Mass on a Spring C2: Simple Harmonic Motion. Simple Harmonic Motion. Announcements Week 12D1

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1 Simple Harmonic Motion 8.01 Week 1D1 Today s Reading Assignment MIT 8.01 Course Notes Chapter 3 Simple Harmonic Motion Sections Announcements Sunday Tutoring in 6-15 from 1-5 pm Problem Set 9 due Nov 19 Tuesday at 9 pm in box outside 6-15 Math Review Nov 19 Tuesday at 9-10:30 pm in 6-15 Exam 3 Tuesday Nov 6 7:30-9:30 pm Conflict Exam 3 Wednesday Nov am, 10-1 noon Nov 0 Drop Date Mass on a Spring C: Simple Harmonic Motion 3 1

2 Hooke s Law Define system, choose coordinate system. Draw free-body diagram. Hooke s Law F spring = kx ˆi d x kx = m dt Concept Q.: Simple Harmonic Motion Which of the following functions x( has a second derivative which is proportional to the negative of the function d x x? dt x( = 1 at x( = Ae t /T t /T x( = Ae x( = Acos π T t SHM: Angular Frequency Newton s Second Law Simple Harmonic Oscillator Differential Equation (SHO) Particular Solution: Required Condition: Angular Frequency: F x = kx SHM d x dt = k m x x( = C cos((π / T ) d x dt = π T x π / T = k / m = π f = π / T = k / m Notation for particular solution: x( = C cos( 6

3 Mass on a Spring C: Demonstrate Initial Conditions 7 Summary: SHO Equation of Motion: Solution: Oscillatory with Period d x kx = m dt T = π / = π m / k Position: Velocity: Initial Position at t = 0: x = C cos( + Dsin( v x = dx dt = C sin( + Dcos( x 0 x(t = 0) = C Initial Velocity at t = 0: General Solution: v x,0 v x (t = 0) = D x = x 0 cos( + v x,0 sin( Table Problem: 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 blockspring system is released from the equilibrium position x 0 = 0 and with speed v 0 in the negative x-direction. a) What is the position as a function of time? b) What is the x-component of the velocity as a function of time? 3

4 Demo: 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) 10 Phase and Amplitude x( = C cos( + Dsin( x( = Acos( t + φ) C = Acos(φ) D = Asin(φ) A = C + D tanφ = D / C 11 Mass on a Spring: Energy x( = Acos( v x ( = Asin( = k / m, A = x 0 + v x,0 1/ Constant energy oscillates between kinetic and potential energies K( = (1/ )m(v x () = (1/ )m A sin ( K( = (1/ )ka sin ( U ( = (1/ )kx = (1/ )ka cos ( E = K( +U ( = (1/ )ka = (1/)mv x,0 + (1/ )kx 0 = constant 1 4

5 Worked Example: Block-Spring Energy Method A block of mass is attached to spring with spring constant k. The block slides on a frictionless surface. Use the energy method to find the equation of motion for the spring-block system. Energy and Simple Harmonic Motion E = K( +U ( = (1/ )k(x() + (1/ )m(v x () Apply chain rule: de / dt = 0 0 = kx dx dt + mv dv x x dt d x 0 = kx + m x dt 14 Concept Question: SHM 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 stretched 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? 4 1. v x = x 0. T v = x 4 x 0 T 3. v x = k 4. m x 0 v x = k m x 0 5

6 Graphical Representations 16 SHM: Oscillating Systems d x dt = bx x( = C cos( + Dsin( v x = dx dt = C sin( + Dcos( = b 17 Table Problem: Simple Pendulum by the Energy Method 1. Find an expression for the mechanical energy when the pendulum is in motion in terms of θ( and its derivatives, m, l, and g as needed.. Find an equation of motion for θ( using the energy method. 6

7 Worked Example: Simple Pendulum Small Angle Approximation Equation of motion mg sinθ = ml d θ dt Angle of oscillation is small Simple harmonic oscillator sinθ θ d θ dt g l θ Analogy to spring equation d x dt = k m x Angular frequency of oscillation g / l Period T 0 = π π l / g Periodic vs. Harmonic Equation of motion g l sinθ = d θ dt periodic Angle of oscillation is small, linear restoring torque sinθ θ Simple harmonic oscillator d θ dt g l θ SHO Angular frequency for SHO is independent of amplitude g / l Demonstration: U-tube Oscillations 1 7

8 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 r. 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. Table Problem: Rolling and Oscillating Cylinder Attach a solid cylinder of mass M and radius R to a horizontal massless spring with spring constant k so that it can roll without slipping along a horizontal surface. At time t, the center of mass of the cylinder is moving with speed V cm and the spring is compressed a distance x from its equilibrium length. What is the period of simple harmonic motion for the center of mass of the cylinder? 3 8