Keywords: Simple Harmonic Motion; Oscillations
Introduction:
Have you ever felt you were the slave of a clock? Clocks are mechanisms that include a pendulum or balance wheel whose repeated patterns of movement define equal time intervals, one after another. Such repeated movements are called periodic motion. Periodic motion may occur when a particle or body is confined to a limited region of space by the forces acting on it and does not have sufficient energy to escape.In this lesson you will study the special kind of periodic motion that results when the net force acting on a particle, often called the restoring force, is directly proportional to the particle's displacement from its equilibrium position; this is known as simple harmonic motion. Actually, simple harmonic motion is an idealization that applies only when friction, finite size, and other small effects in real physical systems are neglected. But it is a good enough approximation that it ranks in importance with other special kinds of motion (free fall, circular, and rotational motion) that you have already studied. Systems that can be analyzed in terms of simple harmonic oscillations include cars without shock absorbers, a child's swing, violin strings, and, more importantly, sound waves and certain electrical circuits that you will study in later lessons.
Readings on the InfoMall:
There is probably no topic in general physics of greater interest to physicists than simple harmonic motion. Knowing only a few facts about a SHM system, everything else can be computed, as you will soon learn. To prepare yourself for mastering SHM to the degree that will satisfy your instructor, you need to explore at least three different SHM topics on the InfoMall.
Topic One: Robert Hooke-find out about him as a person and relate his work to SHM. Give the citations for all the references you use.
Topic Two: Find a derviation of the equations for simple harmonic motion from Newton's second law. Paste them here and give the citation for their origin.
Topic Three: Find a derivation for the energy relationships for simple harmonic motion. Paste them here and give the citation for their origin.
11-1: HARMONIC OSCILLATIONS IN ONE DIMENSION
Keywords: Calculus; Period; Amplitude; Frequency; Fundamental Motion; Properties Of Waves; Simple Harmonic Motion; Newton's Second Law; Differential Equations
OBJECTIVES:
Write Newton's second law for a particle undergoing simple harmonic motion in one dimension and write the general solution for the resulting differential equation.
Define the following terms as they relate to a simple harmonic oscillator:
simple harmonic motion, amplitude, frequency (herz),
phase constant (or phase angle), angular frequency, period,
spring constant, restoring force.
Given the necessary information about a system oscillating harmonically in one dimension, solve for any of the following:
position as a function of time,
angular frequency,
period,
amplitude,
frequency,
velocity,
acceleration,
mass,
restoring force.
Commentary:
The general equation for simple harmonic motion along the x-axis results from a straightforward application of Newton's second law to a particle of mass m acted on by a force F = -kx, where x is the displacement from equilibrium.
Since the acceleration a = dv/dt = d2x/dt2,
Newton's second law becomes-kx = m d2x/dt2,
which is called a second-order differential equation because it contains a second derivative.
We can combine the constants k and m by making the substitution k/m = 2, and rewrite this equation as
d2x/dt2= -2x. (1)
Your calculus background may not have acquainted you with differential equations; hence, we will discuss them briefly here without getting too fancy or formal.
Equation (1) is not like an algebraic equation for which certain constant values of x satisfy the equality.
The solution of Eq. (1) is a function of the time.
Although the function
x = A cos(t + ) = A cos cos t - A sin sin t (2)
can be thought of simply as being arrived at by a very clever guess, it can be shown (by advanced mathematical techniques) to be the most general possible solution of Eq. (1).
Equation (2) can also be written in terms of two new constants B and C as
x(t) = B cos t + C sin t. (3)
(What are the relations among B, C, A, and ?)
The velocity is then
v(t) = dx(t)/dt = -B sin t + C cos t. (4)
These last two equations are especially helpful.
For instance,
if you are told that a particle begins its simple harmonic motion from rest at the point x0, you know that x(0) = x0 and v(0) = 0; hence, since cos(0) = 1 and sin(0) = 0 you immediately have B = x0 and C = 0.
If the particle starts at the origin(x=0) with velocity v0, then you can conclude that B = 0 and C = v0.
