Ch 10 SHM & Elasticity

Chapter 10 Simple Harmonic Motion and Elasticity

Learning Objectives Oscillations Simple harmonic motion (dynamics and energy relationships) Students should understand simple harmonic motion, so they can: Sketch or identify a graph of displacement as a function of time, and determine from such a graph the amplitude, period, and frequency of the motion. Write down an appropriate expression for displacement of the form A sin(  t) or A cos(  t) to describe the motion. State the relations between acceleration, velocity, and displacement, and identify points in the motion where these quantities are zero or achieve their greatest positive and negative values. State and apply the relation between frequency and period. State how the total energy of an oscillating system depends on the amplitude of the motion, sketch or identify a graph of kinetic or potential energy as a function of time, and identify points in the motion where this energy is all potential or all kinetic. Calculate the kinetic and potential energies of an oscillating system as functions of time, sketch or identify graphs of these functions, and prove that the sum of kinetic and potential energy is constant.

Learning Objectives Mass on a spring Students should be able to apply their knowledge of simple harmonic motion to the case of a mass on a spring, so they can: Apply the expression for the period of oscillation of a mass on a spring. Analyze problems in which a mass hangs from a spring and oscillates vertically. Analyze problems in which a mass attached to a spring oscillates horizontally. Pendulum and other oscillations Students should be able to apply their knowledge of simple harmonic motion to the case of a pendulum, so they can: Apply the expression for the period of a simple pendulum. State what approximation must be made in deriving the period .

Table of Contents Ideal Spring and Simple Harmonic Motion Simple Harmonic Motion and the Reference Circle Energy and Simple Harmonic Motion The Pendulum Damped Harmonic Motion Driven Harmonic Motion and Resonance Elastic Deformation (?) Stress, Strain, and Hooke’s Law

Chapter 10 Simple Harmonic Motion and Elasticity Section 1: Ideal Spring and Simple Harmonic Motion

Ideal Spring The more you try to stretch the spring The greater the force you will have to apply spring constant Units: N/m

Example 1 A Tire Pressure Gauge The spring constant of the spring is 320 N/m and the bar indicator extends 2.0 cm. What force does the air in the tire apply to the spring?

Ideal Spring A deformed spring held in place at location “x” is therefore not moving From Newton’s Laws, if the object’s motion is constant sum of the forces is zero Force spring exerts is equal and opposite to the force that is being applied

HOOKE’S LAW: RESTORING FORCE OF AN IDEAL SPRING The restoring force on an ideal spring is

10.1.1. A 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 . When an object of mass of 2 m is hung from the spring, it stretches by a distance 2 y . A second, identical spring is then attached to the free end of the first spring. If the object of mass 2 m is attached to the bottom of the second spring, how far will the bottom of the second spring move downward from its unstretched position? Assume the masses of the springs are negligible when compared to m . a) y /2 b) y c) 3 y /2 d) 2 y e) 4 y

10.1.2. A 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 . When an object of mass of 2 m is hung from the spring, it stretches by a distance 2 y . A second, identical spring is then attached to the fixed support as shown and a light rod is placed across the free ends of the springs. If the object of mass 2 m is attached to the middle of the rod, how far will the rod move downward? Assume the masses of the springs and the rod are negligible when compared to m . a) y /2 b) y c) 3 y /2 d) 2 y e) 4 y

10.1.3. A 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 . When an object of mass of 2 m is hung from the spring, it stretches by a distance 2 y . The same spring is then set up in a pulley system as shown in the drawing. Here an object of mass m is suspended from a light rope that passes over each pulley and connects to the two ends of the spring. By what distance is the spring stretched in this configuration? Express your answer in terms of the original stretch distance y . a) y /2 b) y c) 3 y /2 d) 2 y e) 4 y

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 2 W while the object oscillates. e) The force varies between W and 2 W while the object oscillates.

