Simple Harmonic Oscillations Due: 12:00pm on Thursday, September 8, 2011 Note: You will receive no credit for late submissions. To learn more, read your instructor's Grading Policy [Switch to Standard Assignment View] View]
Good Vibes: Introduction to Oscillations Learning Goal: To learn the basic terminology and relationships among the main characteristics of simple harmonic motion.
Motion that repeats itself over and over is called periodic called periodic motion. motion. There are many examples of periodic motion: the earth revolving around the sun, an elastic ball bouncing up and down, or a block attached to a spring oscillating back and forth. The last example differs from the first two, in that it represents a special kind of periodic motion called simple called simple harmonic motion. motion. The conditions that lead to simple harmonic motion are as follows:
There must be a position of stable stable equilibrium. equilibrium. There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must b e directly proportional to the magnitude of the object's displacement from its equilibrium equilibrium position. Mathematically, the restoring force
is given by
, where
is the
displacement from from equilibrium and is a constant that depends on the properties of the oscillating system. The resistive forces in the system must be reasonably small.
In this problem, we will introduce some of the basic quantities that describe oscillations and the relationships among them. Consider a block of mass
attached to a spring with force constant , as shown in the figure
. The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at released,
. If the block is pulled to the right a distance
and then
will be the amplitude of the resulting oscillations.
Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block. Part A
After the block is released from ANSWER:
, it will
remain at rest. move to the left until it reaches equilibrium and stop there. move to the left until it reaches
and stop there.
move to the left until it reaches right.
and then begin to move to the
Correct As the block begins its motion to the left, it accelerates. Although the restoring force decreases as the block approaches equilibrium, it still pulls the block to the left, so by the time the equilibrium position is reached, the block has gained some speed. It will, therefore, pass the equilibrium position and keep moving, compressing the spring. The spring will now be pushing the block to the right, and the block will slow down, temporarily coming to rest at After
. is reached, the block will begin its motion to the right, pushed by the spring.
The block will pass the equilibrium position and continue until it reaches , completing one cycle of motion. The motion will then repeat; if, as we've assumed, there is no friction, the motion will repeat indefinitely. The time it takes the block to complete one cycle is called the period . Usually, the period is denoted
and is measured in seconds.
The frequency, denoted , is the number of cycles that are completed per unit of time: . In SI units, is measured in inverse seconds, or hertz (
).
Part B
If the period is doubled, the frequenc y is ANSWER:
unchanged. doubled. halved. Correct
Part C
An oscillating object takes 0.10 to complete one cycle; that is, its period is 0.10 . What is its frequency ? Express your answer in hertz. ANSWER:
10 = Correct
Part D
If the frequency is 40 , what is the period Express your answer in seconds. ANSWER:
0.025 = Correct
?
The following questions refer to the figure that graphically depicts the oscillations of the block on the spring. Note that the vertical axis represents the x coordinate of the oscillating object, and the horizontal axis represents time. Part E
Which points on the x axis are located a distance ANSWER:
from the equilibrium position?
R only Q only both R and Q Correct
Part F
Suppose that the period is time interval ANSWER:
? K and L K and M K and P L and N M and P
. Which of the following points on the t axis are separated by the
Correct Now assume that the x coordinate of point R is 0.12 0.0050 .
and the t coordinate of point K is
Part G
What is the period
?
Hint G.1 How to approach the problem
In moving from the point
to the point K, what fraction of a full wavelength is covered?
Call that fraction . Then you can set period
. Dividing by the fraction will give the
.
Express your answer in seconds. ANSWER:
0.02 = Correct
Part H
How much time does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement? Express your answer in seconds. ANSWER:
0.01 = Correct
Part I
What distance does the object cover during one period of oscillation? Express your answer in meters. ANSWER:
0.48 = Answer Requested
Part J
What distance does the object cover between the moments labeled K and N on the graph? Express your answer in meters. ANSWER:
0.36 = Correct
Harmonic Oscillator Acceleration Learning Goal: To understand the application of the general harmonic equation to finding the acceleration of a spring oscillator as a fun ction of time.
