Brenda Rubenstein (Physics PhD)

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Lisa Wilcox
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That notion is what initially drew me to physics and what continues to do so today. Today, Brenda works as a postdoctoral researcher at Lawrence Livermore National Lab (LLNL).

1 PHYSICIST PROFILE Brenda Rubenstein (Physics PhD) Postdoctoral Researcher Lawrence Livermore Nat l Lab Livermore, CA In college, Brenda looked for a career path that would allow her to make a positive change in the world. She was struck by the power of simple ideas in physics. A single person s efforts can amount to something profound. That notion is what initially drew me to physics and what continues to do so today. Today, Brenda works as a postdoctoral researcher at Lawrence Livermore National Lab (LLNL). Brenda says she enjoys working in a national lab: after graduate school, she wanted to balance her time between research and mentoring opportunities. Lab scientists get the best of both worlds in this regard while they often mentor other scientists, they still have the opportunity to conduct their own day to day research.

2 Class 39 4 December 2017 Gravitational Potential Energy and Escape Velocity Periodic Motion Final Exam: Saturday, December 16, 2017, 6:30-9:30 pm

3 Review Session from SMART Learning Commons

4 Gravitational Potential Energy Gravitational forces are always attractive Usual definition is that gravitational potential energy is zero at infinite separation Gravitational potential energy is always negative for finite distance U = G m m 1 2dr = G m m 1 2 R r 2 R

5 Escape Velocity From The Moon How does gravitational potential energy on the surface of the moon compare with gravitational potential energy on the surface of the Earth? Moon is less massive: decreases potential energy magnitude Moon has smaller radius: increases potential energy magnitude Escape velocity is proportional to square root of potential energy Escape velocity suggests use of a Lunar Lander

6 Problem What is the escape velocity from the surface of the Moon? The mass of the Moon is 7.36 x kg. The radius of the Moon is x 10 6 m. (a) 1.79 km/s (b) 2.38 km/s (c) 5.14 km/s (d) 9.89 km/s (e) 11.2 km/s 6

7 Problem What is the escape velocity from the surface of the Moon? The mass of the Moon is 7.36 x kg. The radius of the Moon is x 10 6 m. (b) 2.38 km/s r 2GmMoon v = r v =2.38 km/s 7

8 Escape Velocity From The Moon Earth: kg; m Moon: kg; m Potential Energy Ratio:.045 Escape Velocity Ratio:.213 Gravitational Force Ratio: 0.166

9 Periodic Motion Periodic motion is motion that repeats either for a while or endlessly after some interval of time called the period T. There are many examples of periodic motion Mass on a spring Pendulums Pistons in engines Vibrations of a bridge 9

10 Periodic Motion and Energy Periodic motion may occur with or without the ongoing addition of external energy Generally, a system that exhibits periodic motion has an ability to store and recover kinetic energy. Such a system swaps energy between kinetic and potential forms. Mass on a spring: Potential energy stored on the spring Pendulum: Energy stored as gravitational potential energy Piston in a car: Energy stored in flywheel and car motion Vibrations of a bridge: Energy stored as both spring and gravitational potential energy 10

11 Describing Periodic Motion The best way to describe periodic motion is with periodic functions. The most straightforward periodic functions are sines and cosines. x = a sin wt repeats with a period T = 2π/w, where a is called the amplitude and w is called the angular frequency w = 2πf, where f in Hertz is the frequency. T = 1/f is called the period and is measured in seconds. 11

12 Approximating Functions In linear algebra, a well-behaved, arbitrary function can be approximated by summing a set of functions known as basis functions. An example is a Taylor Series, which approximates an arbitrary function f(t) by a sum of powers of t, each with a different coefficient f(t) f(t 0 )+ 1 df (t) t=t0 (t t 0 )+ 1 d 2 f(t) 2 t=t0 (t t 0 ) ! dt 2! dt Let t 0 =0 e t 1+t t

13 Approximating Periodic Functions A polynomial is not a good way to approximate a periodic function because terms such as t n are not periodic. A better approach is to use naturally periodic functions as basis functions for the series. A Fourier Series uses sines and cosines as the basis functions f(t) a X n=1 f(t) a X (a n cos n!t + b n sin n!t) n=1 (a n cos n!t + n ) 13

14 Fourier Series An example is a Fourier Series to approximate a square periodic function. Each graph has one more term than the graph above it. Only 3 or 4 terms are need for a reasonable approximation Sharp corners require higher frequency terms 14

15 Simple Harmonic Motion A system for which the restoring force towards the equilibrium point is linearly proportional to the displacement from the equilibrium point will exhibit simple harmonic motion (SHM) SHM can be described by a Fourier Series with only one term. Examples of systems that exhibit SHM Mass on a spring Pendulum (for small displacements) There is a connection between SHM and uniform circular motion. Uniform circular motion can be described parametrically by SHM for x and y with a π/2 phase difference between the x and y motions. 15

16 Parametric Circular Motion A circle can be described parametrically in terms of its radius r and an angle that varies from 0 to 2π Uniform velocity circular motion can also be described parametrically in terms of a radius r and time dependent x and y coordinates. 16

17 x = r cos wt y = r sin wt Parametric Circular Motion The x and y motion each separately appear similar to the motion of a mass on a spring. The reason for this similarity is that these two different kinds of motion are both described by the same mathematical functions 17

18 Mass On A Spring Hooke s Law: F = -kx Newton s 2 nd Law: F = ma Second order, homogenous differential equation General solution + two constants determined by boundary conditions Euler s Formula e ix = cos x + i sin x F = ma = kx d 2 x 2 + k dt m x =0 x = Ae i!t + Be i!t x = A sin(!t + ) x = A cos(!t + ) 18

19 Mass On A Spring d 2 x 2 + k dt m x =0 x = A cos(!t) dx =!A sin!t dt d 2 x 2 =!2 A cos!t dt! 2 A cos!t + k A cos!t =0 m r k! = m T = 2 r m! =2 k 19

20 Pendulum Another example of simple harmonic motion is a pendulum. The distance along the arc is Lq The force towards the equilibrium point is mg sin q 20

21 Pendulum m d2 (L ) 2 = mg sin dt d 2 ( ) dt 2 + g sin =0 L sin d 2 ( ) dt 2 + g L =0 r g = cos!t! = L s L T =2 g 21

Chapter 11 Vibrations and Waves

Chapter 11 Vibrations and Waves If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system