Look at the equations and check these results for yourself.
If you have a more complicated case in which the particle starts at x0 with velocity v0, then you can find B and C yourself, using the same method. Try it.
Once you have found B and C, you can then find A and .
All of the terms listed in the objectives for this section are defined in the readings in the different textbooks on the InfoMall.
Familiarize yourself with the symbols used to identify the parameters of harmonic motion, and recognize that different texts may use different symbols.
For example, you may see or (instead of ) used to represent the phase constant.The phase constant determines the initial conditions (displacement and velocity) of the motion. The significance of the angular nature of the phase constant should become somewhat clearer in the next section when we examine the analogy between harmonic motion and uniform circular motion. For now, notice that since is a constant, its value is arbitrary in the general solution of Eq. (1). Even when the initial conditions of the motion (at t = 0) are specified, is only determined to within an integral multiple of 2. That is, if x = A cos(t + ) describes the motion, then so does x = A cos (t + + 2n) for any integer n= 1, 2,3, ....
The equation x = A sin (t + '), with ' = + /2, is an equally valid solution as you can verify for yourself.
Be alert to the difference between frequency, , and angular frequency, .
Both can have dimensions of s-1, but the units of are oscillations/second while those of are radians/second: they are related by = 2.
Problem 11-1: The displacement of an object undergoing simple harmonic motion is given by the equationx(t) = 3.00 sin(8t + /4) where x is in meters, t is in seconds and the argument of the sine function is in radians.
(a) What is the amplitude of motion?
(b) What is the frequency of the motion?
(c) What are the position, velocity, and acceleration of the object at t = 0?
11-2: SIMPLE HARMONIC MOTION AND UNIFORM CIRCULAR MOTION
Keywords: Simple Harmonic Motion; Uniform Circular Motion
OBJECTIVE:
Describe simple harmonic motion as the projection of uniform circular motion along a diameter of the circle, and use the resulting analogy to analyze problems involving linear oscillations.
Commentary:
Viewed edge-on (i.e., projected along a diameter) the motion of a particle moving uniformly with angular speed in a circle of radius A is indistinguishable from a particle oscillating harmonically in one dimension with amplitude A and angular frequency . This fact can be very useful in helping you to remember and apply the parameters of simple harmonic motion.
In the figure to the right, the position of a particle with angular speed about a circle of radius A is projected onto the vertical diameter. We arbitrarily let the projection be at the center of the diameter at t = 0, and call y the displacement of the projection from the center. Then, at some later time t, the particle will have turned through an angle , equal to t, and the projection will have moved a distance
y = A sin t.
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Taking the expression for y and differentiating twice with respect to time, we obtain
dy/dt = A cos t;
d2y/dt2 = -2A sin t.
By definition, d2y/dt2 is acceleration and 2 is a positive constant. From Newton's second law:
F = md2y/dt2,
F = -m2A sin t.
Therefore, since A sin t = y is the displacement of the projection from the center of the circle, our final equation is
F = -m2y = -ky, and the projection moves with simple harmonic motion, with the center of the diameter as the equilibrium position.
The circle of radius A is called the reference circle for the harmonic oscillation of amplitude A.
In the figure above, the period of the motion is the time required for one complete vibration. In this time, then, the projection must move from the center of the diameter, up to a maximum positive displacement, down to a maximum negative displacement, and back to the central point.
In the same time, then, the particle moving with angular speed in a circular path will go just once around the circle. The angle turned through by this particle is 2 rad, and we see, from the definition of angular velocity = /t, that = 2/T, where T is the period of the simple harmonic motion. Also, since the frequency = 1/T, = 2. Thus, in our equation x = A sin t, the coefficient of t is 2 or 2/T.
Problem 11-2: A particle is oscillating harmonically about the origin along the y-axis with amplitude A = 4.0 cm and period T = 2.0 s.
a. What is the angular speed of the corresponding uniform circular motion?
b. What is the minimum time required for the particle to travel from
y1 = -1 cm to y2 = +2 cm?