One end of a spring with spring constant is attached to the wall. The other end is attached to
a block of mass
. The block rests on a frictionless horizontal surface. The equilibrium
position of the left side of the block is defined to be
. The length of the relaxed spring is
. The block is slowly pulled from its equilibrium position to some position axis. At time
, the block is released with zero initial velocity.
The goal of this problem is to determine the acceleration of the block in terms of ,
along the x
, and
as a function of time
.
It is known that a general solution for the position of a harmonic oscillator is ,
where
,
, and
are constants.
Your task, therefore, is to determine the values of
,
, and
in terms of ,
,and
and
then use the connection between
and
to find the acceleration.
Part A
Combine Newton's 2nd law and Hooke's law for a spring to find the acceleration of the block as a function of time. Hint A.1 Physical laws H in t not displayed
Express your answer in terms of ,
, and the coordinate of the block
.
ANSWER:
= Correct The negative sign in the answer is important: It indicates that the restoring force (the tension of the spring) is always directed opposite to the block's displacement. When the block is pulled to the right from the equilibrium position, the restoring force is pulling back, that is, to the left--and vice versa. Part B
Using the fact that acceleration is the second derivative of position, find the acceleration of the block
as a function of time.
Express your answer in terms of
, , and
.
ANSWER:
=
Correct
Part C
Find the angular frequency
.
Hint C.1 Using the previous results H in t not displayed
Express your answer in terms of and
.
ANSWER:
= Correct Note that the angular frequency
and, therefore, the period of oscillations
dependonly on
the intrinsic physical characteristics of the system ( and ). Frequency and period do not depend on the initial conditions or the amplitude of the motion.
Harmonic Oscillator Kinematics
Learning Goal: To understand the application of the general harmonic equation to the kinematics of a spring oscillator.
One end of a spring with spring constant is attached to the wall. The other end is attached to a block of mass . The block rests on a frictionless horizontal surface. The equilibrium position of the left side of the block is defined to be
. The length of the relaxed spring is
. The block is slowly pulled from its equilibrium position to some position axis. At time
along the x
, the block is released with zero initial velocity.
The goal is to determine the position of the block .
as a function of time in terms of
It is known that a general solution for the displacement from equilibrium of a harmonic oscillator is ,
and
where
,
, and
are constants.
Your task, therefore, is to determine the values of
and
in terms of
and
.
Part A
Using the general equation for position of the block
given in the problem introduction, express the initial
in terms of
,
, and
(Greek letter omega).
Hint A.1 Consider
Evaluate the general expression for
when
.
Hint A.2 Some useful trigonometry
Recall that ANSWER:
and =
.
Correct
This result is a good first step. The constant of the block. What about
in this case is simply
? To find the relationship between
consider another initial condition that we know: At
, the initial position
and other variables, let us
, the velocity of the block is zero.
Part B
Find the value of
using the given condition that the initial velocity of the block is zero:
. Hint B.1 How to approach the problem H in t not displayed
Hint B.2 Differentiating harmonic functions H in t not displayed
ANSWER:
0 Correct Part C
What is the equation
for the block?
Hint C.1 Start with the general solution
Use the general solution and the values for
and
Express your answer in terms of ,
.
, and
obtained in the previous parts.
ANSWER:
=
Correct
In this problem the initial velocity is zero, so the quantity is the maximum displacement of the block from the equilibrium position. The magnitude of the maximum displacement is called the amplitude, often denoted rewritten as
. Using this notation, the formula for
can be
. Now, imagine that we have exactly the same physical situation but that the x axis is translated, so that the position of the wall is now defined to be
.
The initial position of the block is the same as before, but in the new coordinate system, the block's starting position is given by
.
Part D
Find the equation for the block's position
in the new coordinate system.
