Dawson High School AP Physics 1 Oscillations Dawson High School AP Physics 1

Acknowledgements © Mark Lesmeister/Pearland ISD Selected graphics from Cutnell and Johnson, Physics 9e: Instructor’s Companion Site, © 2015 John Wiley and Sons. Selected graphic from Serway and Faughn, Holt Physics, © 2002 Holt, Rinehart and Winston Acknowledgements

Introduction

Observing Oscillating Systems Questions to observe: Does the rate of oscillation depend on the amplitude of the motion? Does the rate of oscillation depend on the mass being oscillated? Show oscillating spring and mass systems. (Vertically mounted, or horizontally mounted with a spring on each side. Observing Oscillating Systems

Oscillations Many types of systems undergo oscillation. Horizontal Spring and Mass Essential Knowledge 3.B.3: Restoring forces can result in oscillatory motion. --------------------------------------------------- Oscillation- when a system varies in a repeating pattern, returning to the original state each time. Oscillations

Oscillation is also called periodic motion. Oscillations

Oscillations can occur when there is a restoring force. Essential Knowledge 3.B.3: Restoring forces can result in oscillatory motion. Restoring Force

Describing Oscillations The frequency f is the number of oscillations per unit time. The period T is the time for one oscillation. Horizontal Spring and Mass Describing Oscillations

The maximum displacement from equilibrium is called the amplitude.

Review: Hooke’s Law FELASTIC = -k x FELASTIC x Ask students to explain what each term means. Felastic is the force the spring exerts on the mass. X is the distance the mass is displaced from equilibrium. The negative means that the force is in the opposite direction from the displacement, i.e. a restoring force. K is a measure of how stiff the spring is, i.e. how much force it exerts for a given displacement. x Review: Hooke’s Law

Lab: Period of a Spring and Mass System Determine quantitatively the effect on the period when: the mass is increased. the amplitude is varied. Source: Wikipedia HorizontalSpring and Mass

Simple Harmonic Motion of a Spring and Mass System

Simple Harmonic Motion When the restoring force is proportional to the displacement, the result is simple harmonic motion. Essential Knowledge 3.B.3: Restoring forces can result in oscillatory motion. When a linear restoring force is exerted on an object displaced from an equilibrium position, the object will undergo a special type of motion called simple harmonic motion. Examples should include gravitational force exerted by the Earth on a simple pendulum and a mass-spring oscillator. Simple Harmonic Motion

Simple Harmonic Motion Simple Harmonic Motion Plot Simple harmonic motion forms a sinusoidal graph. Simple Harmonic Motion

Simple Harmonic Motion Simple Harmonic Motion Plot Simple Harmonic Motion

Essential Knowledge 3. B. 3. c Essential Knowledge 3.B.3.c. Minima, maxima, and zeros of position, velocity, and acceleration are features of harmonic motion. Students should be able to calculate force and acceleration for any given displacement for an object oscillating on a spring. ---------------------------------------------------------------------- The force and acceleration are maximum at maximum displacement. The velocity is zero. The force and acceleration are 0 at equilibrium. The velocity is a maximum (since it immediately starts to slow down after leaving that point.)

Maxima and minima for SHM The restoring force, and thus the acceleration, are at a maximum when displacement is maximum. Maxima and minima for SHM

Maxima and Minima for SHM The velocity is at a maximum when the displacement is zero. Maxima and Minima for SHM

Graphs for SHM Displacement: Velocity: Acceleration Draw corresponding velocity and acceleration graphs for given SHM displacement. Graphs for SHM

Simple Harmonic Motion The frequency and period depend on the setup, and are independent of the amplitude. Simple Harmonic Motion

Frequency and Period of a Spring and Mass System The frequency and period for a spring and mass are Essential Knowledge 3.B.3.a. For a spring that exerts a linear restoring force the period of a mass-spring oscillator increases with mass and decreases with spring stiffness. Frequency and Period of a Spring and Mass System

Energy of a Spring and Mass System If there is no friction, mechanical energy is conserved. A -A Energy of a Spring and Mass System

x

In an ideal system, the mass-spring system would oscillate indefinitely. Damping occurs when friction retards the motion. Damping causes the system to come to rest after a period of time. If we observe the system over a short period of time, damping is minimal, and we can treat the system like as ideal. (0) What will spring do if there is no friction? Does the energy get “used up”? What will it look like if there is friction? Damping

