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Homework Question 3.44) Suppose that the position function of a particle undergoing Simple Harmonic Motion follows from the equation: y*+ n'y=0. A) Write down the general solution for this particular case, this time using the form (GS2). (No work, really...) B) If at time r=2 the particle has position y(2)=0 and velocity y(2)=5n, find the equation of motion y(t) of the particle. Hint: When using the given conditions on position and velocity to find the appropriate A and 4. remember that cosine and sine are periodic of period 2m. C) Give the amplitude, the frequency, the period, and the phase of the motion. Examples of Simple Harmonic Motion in nature are very numerous; at least, if we are willing to count motions that are approximately simply harmonic. Anything turning in a circle at constant speed has simply harmonic coordinates, the voltage of alternating current in your home outlets is harmonic (it is sinusoidal), the pistons in your motor engine closely follow harmonic motion (they are attached at one end to a rotating piece, like the pistons of old vapor trains were attached to the large wheels), the amplitudes of the fields of electromagnetic waves such as those sent by radio stations are superpositions of simple harmonic parts, etc. Well known examples from elementary physics include pendulum motion (to good approximation when the oscillation is small) as well as oscillating masses attached on springs when friction is negligible. We describe this latter. In the mid-seventeenth century, Robert Hooke discovered that under regular conditions, the force exerted by a spring can be approximated as being proportional to its extension or compression: F=-kx, (F) where k>0 is a constant intrinsic to the spring called the spring constant or spring modulus, and x is the displacement of the end of the spring from its equilibrium position. Equation (F) is called Hooke's Law and is valid for both positive and negative displacements x from equilibrium, that is, for both extensions and compressions of the spring. Equilibrium. Extension Immig x=0 F=-kxco Im my Compression. F=-kxxb fmmy X

Homework Question 3.44) Suppose that the position function of a particle undergoing Simple Harmonic Motion follows from the equation: y*+ n'y=0. A) Write down the general solution for this particular case, this time using the form (GS2). (No work, really...) B) If at time r=2 the particle has position y(2)=0 and velocity y(2)=5n, find the equation of motion y(t) of the particle. Hint: When using the given conditions on position and velocity to find the appropriate A and 4. remember that cosine and sine are periodic of period 2m. C) Give the amplitude, the frequency, the period, and the phase of the motion. Examples of Simple Harmonic Motion in nature are very numerous; at least, if we are willing to count motions that are approximately simply harmonic. Anything turning in a circle at constant speed has simply harmonic coordinates, the voltage of alternating current in your home outlets is harmonic (it is sinusoidal), the pistons in your motor engine closely follow harmonic motion (they are attached at one end to a rotating piece, like the pistons of old vapor trains were attached to the large wheels), the amplitudes of the fields of electromagnetic waves such as those sent by radio stations are superpositions of simple harmonic parts, etc. Well known examples from elementary physics include pendulum motion (to good approximation when the oscillation is small) as well as oscillating masses attached on springs when friction is negligible. We describe this latter. In the mid-seventeenth century, Robert Hooke discovered that under regular conditions, the force exerted by a spring can be approximated as being proportional to its extension or compression: F=-kx, (F) where k>0 is a constant intrinsic to the spring called the spring constant or spring modulus, and x is the displacement of the end of the spring from its equilibrium position. Equation (F) is called Hooke's Law and is valid for both positive and negative displacements x from equilibrium, that is, for both extensions and compressions of the spring. Equilibrium. Extension Immig x=0 F=-kxco Im my Compression. F=-kxxb fmmy X