Chapter 10 Simple Harmonic Motion and Elasticity Section 2: Simple Harmonic Motion and the Reference Circle

period T: the time required to complete one cycle frequency f: the number of cycles per second (measured in Hz) amplitude A: the maximum displacement

Calculus Alert! Derivative of position with respect to time is speed Derivative of cosine function is –sine function Derivative of the cosine function is sine

Example 3 The Maximum Speed of a Loudspeaker Diaphragm The frequency of motion is 1.0 KHz and the amplitude is 0.20 mm. What is the maximum speed of the diaphragm? Where in the motion does this maximum speed occur? (a) The maximum speed occurs midway between the ends of its motion.

FINDING THE FREQUENCY OF VIBRATION

Example 6 A Body Mass Measurement Device The device consists of a spring-mounted chair in which the astronaut sits. The spring has a spring constant of 606 N/m and the mass of the chair is 12.0 kg. The measured period is 2.41 s. Find the mass of the astronaut.

10.2.1. A 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 . When an object of mass of 2 m is hung from the spring, it stretches by a distance 2 y . With the object of mass 2 m attached to the spring, the object is pulled slightly downward and released. It then moves in simple harmonic motion with a period T . This arrangement is used to determine the acceleration due to gravity, g . Express g in terms of the parameters given to show how this measurement may be accomplished. a) g = 4 my T  2 b) g = 4 my T 2 c) g = 2  y T 2 d) g = 4  y T  2 e) g = my T  2

10.2.2. A steel ball is hung from a vertical ideal spring where it oscillates in simple harmonic motion with an amplitude of 0.157 m and an angular frequency of  rad/s. Which one of the following expressions represents the acceleration, in m/s 2 , of the ball as a function of time? a) a =  1.55 cos(  t ) b) a =  1.55 cos 2 (  t ) c) a =  0.157 cos(  t ) d) a =  0.493 cos 2 (  t ) e) a =  0.493 cos(  t )

10.2.3. Object A is attached to ideal spring A and is moving in simple harmonic motion. Object B is attached to ideal spring B and is moving in simple harmonic motion. The period and the amplitude of object B are both two times the corresponding values for object A. How do the maximum speeds of the two objects compare? a) The maximum speed of A is one fourth that of object B. b) The maximum speed of A is one half that of object B. c) The maximum speed of A is the same as that of object B. d) The maximum speed of A is two times that of object B. e) The maximum speed of A is four times that of object B.

10.2.4. A steel ball is hung from a vertical ideal spring where it oscillates in simple harmonic motion with a period T . At time t = 0 s, the ball is at its maximum displacement, A , from its equilibrium position. In terms of the period, at what time will the ball be at y = 0.75 A ? a) 0.33 T b) 0.25 T c) 0.19 T d) 0.15 T e) 0.12 T

10.2.5. A 2.00-kg 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. After being displaced a small distance and released from rest, the block is observed to oscillate at a frequency of 2.77 Hz. The spring is then cut in half. One of the halves is then connected between the wall and the block. What will be the frequency when the block is displaced and released from rest again? a) 5.54 Hz b) 3.91 Hz c) 2.77 Hz d) 1.96 Hz e) 1.39 Hz

Chapter 10 Simple Harmonic Motion and Elasticity Section 3: Energy and Simple Harmonic Motion

A compressed spring can do work, but we need calculus to find it…

DEFINITION OF ELASTIC POTENTIAL ENERGY The elastic potential energy is the energy that a spring has by virtue of being stretched or compressed. For an ideal spring, the elastic potential energy is SI Unit of Elastic Potential Energy: joule (J)

Example 8 Changing the Mass of a Simple Harmonic Oscilator A 0.20-kg ball is attached to a vertical spring. The spring constant is 28 N/m. When released from rest, how far does the ball fall before being brought to a momentary stop by the spring?

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 x 0 and given an initial speed of v 0 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 x 0 . c) The amplitude will be equal to x 0 . d) The amplitude will be greater than x 0 . 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.

10.3.3. 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 x 0 and given an initial speed of v 0 as it is released. Which one of the following parameters must be known in addition to x 0 and v 0 to determine the amplitude of the subsequent simple harmonic motion? a) period b) spring constant c) mass of the block d) the direction of the initial velocity of the block e) the direction of the initial displacement of the block

10.3.4. 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 x 0 = 0.050 m and released from rest. The period of the subsequent periodic motion of the block is 0.64 s. At what distance from the unstrained position is the speed of the block equal to 0.30 m/s? a) 0.050 m b) 0.040 m c) 0.030 m d) 0.020 m e) 0.010 m

10.3.5. Block A has a mass m and block B has a mass 2 m . 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 µ k . 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.