10.1.4. An ideal spring is hung vertically from a device that displays the force exerted on it. A heavy object is then hung from the spring and the display on the device reads W, the weight of the spring plus the weight of the object, as both sit at rest. The object is then pulled downward a small distance and released. The object then moves in simple harmonic motion. What is the behavior of the display on the device as the object moves? a) The force remains constant while the object oscillates. b) The force varies between W and +W while the object oscillates. c) The force varies between a value near zero newtons and W while the object oscillates. d) The force varies between a value near zero newtons and 2W while the object oscillates. e) The force varies between W and 2W while the object oscillates.

10. 3. 1. An ideal spring is hung vertically from a fixed support 10.3.1. An ideal spring is hung vertically from a fixed support. When an object of mass m is attached to the end of the spring, it stretches by a distance y. The object is then lifted and held to a height y +A, where A << y. Which one of the following statements concerning the total potential energy of the object is true? a) The total potential energy will be equal to zero joules. b) The total potential energy will decrease and be equal to the gravitational potential energy of the object. c) The total potential energy will decrease and be equal to the elastic potential energy of the spring. d) The total potential energy will decrease and be equal to the sum of elastic potential energy of the spring and the gravitational potential energy of the object. e) The total potential energy will increase and be equal to the sum of elastic potential energy of the spring and the gravitational potential energy of the object.

10.3.2. A block is attached to the end of a horizontal ideal spring and rests on a frictionless surface. The other end of the spring is attached to a wall. The block is pulled away from the spring’s unstrained position by a distance x0 and given an initial speed of v0 as it is released. Which one of the following statements concerning the amplitude of the subsequent simple harmonic motion is true? a) The amplitude will depend on whether the initial velocity of the block is in the +x or the x direction. b) The amplitude will be less than x0. c) The amplitude will be equal to x0. d) The amplitude will be greater than x0. e) The amplitude will depend on whether the initial position of the block is in the +x or the x direction relative to the unstrained position of the spring.

10. 3. 5. Block A has a mass m and block B has a mass 2m 10.3.5. Block A has a mass m and block B has a mass 2m. Block A is pressed against a spring to compress the spring by a distance x. It is then released such that the block eventually separates from the spring and it slides across a surface where the friction coefficient is µk. The same process is applied to block B. Which one of the following statements concerning the distance that each block slides before stopping is correct? a) Block A slides one-fourth the distance that block B slides. b) Block A slides one-half the distance that block B slides. c) Block A slides the same distance that block B slides. d) Block A slides twice the distance that block B slides. e) Block A slides four times the distance that block B slides.

Pendulum Motion

The forces on the bob are the weight mg and the tension T. The restoring force is the tangential component of the weight, -mg sin θ . Use the radial and tangential directions for axes. The radial force is T-mg cos θ. What does the radial force have to add up to? Pendulum Motion

Pendulum Motion For small angles, The restoring force is The motion is SHM Pendulum Motion

The mass cancels out when Newton’s 2nd Law is applied: Pendulum Motion

Pendulum Motion The frequency and period are Essential Knowledge 3.B.3.b. For a simple pendulum oscillating the period increases with the length of the pendulum. Pendulum Motion

From Holt Physics © Holt, Rinehart

10.4.1. You would like to use a simple pendulum to determine the local value of the acceleration due to gravity, g. Consider the following parameters: (1) pendulum length, (2) mass of the object at the free end of the pendulum, (3) the period of the pendulum as it swings in simple harmonic motion, (4) the amplitude of the motion. Which of these parameters must be measured to find a value for g? a) 1 only b) 2 only c) 3 and 4 only d) 1 and 3 only e) 1, 2, and 4 only

10. 4. 2. At the surface of Mars, the acceleration due to gravity is 3 10.4.2. At the surface of Mars, the acceleration due to gravity is 3.71 m/s2. On Earth, a pendulum that has a period of one second has a length of 0.248 m. What is the length of a pendulum on Mars that oscillates with a period of one second? a) 0.0940 m b) 0.143 m c) 0.248 m d) 0.296 m e) 0.655 m