11-3: ENERGY CONSIDERATIONS IN SIMPLE HARMONIC MOTION
Keywords: Simple Harmonic Motion; Conservation Of Energy; Kinetic Energy; Potential Energy
OBJECTIVE:
Use conservation of energy and the equation for simple harmonic motion to analyze the relation between the kinetic and potential energy of a simple harmonic oscillator.
PREREQUISITES:
Defining potential and kinetic energy and identifying systems to which the principle of conservation of total mechanical energy may be applied
Commentary
Harmonic motion can be analyzed in terms of a particle with a fixed total energy oscillating in a potential-energy "well," i.e., in the presence of a potential that increases in either direction away from the equilibrium point. In fact, this approach was used in the text in the initial development of the concept of the restoring force. Now that we have derived a general solution to the equation of simple harmonic motion and can write expressions for displacement and velocity as functions of time, we are in a position to verify that the sum of kinetic and potential energy is, in fact, constant for a simple harmonic oscillator.
Find equations in a textbook on the InfoMall that express the potential and kinetic energy of simple harmonic motion as functions of time:
Enter those equations here:
By applying the identity sin2 + cos2 = 1,
show the total energy is constant and equal to (1/2)kA2.
This leads to several other useful results, such as an equation expressing velocity as a function of position, find or derive such an equation and enter it here:
Finally the equation A = SQR RT (2E/k) relates the amplitude of the SHM to the total energy. It can be used to find the amplitude of the motion if the total energy is known and vice versa.
Problem 11-3: For the oscillating particle described in Problem 12-2, what is the linear speed as a function of position?
Problem 11-4: A 1.0-kg block rests on a frictionless table and is attached to a spring with a force constant of 250 N/m. The object is displaced 20 cm and given an initial velocity of 3 m/s away from the equilibrium position.
(a) What is the total energy of the resulting motion?
(b) What is the frequency of the motion?
(c) What is the amplitude of the motion?
11-4: APPLICATIONS OF SIMPLE HARMONIC MOTION
Keywords: Pendulums; Springs
OBJECTIVES:
Analyze the motion of a body to determine if it can be described either exactly or approximately in terms of simple harmonic motion and identify the conditions under which approximations to simple harmonic motion are valid.
Use given information concerning simple mechanical systems, together with Newton's second law and/or conservation of energy, to solve for any of the kinematic or dynamic variables of simple harmonic motion.
PREREQUISITES: Defining angular velocity, acceleration, displacement, and torque Applying Newton's second law for rotation to solve simple problems
Commentary
Using the equations:
F(x) = -kx, (1)
m(d2x/dt2) = -kx. (2)
we will outline a method for analyzing a mechanical system to determine whether it can be characterized by simple harmonic motion.
I. Determine the net force acting on the particle.
(a) Identify forces acting on the particle by drawing a free-body diagram. Choose a convenient coordinate system.
(b) Find the net force acting on the particle as a function of its position in the chosen coordinate system.
II. Describe the particle's displacement from the equilibrium position.
(a) Find the position where the net force is equal to zero; this is the equilibrium position of the particle.
(b) If necessary, introduce a new coordinate system with its origin at the equilibrium position.
III. Use the coordinate system introduced in II(b) to state Eqs. (1) and (2).
(a) Express the net force as a function of the new coordinates. Compare this with Eq. (1).
(b) Express the acceleration in terms of the second time derivative of the new coordinates.
(c) Use the expressions derived in steps (a) and (b) to state Newton's second law (F = ma) in terms of the new coordinates. Compare its form with Eq. (2).
Problem 11-4: A particle of mass m is constrained to move on a vertical frictionless track. It is attached to one end of the massless spring with spring constant k and unextended length l0 = 0 m (small compared to other lengths in the problem). The other end of the spring is hooked to a peg at the distance d from the track as shown in the diagram. See Figure (a) Show that the particle carries out simple harmonic motion when displaced from its equilibrium position. (b) Find the period of oscillation of the particle.