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Integrative Activity Simple Harmonic Motion This small self-learning project is a mandatory part of the course. Suppose we want to find a function y = f(t) such that its derivatives would satisfy the equation y"+o²y=0 (DE) where a (a Greek lowercase letter called omega) is a positive constant. Such an equation involving a function y = f(t) and its derivatives together is called a differential equation. Solving a differential equation does not consist in finding a number that fits an equation, the goal is rather to find a function that fits an equation. In the above, the goal is to find which function y = f(t) satisfies equation (DE). In general, finding functions that solve a differential equation is quite hard. We sometimes offer a course on differential equations in your last term! For the specific differential equation y"+²y=0 that we have at hand, a solution is not too difficult. As you will verify below in question 3.41B, any function of the form y(t)= acos(at)+b sin(at) (GS1) where a and b are arbitrary constants, does satisfy differential equation (DE). Note that y(t) = acos(at)+bsin(at) represents infinitely many possible functions due to the infinitely many values that the arbitrary constants a and b can take. Moreover, it is shown in more advanced mathematics courses that there are no other possible functions satisfying equation (DE), that is, the infinitely many functions of equation (S1) give precisely all the possible functions that are solutions of the differential equation (DE). We call (GS1) a general solution of (DE). With some trigonometry, it can be shown that another way to write the general solution of (DE) is (GS2) y(t) = Acos(at + p) where A and are constants and A is taken positive. As you will see in question 3.42 below, one may switch between the two equivalent forms (GS1) and (GS2) of the general solution. There are many physical situations that lead to a differential equation such as (DE), called the (differential) equation of Simple Harmonic Motion; we will see one such example below. A system whose position as a function of time must satisfy the equation of Simple Harmonic Motion (DE), that is, whose position as a function of time will be given by a particular instance of equations (GS1) or (GS2), is said to undergo Simple Harmonic Motion. Note that in (GS2) corresponds to a "phase shift" (reason for the choice of the greak letter phi) compared to where the usual cosine function starts: dose Pift Amplitude RTY To aut
Transcribed Image Text:Integrative Activity Simple Harmonic Motion This small self-learning project is a mandatory part of the course. Suppose we want to find a function y = f(t) such that its derivatives would satisfy the equation y"+o²y=0 (DE) where a (a Greek lowercase letter called omega) is a positive constant. Such an equation involving a function y = f(t) and its derivatives together is called a differential equation. Solving a differential equation does not consist in finding a number that fits an equation, the goal is rather to find a function that fits an equation. In the above, the goal is to find which function y = f(t) satisfies equation (DE). In general, finding functions that solve a differential equation is quite hard. We sometimes offer a course on differential equations in your last term! For the specific differential equation y"+²y=0 that we have at hand, a solution is not too difficult. As you will verify below in question 3.41B, any function of the form y(t)= acos(at)+b sin(at) (GS1) where a and b are arbitrary constants, does satisfy differential equation (DE). Note that y(t) = acos(at)+bsin(at) represents infinitely many possible functions due to the infinitely many values that the arbitrary constants a and b can take. Moreover, it is shown in more advanced mathematics courses that there are no other possible functions satisfying equation (DE), that is, the infinitely many functions of equation (S1) give precisely all the possible functions that are solutions of the differential equation (DE). We call (GS1) a general solution of (DE). With some trigonometry, it can be shown that another way to write the general solution of (DE) is (GS2) y(t) = Acos(at + p) where A and are constants and A is taken positive. As you will see in question 3.42 below, one may switch between the two equivalent forms (GS1) and (GS2) of the general solution. There are many physical situations that lead to a differential equation such as (DE), called the (differential) equation of Simple Harmonic Motion; we will see one such example below. A system whose position as a function of time must satisfy the equation of Simple Harmonic Motion (DE), that is, whose position as a function of time will be given by a particular instance of equations (GS1) or (GS2), is said to undergo Simple Harmonic Motion. Note that in (GS2) corresponds to a "phase shift" (reason for the choice of the greak letter phi) compared to where the usual cosine function starts: dose Pift Amplitude RTY To aut
