Simple Harmonic Oscillator 8.01 Week 13D1 Today’s Reading Assignment Young and Freedman:

Announcements Math Review Tuesday Nov 29 from 9-11 pm Pset 11 due Thursday Dec 1 at 9 pm W013D2 Reading Assignment Young and Freedman:

3 Mass on a Spring C2: Simple Harmonic Motion

Hooke’s Law Define system, choose coordinate system. Draw free-body diagram. Hooke’s Law

5 Mass on a Spring (1) (2) (3)(4)

Concept Question: Simple Harmonic Oscillator Which of the following functions x(t) has a second derivative which is proportional to the negative of the function

Concept Question Answer: Simple Harmonic Oscillator Answer 4. By direct calculation, when

Demo slide: spray paint oscillator C4 Illustrating choice of alternative representations for position as a function of time (amplitude and phase or sum of sin and cos) 8

9 Simple Harmonic Motion

Equation of Motion: Solution: Oscillatory with Period Position: Velocity: Initial Position at t = 0: Initial Velocity at t = 0: General Solution:

Worked Example: Simple Harmonic Motion Block-Spring A block of mass m, attached to a spring with spring constant k, is free to slide along a horizontal frictionless surface. At t = 0 the block- spring system is stretched an amount x 0 > 0 from the equilibrium position and released from rest. What is the x -component of the velocity of the block when it first comes back to the equilibrium?

Worked Example: Simple Harmonic Motion Block-Spring At t = 0, the block is at x(t = 0) = x 0, and it is released from rest so v x (t = 0) = 0. Therefore the initial conditions are So the position and x-component of the velocity of the block are given by of the block as a function of time is The particle first returns to equilibrium at t = T/4, therefore

13 Mass on a Spring: Energy Constant energy oscillates between kinetic and potential energies

14 Energy and Simple Harmonic Motion

Concept Question: Simple Harmonic Motion Velocity A block of mass m is attached to a spring with spring constant k is free to slide along a horizontal frictionless surface. At t = 0 the block-spring system is compressed an amount x 0 < 0 from the equilibrium position and is released from rest. What is the x -component of the velocity of the block when it first comes back to the equilibrium?

CQ Answer: Simple Harmonic Motion Velocity Answer 3. The particle starts with potential energy. When it first returns to equilibrium it now has only kinetic energy. Since the energy of the block-spring system is constant: Take positive root because object is moving in positive x - direction when it first comes back to equilibrium, and x o 0, therefore

Table Problem: Block-Spring System with No Friction A block of mass m slides along a frictionless horizontal surface with speed v x,0. At t = 0 it hits a spring with spring constant k and begins to slow down. How far is the spring compressed when the block has first come momentarily to rest?

Kinetic Energy vs. Potential Energy State Kinetic energy Potential energy Mechanical energy Initial Final

Source 19

Energy Diagram Choose zero point for potential energy: Potential energy function: Mechanical energy is represented by a horizontal line Graph of Potential energy function U ( x ) vs. x

Concept Question: Harmonic Energy Diagram The position of a particle is given by Where was the particle at t = 0 ? 1) 1 2) 2 3) 3 4) 4 5) 5 6) 1 or 5 7) 2 or 4 21

Concept Question: Harmonic Energy Diagram Answer The position and x-component of velocity of a particle is given by Hence Thus the particle starts out with a positive x- coordinate and the initial x-component of the velocity is negative therefore it is moving toward the origin. It has both non-zero potential and kinetic energies and hence it is not at it’s maximal position so at it is located at position 4. 22

23 Oscillating Systems: Simple Harmonic Motion

Demo: Air Spring with Oscillating steel ball C12 24

Worked Example: fluid oscillations in a U-tube A U-tube open at both ends to atmospheric pressure is filled with an incompressible fluid of density . The cross-sectional area A of the tube is uniform and the total length of the column of fluid is L. A piston is used to depress the height of the liquid column on one side by a distance x, and then is quickly removed. What is the frequency of the ensuing simple harmonic motion? Assume streamline flow and no drag at the walls of the U- tube. The gravitational constant is g. 25

Demonstration: U-tube oscillations 26

Table Problem: Earthquake 27 In 1993, to try to protect its historic courthouse from damage due to earthquakes, the city of San Francisco designed a special foundation. The building rests on N identical supports. Each support has a small diameter spherical cap anchored to the ground resting in an inverted spherical cup of radius R attached to the building. As a result, if the building undergoes a small horizontal displacement x, it rises by an amount as shown in the diagram. Building has mass M. a) Assume that there is no friction. Find the differential equation which describes the dynamics of small displacements x(t) of the center of the cups from the centers of the caps. b) A sudden shock sets the building moving with the initial conditions x(0) = 0 and. Find an expression for x(t) for all subsequent times.