Physics I 95.141 LECTURE 22 11/29/10

Administrative Notes Exam III In Class Wednesday, 9am Chapters 9-11 9 problems posted on Website in Practice Exam Section. At least 1 of these problems will be on EXAM III. If you have questions, start a discussion thread on Facebook, that way my response is seen by everyone in the class. Review Session Tuesday Night, 6:00pm, OH150

Outline Oscillations Simple Harmonic Motion What do we know? Units Work by Constant Force Scalar Product of Vectors Work done by varying Force Work-Energy Theorem Conservative, non-conservative Forces Potential Energy Mechanical Energy Conservation of Energy Dissipative Forces Gravitational Potential Revisited Power Momentum and Force Conservation of Momentum Collisions Impulse Conservation of Momentum and Energy Elastic and Inelastic Collisions2D, 3D Collisions Center of Mass and translational motion Angular quantities Vector nature of angular quantities Constant angular acceleration Torque Rotational Inertia Moments of Inertia Angular Momentum Vector Cross Products Conservation of Angular Momentum Oscillations Simple Harmonic Motion What do we know? Units Kinematic equations Freely falling objects Vectors Kinematics + Vectors = Vector Kinematics Relative motion Projectile motion Uniform circular motion Newton’s Laws Force of Gravity/Normal Force Free Body Diagrams Problem solving Uniform Circular Motion Newton’s Law of Universal Gravitation Weightlessness Kepler’s Laws

Review of Lecture 21 Discussed cross product definition of angular momentum and torque Why would we ever use cross products instead of simpler scalar expressions? 3D vectors Point masses not moving in uniform circle Conservation of Angular Momentum Newton’s 2nd Law for rotational motion No external torque, angular momentum conserved.

Oscillations (Chapter 14) Imagine we have a spring/mass system, where the mass is attached to the spring, and the spring is massless. A vmax A

Oscillations So we can say that the mass will move back and forth (it will oscillate) with an Amplitude of oscillation A. Can we describe what is going on mathematically? Would like to determine equation of motion of the mass. In order to do this, we need to know the force acting on the block. Force depends on position: Hooke’s Law

Oscillations

Possible Solutions What if x=At? What if x=Aebt?

Possible Solutions What if x=Acos(bt)? What if x=Acos(bt+Φ)?

Possible Solutions What if If we start with the mass displaced from equilibrium by a distance A at t=0, then we can determine x(t).

What does motion vs. time look like? Plot x at t=0, π/2ω, π/ω, 3π/2ω, 2π/ω, 5π/2ω x A t -A

Harmonic Motion: terminology Displacement: Distance from equilibrium Amplitude of oscillation: max displacement of object from equilibrium Cycle: one complete to-and-fro motion, from some initial point back to original point. Period: Time it takes to complete one full cycle Frequency: number of cycles in one second

Simple Harmonic Motion A form of motion where the only force on the object is the net restoring force, which is proportional to the negative of the displacement. Such a system is often referred to as a simple harmonic oscillator The simple harmonic oscillator’s motion is described by:

What is Φ?

More terminology So the Φ term is known as the phase of the oscillation. It basically shifts the x(t) plot in time. The term ω, which for a spring mass system, is equal to , is known as the angular frequency.

Velocity and Acceleration for SHO If we know x(t), we can calculate v(t) and a(t)

Velocity and Acceleration for SHO If

Example A SHO oscillates with the following properties: Amplitude=3m Period = 2s Give the equation of motion for the SHO

Example A SHO oscillates with the following properties: Amplitude=3m Period = 2s At t=0s, x=3m. Give the equation of motion for the SHO

Example A SHO oscillates with the following properties: Amplitude=3m Period = 2s At t=0s, x=1.5m Give the equation of motion for the SHO

Example A SHO oscillates with the following properties: Amplitude=3m Period = 2s At t=0s, v=2m/s. Give the equation of motion for the SHO

Energy of SHO The total energy of a simple harmonic oscillator comes from the potential energy in the spring, and the kinetic energy of the mass.

Example A spring mass system with m=4kg and k=400N/m is displaced +0.2m from equilibrium and released. A) What is the equation of motion for the mass?

Example A spring mass system with m=4kg and k=400N/m is displaced +0.2m from equilibrium and released. B) What is the total energy of the system?

Example A spring mass system with m=4kg and k=400N/m is displaced +0.2m from equilibrium and released. C) What is the Kinetic Energy and Potential Energy of the system at t=2s?

SHO and Circular Motion y A You will notice that we use the same variable for both angular velocity and angular frequency of a simple harmonic oscillator. If we imagine an object moving with uniform circular motion (angular velocity=ω) on a flat surface. Starting, at t=0s, at θ=0. We know that θ(t)=ωt We can write the x-position of the object as: And the y-position as: θ x

The pendulum A simple pendulum consists of a mass (M) attached to a massless string of length L. We know the motion of the mass, if dropped from some height, resembles simple harmonic motion: oscillates back and forth. Is this really SHO? Definition of SHO is motion resulting from a restoring force proportional to displacement.

Simple Pendulum L θ Δx We can describe displacement as: The restoring Force comes from gravity, need to find component of force of gravity along x Need to make an approximation here for small θ… θ Δx

Simple Pendulum L Now we have an expression for the restoring force From this, we can determine the effective “spring” constant k And we can determine the natural frequency of the pendulum θ Δx

Simple Pendulum L θ Δx If we know We can determine period T And we can the equation of motion for displacement in x …or θ θ Δx