Physics 218 Chapter 15 Prof. Rupak Mahapatra Physics 218, Lecture XXII

Checklist for Today Midterm 3 average 61 Collect your exams from your TAs. Last lecture next Monday Will cover up to Chpater 18 Ch 14 and 15 Home work due Wed this week Ch 18 Home work due Mon next week Physics 218, Lecture XXII

Angular Quantities Momentum  Angular Momentum L Energy Position  Angle q Velocity  Angular Velocity w Acceleration  Angular Acceleration a Force  Torque t Mass  Moment of Inertia I Today we’ll finish: Momentum  Angular Momentum L Energy Physics 218, Lecture XXII

Rotational Kinetic Energy KEtrans = ½mv2  KErotate = ½Iw2 Conservation of Energy must take rotational kinetic energy into account Physics 218, Lecture XXII

Rotation and Translation Objects can both Rotate and Translate Need to add the two KEtotal = ½ mv2 + ½Iw2 Rolling without slipping is a special case where you can relate the two V = wr Physics 218, Lecture XXII

Rolling Down an Incline You take a solid ball of mass m and radius R and hold it at rest on a plane with height Z. You then let go and the ball rolls without slipping. What will be the speed of the ball at the bottom? What would be the speed if the ball didn’t roll and there were no friction? Note: Isphere = 2/5MR2 Z Physics 218, Lecture XXII

A bullet strikes a cylinder A bullet of speed V and mass m strikes a solid cylinder of mass M and inertia I=½MR2, at radius R and sticks. The cylinder is anchored at point 0 and is initially at rest. What is w of the system after the collision? Is energy Conserved? Physics 218, Lecture XXII

Rotating Rod A rod of mass uniform density, mass m and length l pivots at a hinge. It has moment of inertia I=ml/3 and starts at rest at a right angle. You let it go: What is w when it reaches the bottom? What is the velocity of the tip at the bottom? Physics 218, Lecture XXII

Person on a Disk A person with mass m stands on the edge of a disk with radius R and moment ½MR2. Neither is moving. The person then starts moving on the disk with speed V. Find the angular velocity of the disk Physics 218, Lecture XXII

Same Problem: Forces Same problem but with Forces Physics 218, Lecture XXII

Chapter 18: Periodic Motion This time: Oscillations and vibrations Why do we care? Equations of motion Simplest example: Springs Simple Harmonic Motion Next time: Energy } Concepts } The math Physics 218, Lecture XXIV

Physics 218, Lecture XXIV

What is an Oscillation? The good news is that this is just a fancy term for stuff you already know. It’s an extension of rotational motion Stuff that just goes back and forth over and over again “Stuff that goes around and around” Anything which is Periodic Same as vibration No new physics… Physics 218, Lecture XXIV

Examples Lots of stuff Vibrates or Oscillates: Radio Waves Guitar Strings Atoms Clocks, etc… In some sense, the Moon oscillates around the Earth Physics 218, Lecture XXIV

Why do we care? Lots of engineering problems are oscillation problems Buildings vibrating in the wind Motors vibrating when running Solids vibrating when struck Earthquakes Physics 218, Lecture XXIV

What’s Next You’ll see why we do this later First we’ll “model” oscillations with a mass on a spring You’ll see why we do this later Then we’ll talk about what happens as a function of time Then we’ll calculate the equation of motion using the math Physics 218, Lecture XXIV

Simplest Example: Springs What happens if we attach a mass to a spring sitting on a table at it’s equilibrium point (I.e., x = 0) and let go? What happens if we attach a mass, then stretch the spring, and then let go? k Physics 218, Lecture XXIV

Questions What are the forces? Hooke’s Law: F= -kx Does this equation describe our motion? x = x0 + v0t + ½at2 Physics 218, Lecture XXIV

The forces No force Force in –x direction Force in +x direction Physics 218, Lecture XXIV

More Detail Time Physics 218, Lecture XXIV

Some Terms Amplitude: Max distance Period: Time it takes to get back to here Physics 218, Lecture XXIV

Overview of the Motion It will move back and forth on the table as the spring stretches and contracts At the end points its velocity is zero At the center its speed is a maximum Physics 218, Lecture XXIV

Simple Harmonic Motion Call this type of motion Simple Harmonic Motion (Kinda looks like a sine wave) Next: The equations of motion: Use SF = ma = -kx (Here comes the math. It’s important that you know how to reproduce what I’m going to do next) Physics 218, Lecture XXIV

Equation of Motion A block of mass m is attached to a spring of constant k on a flat, frictionless surface What is the equation of motion? k Physics 218, Lecture XXIV

Summary: Equation of Motion Mass m on a spring with spring constant k: x = A sin(wt + f) Where w2 = k/m A is the Amplitude is the “phase” (phase just allows us to set t=0 when we want) Physics 218, Lecture XXIV

Simple Harmonic Motion At some level sinusoidal motion is the definition of Simple Harmonic Motion A system that undergoes simple harmonic motion is called a simple harmonic oscillator Physics 218, Lecture XXIV

Understanding Phase: Initial Conditions A block with mass m is attached to the end of a spring, with spring constant k. The spring is stretched a distance D and let go at t=0 What is the position of the mass at all times? Where does the maximum speed occur? What is the maximum speed? Physics 218, Lecture XXIV

Paper which tells us what happens as a function of time Check: This looks like a cosine. Makes sense… Spring and Mass Paper which tells us what happens as a function of time Physics 218, Lecture XXIV

Example: Spring with a Push We have a spring system Spring constant: K Mass: M Initial position: X0 Initial Velocity: V0 Find the position at all times Physics 218, Lecture XXIV

Simple Harmonic Motion What is MOST IMPORTANT? Simple Harmonic Motion X= A sin(wt + f) What is the amplitude? What is the phase? What is the angular frequency? What is the velocity at the end points? What is the velocity at the middle? Physics 218, Lecture XXIV