10/17/2013PHY 113 C Fall Lecture 151 PHY 113 A General Physics I 11 AM – 12:15 PM TR Olin 101 Plan for Lecture 15: Chapter 15 – Simple harmonic motion 1.Object attached to a spring; pendulum  Displacement as a function of time  Kinetic and potential energy 2.Driven harmonic motion; resonance

10/17/2013 PHY 113 C Fall Lecture 152

10/17/2013PHY 113 C Fall Lecture 153 From Webassign Assignment #13 A uniform beam of length L = 7.45 m and weight N is carried by two workers, Sam and Joe, as shown in the figure below. Determine the force that each person exerts on the beam. X mg F Sam F Joe d1d1 d2d2

10/17/2013PHY 113 C Fall Lecture 154 Illustration of “simple harmonic motion”.

10/17/2013PHY 113 C Fall Lecture 155 Hooke’s law F s = -k(x-x 0 ) x 0 =0 Behavior of materials : Young’s modulus Simple harmonic motion; mass and spring

10/17/2013PHY 113 C Fall Lecture 156 Microscopic picture of material with springs representing bonds between atoms Measurement of elastic response: k

10/17/2013PHY 113 C Fall Lecture 157 F s (N) x U s (J) x F s = -kx U s = ½ k x 2 General potential energy curve: U (J) x U s = ½ k (x-1) 2 U(x) Potential energy associated with Hooke’s law: Form of Hooke’s law for ideal system:

10/17/2013PHY 113 C Fall Lecture 158 Motion associated with Hooke’s law forces Newton’s second law: F = -k x = m a  “second-order” linear differential equation

10/17/2013PHY 113 C Fall Lecture 159 How to solve a second order linear differential equation: Earlier example – constant force F 0  acceleration a 0 x(t) = x 0 +v 0 t + ½ a 0 t 2 2 constants (initial values)

10/17/2013PHY 113 C Fall Lecture 1510 Hooke’s law motion: Forms of solution: where: 2 constants (initial values)

Verification: Differential relations: Therefore: 10/17/2013PHY 113 C Fall Lecture 1511

10/17/2013PHY 113 C Fall Lecture 1512

10/17/2013PHY 113 C Fall Lecture 1513 iclicker exercise: Which of the following other possible guesses would provide a solution to Newton’s law for the mass- spring system:

10/17/2013PHY 113 C Fall Lecture 1514 iclicker question How do you decide whether to use A.Either can be a solution, depending on initial position and/or velocity B.Pure guess

10/17/2013PHY 113 C Fall Lecture 1515

10/17/2013PHY 113 C Fall Lecture 1516

10/17/2013PHY 113 C Fall Lecture 1517

10/17/2013PHY 113 C Fall Lecture 1518 iclicker exercise: A certain mass m on a spring oscillates with a characteristic frequency of 2 cycles per second. Which of the following changes to the mass would increase the frequency to 4 cycles per second? (a) 2m (b) 4m (c) m/2 (d) m/4

10/17/2013PHY 113 C Fall Lecture 1519 Energy associated with simple harmonic motion

10/17/2013PHY 113 C Fall Lecture 1520 Energy diagram: x U(x)=1/2 k x 2 E=1/2 k A 2 A K

10/17/2013PHY 113 C Fall Lecture 1521 Simple harmonic motion for a pendulum:  L Approximation for small 

10/17/2013PHY 113 C Fall Lecture 1522 Pendulum example:  L Suppose L=2m, what is the period of the pendulum?

10/17/2013PHY 113 C Fall Lecture 1523 The notion of resonance: Suppose F=-kx+F 0 sin(  t) According to Newton:

10/17/2013PHY 113 C Fall Lecture 1524 Physics of a “driven” harmonic oscillator: “driving” frequency “natural” frequency   =2rad/s) (mag)

10/17/2013PHY 113 C Fall Lecture 1525 Examples: Suppose a mass m=0.2 kg is attached to a spring with k=1.81N/m and an oscillating driving force as shown above. Find the steady-state displacement x(t). F(t)=1 N sin (3t) Note: If k=1.90 N/m then:

10/17/2013PHY 113 C Fall Lecture 1526 Tacoma Narrows bridge resonance Images from Wikipedia Rebuilt bridge in Tacoma WA

10/17/2013PHY 113 C Fall Lecture 1527 Bridge in July, 1940

10/17/2013PHY 113 C Fall Lecture 1528 Collapse of bridge on Nov. 7,

10/17/2013PHY 113 C Fall Lecture 1529 Effects of gravity on spring motion x=0

10/17/2013PHY 113 C Fall Lecture 1530 Example Suppose a mass of 2 kg is attached to a spring with spring constant k=20 N/m. At t=0 the mass is displaced by 0.4 m and released from rest. What is the subsequent displacement as a function of time? What is the total energy?:

10/17/2013PHY 113 C Fall Lecture 1531 Example Suppose a mass of 2 kg is attached to a spring with spring constant k=20 N/m. The mass is attached to a device which supplies an oscillatory force of the form What is the subsequent displacement as a function of time?