Hint D.1 Equilibrium position
Changing the origin of the coordinate system has no effect on the physical parameters of the problem (e.g., the frequency or the amplitude of the block's oscillations). The initial velocity is still zero. The only difference is that now the block is oscillating around whereas before it was oscillating around between
. What is the difference, at any moment ,
in the new coordinate system and
in the old coordinate system?
ANSWER:
= Correct Use this relationship and the expression for system, to derive
, the block's position in the old coordinate
.
Express your answer in terms of
,
,
(Greek letter omega), and .
ANSWER:
=
Correct
Position, Velocity, and Acceleration of an Oscillator Learning Goal: To learn to find kinematic variables from a graph of position vs. time. The graph of the position of an oscillating object as a function of time is shown.
Some of the questions ask you to determine ranges on the graph over which a statement is
true. When answering these questions, choose the most complete answer. For example, if the answer "B to D" were correct, then "B to C" would technically also be correct--but you will only recieve credit for choosing the most complet e answer. Part A
Where on the graph is ANSWER:
?
A to B A to C C to D C to E B to D A to B and D to E Correct
Part B
Where on the graph is ANSWER:
?
A to B A to C C to D C to E B to D A to B and D to E Correct
Part C
Where on the graph is ANSWER:
A only C only
?
E only A and C A and C and E B and D Correct Part D
Where on the graph is the velocity Hint D.1
?
Finding instantaneous velocity H in t not displayed
ANSWER:
A to B A to C C to D C to E B to D A to B and D to E Correct
Part E
Where on the graph is the velocity ANSWER:
A to B A to C C to D C to E B to D A to B and D to E
?
Correct Part F
Where on the graph is the velocity Hint F.1
?
How to tell if H in t not displayed
ANSWER:
A only B only C only D only E only A and C A and C and E B and D
Correct Part G
Where on the graph is the acceleration Hint G.1
?
Finding acceleration H in t not displayed
ANSWER:
A to B A to C C to D C to E B to D A to B and D to E
Correct Part H
Where on the graph is the acceleration ANSWER:
?
A to B A to C C to D C to E B to D A to B and D to E Correct
Part I
Where on the graph is the acceleration Hint I.1
?
How to tell if H in t not displayed
ANSWER:
A only B only C only D only E only A and C A and C and E B and D Correct
Relating Two General Simple Harmonic Motion Solutions Learning Goal: To understand how the two standard ways to write the general solution to a harmonic oscillator are related. There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t:
1.
and
2.
.
Either of these equations is a general solution o f a second-order differential equation ( ); hence both must have at least two--arbitrary constants--parameters that can be adjusted to fit the solution to the particular motion at hand. (Some texts refer to these arbitrary constants as boundary values.) Part A
What are the arbitrary constants in Equation 1? Hint A.1 What is considered a constant?
A constant is something that is defined by the physical situation under consideration (e.g., a spring constant, acceleration due to gravity, frequency of oscillation, or mass) and it does not change even if the motion is different owing to different initial conditions. Hint A.2 What is considered arbitrary?
Arbitrary constants are used to "fit" the general solution to a particular set of initial conditions, such as how far you pull the oscillator from its equilibrium position, and how fast it is moving when you let it go. ANSWER:
only only and and and and Correct
Part B
and
What are the arbitrary constants in Equation 2? ANSWER:
only only only and and and
Correct Because both Equation 1 and Equation 2 are general solutions, they can both represent any set of initial conditions (i.e., initial position and velocity). Therefore, one equation could be expressed in terms of the other. Part C
Find analytic expressions for the arbitrary constants in terms of the constants as given parameters.
and
in Equation 2 (found in Part B)
and in Equation 1 (found in Part A), which are now considered
Hint C.1 A useful trig identity
What is the angle-sum trigonometric identity for ? Hint: If you can remember the general form but are unsure of the sign or whether a particular term is cos or sin, try your expression for simple values like 0 and/or Give your answer in terms of
,
. ,
, and
.