10.3.6. In designing a spring loaded cannon, determine the spring constant required to launch a 2.0-kg ball with an initial speed of 1.2 m/s from a position where the spring is displaced 0.15 m from its equilibrium position. a) 16 N/m b) 32 N/m c) 64 N/m d) 130 N/m e) 180 N/m

Chapter 10 Simple Harmonic Motion and Elasticity Section 4: The Pendulum

Simple Pendulum A simple pendulum consists of a particle attached to a frictionless pivot by a cable of negligible mass.

Problem: Keeping Time Determine the length of a simple pendulum that will swing back and forth in simple harmonic motion with a period of 1.00 min.

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 3.71 m/s 2 . 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

10.4.3. As a pendulum swings in simple harmonic motion at the surface of the Earth, the angle the pendulum makes relative to its equilibrium position is given by  ( t ) = (0.140 rad) cos (5.72 t ) where t is in seconds. What is the length of this pendulum? a) 0.140 m b) 0.250 m c) 0.300 m d) 0.439 m e) 0.801 m

Chapter 10 Simple Harmonic Motion and Elasticity Section 5: Damped Harmonic Motion

Damped Harmonic Motion In simple harmonic motion, an object oscillated with a constant amplitude. In reality, friction or some other energy dissipating mechanism is always present and the amplitude decreases as time passes. This is referred to as damped harmonic motion .

Damped Harmonic Motion simple harmonic motion underdamped critically damped overdamped

10.5.1. A block is attached to the end of a horizontal ideal spring and rests on a rough surface. The other end of the spring is attached to a wall. The block is pulled away from the spring’s unstrained position and released from rest. The block begins oscillating, but the amplitude is observed to decrease with time. After a relatively short time, the block stops oscillating. For which of the following conditions, would the harmonic motion of this system be described as underdamped? a) The frictional force is greater than the elastic force. b) The frictional force is less than the elastic force. c) The work done by the frictional force is greater than the elastic potential energy. d) The kinetic energy of the block is greater than the elastic potential energy. e) This kind of system would not show damped harmonic motion.

10.5.2. The amplitude of a damped harmonic oscillator decreases two percent with each cycle. How much energy is lost to frictional forces during each cycle? a) 4.0 % b) 3.0 % c) 2.0 % d) 1.4 % e) 1.0 %

Chapter 10 Simple Harmonic Motion and Elasticity Section 6: Driven Harmonic Motion and Resonance

Driven Harmonic Motion When a force is applied to an oscillating system at all times, the result is driven harmonic motion . Here, the driving force has the same frequency as the spring system and always points in the direction of the object’s velocity.

RESONANCE Resonance is the condition in which a time-dependent force can transmit large amounts of energy to an oscillating object, leading to a large amplitude motion. Resonance occurs when the frequency of the force matches a natural frequency at which the object will oscillate.

10.6.1. For many centuries, soldiers have been trained to march side by side with their steps matching all of the other soldiers. At time t = 0 s, the soldiers all step with their left feet and continue marching with the same period. When they come upon a bridge, how should they change their marching, if at all? a) They should continue marching with the same period. b) Every other row of soldiers should take two steps for every one taken by the rows of soldiers ahead and behind them. c) Every other row of soldiers should take a step with their right foot when the rows of soldiers ahead and behind them are stepping with their left feet, but keep marching with the same period. d) They should all walk at their own pace so that they do not cause resonance on the bridge.

Chapter 10 Simple Harmonic Motion and Elasticity Section 7: Elastic Deformation AP? SKIP

Elastic Deformation Because of these atomic-level “springs”, a material tends to return to its initial shape once forces have been removed. ATOMS FORCES

STRETCHING, COMPRESSION, AND YOUNG’S MODULUS Young’s modulus has the units of pressure: N/m 2

Example 12 Bone Compression In a circus act, a performer supports the combined weight (1080 N) of a number of colleagues. Each thighbone of this performer has a length of 0.55 m and an effective cross sectional area of 7.7 ×10 -4 m 2 . Determine the amount that each thighbone compresses under the extra weight.

