DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS

TSOKOS LESSON 4-1 SIMPLE HARMONIC MOTION

Introductory Video: Simple Harmonic Motion

IB Assessment Statements Topic 4.1, Kinematics of Simple Harmonic Motion (SHM): 4.1.1. Describe examples of oscillations. 4.1.. Define the terms displacement, amplitude, frequency, period and phase difference. 4.1.3. Define simple harmonic motion (SHM) and state the defining equation as a = ω x. 4.1.4. Solve problems using the defining equation for SHM.

IB Assessment Statements Topic 4.1, Kinematics of Simple Harmonic Motion (SHM): 4.1.5. Apply the equations, as solutions to the defining equation for SHM. v v x x v x x 0 0 0 cost x 0 cost sin t x

IB Assessment Statements Topic 4.1, Kinematics of Simple Harmonic Motion (SHM): 4.1.6. Solve problems, both graphically and by calculation, for acceleration, velocity and displacement during SHM.

IB Assessment Statements Topic 4., Energy Changes During Simple Harmonic Motion (SHM): 4..1. Describe the interchange between kinetic energy and potential energy during SHM.

IB Assessment Statements Topic 4., Energy Changes During Simple Harmonic Motion (SHM): 4... Apply the expressions, E K for the kinetic energy of a particle undergoing SHM E T for the total energy and, E P for the potential energy. 1 m x0 x 1 m x0 1 m x

IB Assessment Statements Topic 4., Energy Changes During Simple Harmonic Motion (SHM): 4..3. Solve problems, both graphically and by calculation, involving energy changes during SHM.

IB Assessment Statements Topic 4.3, Forced Oscillations and Resonance: 4.3.1. State what is meant by damping. 4.3.. Describe examples of damped oscillations. 4.3.3. State what is meant by natural frequency of vibration and forced oscillations.

IB Assessment Statements Topic 4.3, Forced Oscillations and Resonance: 4.3.4. Describe graphically the variation with forced frequency of the amplitude of vibration of an object close to its natural frequency of vibration. 4.3.5. State what is meant by resonance. 4.3.6. Describe examples of resonance where the effect is useful and where it should be avoided.

Objectives By the end of this lesson you should be able to: Understand that in simple harmonic motion there is continuous transformation of energy from kinetic energy into elastic potential energy and vice versa; Discuss the properties of simple harmonic motion from graphs; Understand the terms displacement, amplitude, period, frequency, angular frequency, and phase;

Objectives Recognize the occurrence of simple harmonic motion through defining the relation, a=-ω x; x Acos( t Use the equations: v Asin( t ) ) Solve problems with kinetic energy and elastic potential energy in simple harmonic motion; v T A x

Objectives Objectives. By the end of this class you should be able to: Describe the effect of damping on an oscillating system; Discuss qualitatively the effect of a periodic external force on an oscillating system Understand the meaning of resonance and give examples of its occurrence;

Oscillation vs. Simple Harmonic Motion An oscillation is any motion in which the displacement of a particle from a fixed point keeps changing direction and there is a periodicity in the motion i.e. the motion repeats in some way. In simple harmonic motion, the displacement from an equilibrium position and the force/acceleration are proportional and opposite to each other.

Simple Harmonic Motion: Spring

Simple Harmonic Motion Understand that in simple harmonic motion there is continuous transformation of energy from kinetic energy into elastic potential energy and vice versa

Simple Harmonic Motion: Spring no displ, no energy, no accl max displ, max PE, max accl, zero KE half max displ, half max PE, half max accl, half max KE max displ, max PE, max accl, zero KE zero displ, zero PE, zero accl, max KE

Simple Harmonic Motion: Spring E Total PE KE 0 0 E Total 1 kx 0 E Total 1 kx 1 mv E Total 0 1 mv E Total 1 kx 0

Simple Harmonic Motion: Spring The spring possesses an intrinsic restoring force that attempts to bring the object back to equilibrium: F kx This is Hooke s Law k is the spring constant (kg/s ) The negative sign is because the force acts in the direction opposite to the displacement -- restoring force

Simple Harmonic Motion: Spring Meanwhile, the inertia of the mass executes a force opposing the spring, F=ma: spring executing force on mass F kx mass executing force on spring F ma

Simple Harmonic Motion: Spring These forces remain in balance throughout the motion: ma kx The relationship between acceleration and displacement is thus, k a x m

Simple Harmonic Motion: Spring k a x m Satisfies the requirement for SHM that displacement from an equilibrium position and the force/acceleration are proportional and opposite to each other