ANSWER:
=
Correct
Hint C.2 How to use the trig indentity
The left side of the equation,
, is in the form of Equation 1, whereas the right side
is in the form of Equation 2. If you make a proper substitution for and first on the left and then correspondingly on the right side you can solve for the arbitrary constants in Equation 2. Give your answers for the coefficients of your answers in terms of
and .
and
, separated by a comma. Express
ANSWER:
,
=
, Answer Requested
Part D
Find analytic expressions for the arbitrary constants in terms of the constants as given parameters.
and
in Equation 2 (found in Part B), which are now considered
Hint D.1 Find a relationship between
Examine the sum
and in Equation 1 (found in Part A)
,
and
where you substitute the preceding answers for
and
.
Hint D.1.1 Useful trig identity
Recall that
.
Give your answer in terms of
and other given quantities.
ANSWER:
=
Correct
Hint D.2 Useful trig indentityfor finding
Try using
. Of course you don't want functions of on the right; however, you do want right. Express the amplitude
and phase (separated by a comma) in terms of
and
and
on the
.
ANSWER:
,
=
, Answer Requested
This problem was very mathematical. To understand its utility, realize that for a typical mechanical oscillator, the variables have the following meaning:
is the amplitude of oscillation, is the initial phase angle, is the initial position at
, and
is the initial velocity at
divided by
.
Therefore, if you are given the initial amplitude and phase, you can find the initial position and velocity. Similarly, if you are given the initial position and velocity you can find the initial amplitude (whose square is related to the total energ y) and the phase angle, which permits you to answer questions like "When does the particle reach its maximum displacement?" or "When does the particle first return to
?"
Analyzing Simple Harmonic Motion This applet shows two masses on springs, each accompanied b y a graph of its position versus time. Part A
What is an expression for , the position of mass I as a function of time? Assume that position is measured in meters and time is measured in seconds. Hint A.1 How to approach the problem H in t not displayed
Hint A.2 Find the amplitude H in t not displayed
Hint A.3 Find the angular frequency H in t not displayed
Express your answer as a function of . Express numerical constants to three significant figures. ANSWER:
=
Correct
Part B
What is , the position of mass II as a function of time? Assume that position is measured in meters and time is measured in seconds. Hint B.1 How to approach the problem H in t not displayed
Hint B.2 Find the amplitude H in t not displayed
Hint B.3 Find the angular frequency H in t not displayed
Express your answer as a function of . Express numerical constants to three significant figures. ANSWER:
=
Correct
Period of a Mass-Spring System Ranking Task Different mass crates are placed on top of springs of uncompressed length
and stiffness .
The crates are released and the springs compress to a length
before bringing the crates back up to their original positions.
Part A
Rank the time required for the crates to return to their initial positions from largest to smallest. Hint A.1 Formula for the period
The period is defined as the time it takes for an oscillator to go through one complete cycle of its motion. Therefore, the time for each crate to return to its initial position is one period. The period of a mass-spring system is given by
. Therefore, if can be determined from the provided information, a ranking can be determined. If cannot be determined, the ranking cannot be determined based on the information provided. Hint A.2 Determining the mass
At equilibrium, the force of the spring upward is equal to the force of gravity downward: .
Solving for the mass we get
. Since the crate oscillates with equal amplitude above and below the equilibrium position, the compression of the spring at equilibrium is one -half the total distance the crate falls before beginning to move back upward; that is,
. Combining these two ideas results in
. Expressing in terms of known quantities, and substituting mass into the period formula, will allow you to determine the correct ranking. Hint A.3 Determining
As defined in the problem, is the uncompressed length of the spring and is the maximum compression of the spring. The total distance the crate falls before beginning to move back upward is given by . Rank from largest to smallest. To rank items as equivalent, overlap them. ANSWER:
View Correct
Simple Harmonic Motion Conceptual Question An object of mass is attached to a vertically oriented spring. The object is pulled a short distance below its equilibrium position and released from rest. Set the origin of the coordinate system at the equilibrium position of the object and choose upward as the positive direction. Assume air resistance is so small that it can be ignored.
