] Period  T = 2π/ω  T = 2π √[{m/k}] Frequency  f = 1/T = ω/2π = √[{k/m}]/ 2π Position x = A Cos{ωt} Angular Velocity ω = √[{k/m}] = 2π / T Acceleration α= – ω 2 x Tacoma Narrows Bridge

Pendulums Simple Pendulum A simple pendulum is one which can be considered to be a point mass suspended from a string or rod of negligible mass. It is a resonant system with a single resonant frequency. For small amplitudes, the period of such a pendulum can be approximated by:resonantperiod

small angles θ For small angles θ the solution is:

Physical Pendulum Again for small angles < 15 degrees or.26 radians

Rod Pendulum A physical pendulum in the form of a uniform rod suspended by its end has a period given by:physical pendulumuniform rod Note that the period is independent of the mass and radius of the rod.

Practice Review Problems

Lab V-E14Pendulum Periods Lab Hooke’s Law - Kinematics