SHEAR DEFORMATION AND THE SHEAR MODULUS The shear modulus has the units of pressure: N/m 2

Example 14 J-E-L-L-O You push tangentially across the top surface with a force of 0.45 N. The top surface moves a distance of 6.0 mm relative to the bottom surface. What is the shear modulus of Jell-O?

VOLUME DEFORMATION AND THE BULK MODULUS The Bulk modulus has the units of pressure: N/m 2

10.7.1. While setting up a simple pendulum, Jon hangs a ball on one end of a steel wire of diameter d and attaches the other end to the ceiling. While the ball is stationary, Jon measures the length of the wire and finds that the length has increased by a distance y . Which one of the following options should Jon follow to minimize the amount the length changes? a) Cut the wire in half and use one half to hang the ball from the ceiling. b) Replace the wire with one of identical dimensions, but composed of tungsten. c) Replace the ball with one with one half the mass of the original ball. d) Connect the pendulum to a fixture that allows the pendulum to swing closer to the ground. e) Replace the wire with one made of the same material and of the same length, but with a diameter 2 d .

10.7.2. Two rods are made out of brass and have the same length. The cross section of one of the rods is circular with a diameter 2 a . The other rod has a square cross section, where each side of the square is a length 2 a . One end of the rods is attached to an immovable fixture which allows the rods to hang vertically. To the free end of each rod, a block of mass m is attached. Which rod, if either, will stretch more after the block is attached? a) The one with the circular cross section will stretch more. b) The one with the square cross section will stretch more. c) Both will stretch by the same amount. d) One cannot say which will stretch more without knowing the numerical values of a and m .

10.7.3. A cube made of brass ( B = 6.7 × 10 10 N/m 2 ) is taken by submarine from the surface where the pressure is 1.01 × 10 5 N/m 2 to the deepest part of the ocean at a depth of 1.10 × 10 4 m where it is exposed to a pressure is 1.25 × 10 8 N/m 2 . What is the percent change in volume as a result of this movement? a) 0.413 % b) 0.297 % c) 0.180 % d) 0.114 % e) This cannot be calculated without knowing the initial dimensions of the cube.

10.7.4. A 1.00-m long wire with a diameter of 0.02 m stretches by 0.03 m when a 20 000-N force is applied to one end as the other is held fixed. Consider an identical wire, except that it has a diameter of 0.04 m. If a 20 000-N force is applied to it, how much will this new wire stretch? a) 0.06 m b) 0.015 m c) 0.0075 m d) 0.12 m e) 0.03 m

10.7.5. A 1.00-m long wire with a diameter of 0.02 m stretches by 0.03 m when a 20 000-N force is applied to one end as the other is held fixed. Consider an identical wire, except that it has a length of 2.00 m. If a 20 000-N force is applied to it, how much will this new wire stretch? a) 0.06 m b) 0.015 m c) 0.0075 m d) 0.12 m e) 0.03 m

10.7.6. A cylindrical, 0.500-m rod has a diameter of 0.02 m. The rod is stretched to a length of 0.501 m by a force of 3000 N. What is the Young’s modulus of the material? a) 1.5 × 10 8 N/m 2 b) 1.2 × 10 9 N/m 2 c) 7.5 × 10 7 N/m 2 d) 4.8 × 10 9 N/m 2 e) More information is needed to calculate the Young’s modulus.

Chapter 10 Simple Harmonic Motion and Elasticity Section 8: Stress, Strain, and Hooke’s Law

In general the quantity F/A is called the stress . The change in the quantity divided by that quantity is called the strain: Stress is directly proportional to strain. F = -kx Strain is a unitless quantitiy. SI Unit of Stress: N/m 2 HOOKE’S LAW FOR STRESS AND STRAIN

Ch 10 SHM & Elasticity

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Ch 10 SHM & Elasticity