Refer to these graphs following questions.
when answering the
Part A
Beginning the instant the object is released, select the graph that best matches the position vs. time graph for the object. Hint A.1 How to approach the problem H in t not displayed
Hint A.2 Find the initial position H in t not displayed
ANSWER:
Correct
Part B
Beginning the instant the object is released, select the graph that best matches the velocity vs. time graph for the object. Hint B.1 Find the initial velocity H in t not displayed
Hint B.2 Find the velocity a short time later H in t not displayed
ANSWER:
Correct
Part C
Beginning the instant the object is released, select the graph that best matches the acceleration vs. time graph for the object. Hint C.1
Find the initial acceleration H in t not displayed
ANSWER:
Correct
± The Fish Scale A vertical scale on a spring balance reads from 0 to 215 from the 0 to 215
. The scale has a length of 13.0
reading. A fish hanging from the bottom of the spring oscillates vertically
at a frequency of 2.00
.
Part A
Ignoring the mass of the spring, what is the mass
of the fish?
Hint A.1 How to approach the problem H in t not displayed
Hint A.2 Calculate the spring constant H in t not displayed
Hint A.3 Calculate the angular frequency H in t not displayed
Hint A.4 Formula for the angular frequency of a mass on a spring H in t not displayed
Express your answer in kilograms. ANSWER:
=
10.5 Correct
Exercise 13.12 A 2.00-kg, frictionless block is attached to an ideal spring with force constant 300 the block has velocity -4.00
and displacement +0.200
Part A
Find (a) the amplitude and (b) the phase angle. ANSWER:
0.383 = Correct
Part B ANSWER:
1.02 = Correct
Part C
Write an equation for the position as a function of time. Assume
in meters and in seconds.
.
. At
ANSWER:
=
Correct
Exercise 13.27 You are watching an object that is moving in SHM. When the object is displaced 0.600 the right of its equilibrium position, it has a velocity of of
to
to the right and an acceleration
to the left.
Part A
How much farther from this point will the object move before it stops momentarily and then starts to move back to the left? ANSWER:
0.240 Correct
Test Your Understanding 13.1: Describing Oscillation An object oscillates back and forth along the -axis. Its equilibrium position is at
.
Part A
At an instant when the object is at and -velocity? ANSWER:
, what are the signs of the object's -acceleration
and and ; not enough information is given to determine the sign of and and and ; not enough information is given to determine the sign of and not enough information is given to determine the sign of
or
Correct In oscillation, the sign of the x-acceleration is always the opposite of the sign o f the displacement. Since , it must be that . We are not told whether the object is moving in the positive -direction, moving in the negative -direction, or instantaneously at rest, so we can't say anything about the sign of the -velocity .
Test Your Understanding 13.2: Simple Harmonic Motion An object oscillates back and forth along the -axis in simple harmonic motion. Its displacement as a function of time is given by positive -acceleration.
. At time
the object has a
Part A
Which of the following is a possible value of the phase angle ANSWER:
?
radians
radians radians more than one of the above Correct The x-acceleration is the second derivative of the displacement:
At t = 0, the x-acceleration is
The amplitude A is positive, so in order to have that
,
,
it must be true that
, and
. Note
. Hence the only
possible answer among the choices provided is
.
Exercise 13.20 An object is undergoing SHM with period 0.255 is instantaneously at rest at 6.20 .
and amplitude 6.20
. At
Part A
Calculate the time it takes the object to go from ANSWER:
6.20
−
7.30×10 = Correct
Score Summary: Your score on this assignment is 94.7%. You received 123.14 out of a possible total of 130 points.
to
-1.40
.
the object