Relating SHM to Motion Around A Circle

Radians One radian is defined as the angle subtended by an arc whose length is equal to the radius l r l r 1

Radians Circumference r l r l r Circumference rad

Angular Velocity t v l t l, l r r v r r t

Angular Acceleration a t a r v r v r a r r r r

Period v T T r T T

Frequency T f f 1 T f

Relating SHM to Motion Around A Circle The period in one complete oscillation of simple harmonic motion can be likened to the period of one complete revolution of a circle. angle swept Time taken = ---------------------- angular speed (ω) T T

Relating SHM to Motion Around A Circle F ma k kx ma m k k a x m m a r a x

Relating SHM to Motion Around A Circle Using, k m We then derive x v T Acos( Asin( t t ) k m )

Relating SHM to Motion Around A Circle These equations yield the following graphs

Definitions Understand the terms displacement, amplitude and period displacement (x) distance from the equilibrium or zero point amplitude (A) maximum displacement from the equilibrium or zero point period (T) time it takes to complete one oscillation and return to starting point

Definitions

Definitions Understand the terms period and frequency frequency (f) How many oscillations are completed in one second, equal to the inverse of the period period (T) Time for one complete oscillation T T 1 f f 1 T f f

Definitions Understand the term phase; phase (φ) the difference between the actual position of a sine wave at t=0 and zero. The value of φ determines the displacement at t=0 x Acos( t )

Stopped Here /1/013

Particle in a Bowl F ma mgsin ma a g sin x a g sin r x a g r g a x r

Particle in a Bowl a a T T g r g r x x r g But only for very small amplitudes

Simple Pendulums a a T g L g L x L L g But only for very small amplitudes

Is It SHM? Note that for particles in bowls and for pendulums, while the acceleration is opposite to the displacement x, it is not proportional to it We will have oscillations, but they will not be simple harmonic motion

Energy in SHM E E PE 1 kx KE 1 1 mv kx 1 cons mv tan t

Energy in SHM If the mass is released from rest when the extension is equal to the amplitude, A, (max PE) 1 kx If we solve for velocity, 1 mv 1 ka v k m A x

Energy in SHM v k m A x Since, k m v v max A A x

Damping

Light Damping

Heavy Damping

Critical Damping

Over-Damping

Forced Oscillations

Resonance The state in which the frequency of the externally applied periodic force equals the natural frequency of the system is called resonance. This results in oscillations with large amplitude.

Resonance

Effects of Resonance on Buildings

Resonance

Σary Review At this point in your existence you should be able to: Understand that in simple harmonic motion there is continuous transformation of energy from kinetic energy into elastic potential energy and vice versa; Discuss the properties of simple harmonic motion from graphs; Understand the terms displacement, amplitude, period, frequency, angular frequency, and phase;

Σary Review Recognize the occurrence of simple harmonic motion through defining the relation, a=-ω x; Use the equations: x Acos( t v v T Asin( t ) ) Solve problems with kinetic energy and elastic potential energy in simple harmonic motion; A x

Σary Review Describe the effect of damping on an oscillating system; Discuss qualitatively the effect of a periodic external force on an oscillating system Understand the meaning of resonance and give examples of its occurrence;

IB Assessment Statements Topic 4.1, Kinematics of Simple Harmonic Motion (SHM): 4.1.1. Describe examples of oscillations. 4.1.. Define the terms displacement, amplitude, frequency, period and phase difference. 4.1.3. Define simple harmonic motion (SHM) and stat the defining equation as a = ω x. 4.1.4. Solve problems using the defining equation for SHM.

IB Assessment Statements Topic 4.1, Kinematics of Simple Harmonic Motion (SHM): 4.1.5. Apply the equations, as solutions to the defining equation for SHM. v v x x v x x 0 0 0 cost x 0 cost sin t x

IB Assessment Statements Topic 4.1, Kinematics of Simple Harmonic Motion (SHM): 4.1.6. Solve problems, both graphically and by calculation, for acceleration, velocity and displacement during SHM.

IB Assessment Statements Topic 4., Energy Changes During Simple Harmonic Motion (SHM): 4..1. Describe the interchange between kinetic energy and potential energy during SHM.

IB Assessment Statements Topic 4., Energy Changes During Simple Harmonic Motion (SHM): 4..3. Solve problems, both graphically and by calculation, involving energy changes during SHM.

Tacoma Narrows Bridge Collapse

Homework #1-38 Skip 11, 19,, 3, 34 Beautiful Resonance