Homework Question 3.44) Suppose that the position function of a particle undergoing Simple Harmonic Motion follows from the equation: y"+ ²y = 0. A) Write down the general solution for this particular case, this time using the form (GS2). (No work, really...) B) If at time =2 the particle has position y(2)=0 and velocity y'(2)=5n, find the equation of motion y(t) of the particle. Hint: When using the given conditions on position and velocity to find the appropriate A and 4, remember that cosine and sine are periodic of period 2m. C) Give the amplitude, the frequency, the period, and the phase of the motion. Examples of Simple Harmonic Motion in nature are very numerous; at least, if we are willing to count motions that are approximately simply harmonic. Anything turning in a circle at constant speed has simply harmonic coordinates, the voltage of alternating current in your home outlets is harmonic (it is sinusoidal), the pistons in your motor engine closely follow harmonic motion (they are attached at one end to a rotating piece, like the pistons of old vapor trains were attached to the large wheels), the amplitudes of the fields of electromagnetic waves such as those sent by radio stations are superpositions of simple harmonic parts, etc. Well known examples from elementary physics include pendulum motion (to good approximation when the oscillation is small) as well as oscillating masses attached on springs when friction is negligible. We describe this latter. In the mid-seventeenth century, Robert Hooke discovered that under regular conditions, the force exerted by a spring can be approximated as being proportional to its extension or compression: F = -kx, (F) where k>0 is a constant intrinsic to the spring called the spring constant or spring modulus, and x is the displacement of the end of the spring from its equilibrium position. Equation (F) is called Hooke's Law and is valid for both positive and negative displacements x from equilibrium, that is, for both extensions and compressions of the spring. Equilibrium. Extension Immig -X=0 bo F=-kxco मल Compression. Fe-kx - x ८० AUT The negative sign in (F) indicates that the spring force F is a restoring force, that is, the force is always in the direction that tends to restore the spring to its equilibrium length. It is also worth insisting that Hooke's Law is only an approximation, valid for small displacements only and under the assumption of negligible friction. When the spring is stretched or compressed too much, the approximate proportionality between F and x is no longer appropriate.
Transcribed Image Text:Homework Question 3.44) Suppose that the position function of a particle undergoing Simple Harmonic Motion follows from the equation: y"+ ²y = 0. A) Write down the general solution for this particular case, this time using the form (GS2). (No work, really...) B) If at time =2 the particle has position y(2)=0 and velocity y'(2)=5n, find the equation of motion y(t) of the particle. Hint: When using the given conditions on position and velocity to find the appropriate A and 4, remember that cosine and sine are periodic of period 2m. C) Give the amplitude, the frequency, the period, and the phase of the motion. Examples of Simple Harmonic Motion in nature are very numerous; at least, if we are willing to count motions that are approximately simply harmonic. Anything turning in a circle at constant speed has simply harmonic coordinates, the voltage of alternating current in your home outlets is harmonic (it is sinusoidal), the pistons in your motor engine closely follow harmonic motion (they are attached at one end to a rotating piece, like the pistons of old vapor trains were attached to the large wheels), the amplitudes of the fields of electromagnetic waves such as those sent by radio stations are superpositions of simple harmonic parts, etc. Well known examples from elementary physics include pendulum motion (to good approximation when the oscillation is small) as well as oscillating masses attached on springs when friction is negligible. We describe this latter. In the mid-seventeenth century, Robert Hooke discovered that under regular conditions, the force exerted by a spring can be approximated as being proportional to its extension or compression: F = -kx, (F) where k>0 is a constant intrinsic to the spring called the spring constant or spring modulus, and x is the displacement of the end of the spring from its equilibrium position. Equation (F) is called Hooke's Law and is valid for both positive and negative displacements x from equilibrium, that is, for both extensions and compressions of the spring. Equilibrium. Extension Immig -X=0 bo F=-kxco मल Compression. Fe-kx - x ८० AUT The negative sign in (F) indicates that the spring force F is a restoring force, that is, the force is always in the direction that tends to restore the spring to its equilibrium length. It is also worth insisting that Hooke's Law is only an approximation, valid for small displacements only and under the assumption of negligible friction. When the spring is stretched or compressed too much, the approximate proportionality between F and x is no longer appropriate.
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