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Many systems that are actually rotational rather than rectilinear can be analyzed in terms of simple harmonic motion through the use of the small-angle approximation sin .The first thing to note is that this is true only if is in radians. Obviously sin 1.00 1. But 1 = 0.0174 rad, and sin(0.0174 rad) = 0.0174. This approximation is good up to about 15.0 or 0.262 rad where we have sin (0.262 rad) = 0.258. This is an error of only about 1%, so the approximation is pretty good; the error in the period of a pendulum when the amplitude is 15.0 is only 0.50%. Thus, even though a system actually does not execute simple harmonic motion, if the angular displacement is kept small enough its motion will be essentially simple harmonic.The listing below summarizes a few of the many common examples of simple harmonic oscillators along with the expressions for Fx or t. You should verify these expressions for yourself. The determination of , , and T for each system is left to you as an exercise.
a. Object on a spring: equilibrium occurs at the height for which the spring
force equals -mg. When the object is displaced, the spring force changes, but
mg remains the same. Restoring force is Fx = -kx.
b. Object fastened to two stretched springs: when the object is displaced, one
spring pulls more, and the other pulls less. Restoring force is Fx =
-(2k)x.
c. Object fastened to two stretched springs, but displaced sideways: if the
displacement is small, the forces exerted by the springs change in direction,
but hardly at all in magnitude; the "spring" forces could also be exerted by
elastic strings or some other medium obeying Hooke's law. Restoring force is
Fx = -2F0 (x/l).
d. Massive object on a "massless flagpole": for small displacement, the motion
is almost linear. Restoring force is Fx = -Kx
e. Object on a string (simple pendulum): the restoring force is the component
of mg perpendicular to the string, -mg sin . For small displacements, the
motion is almost linear, and sin . Restoring force is Fx = -mg =
-mg(x/l).
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f. Small object sliding in a frictionless spherical bowl: same as the simple pendulum. Restoring force is Fx = -mg. Or, use the restoring torque = -mgl with I = l2m and = Ia = I d2/dt2.
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g. Pivoted plank on spring: same as a car with good shocks in front,
very bad shocks in back. As the object bounces up and down, the force of
gravity is constant, but the spring force changes. Restoring torque is = - l
k(l) = -k l 2
h. Object hung from wire and rotating about a vertical axis (torsion
pendulum); some mantlepiece clocks use a pendulum of this kind. Generally, the
wire provides a restoring torque = -k
Although many pendulums are designed to approximate the simple pendulum
by having their mass concentrated at the end of a string or rod, a more general
case is that of the compound pendulum or physical pendulum as illustrated in
the figure below:
A compound pendulum consists of a rigid body of mass m suspended from an axis
0. The center of mass is a distance h from the axis. The torque about the axis
0 on the body is equal to the moment of inertia about the axis, I, times the
angular acceleration:
= I = I(d2/dt2). The restoring torque is provided by the
weight mg:
= -mgn sin . Therefore,
-mgh sin = I(d2/dt2).
If is small, sin and
-mgh = I(d2/dt2),
which is an equation for simple harmonic motion in terms of rotational
variables.
Problem 11-5: One day you visit a friend who has a chair
suspended on two springs. When you sit down on the chair, it oscillates
vertically at 0.50 Hz. After the oscillations have died down, you stand up
slowly, and the chair rises 0.50 m. Next, your friend sits in the chair, and
you find that the oscillations have a period of 2.10 s. Assume that your mass
is 60 kg. See Figure
(a) What is the spring constant for the two springs together?
(b) What is the mass of the chair?
(c) What is the mass of your friend?
(d) While you are sitting in the chair, at a certain instant (t = 0) the chair
is 0.300 m above its equilibrium position, and momentarily at rest. Find the
expression for y(t), its displacement from equilibrium as a function of time.
(e) Under these conditions, what is the maximum kinetic energy of you and the
chair? What is your maximum speed?
Problem 11-6: In the book Tik-Tok of Oz, Queen Anne, Hank the mule, the
Rose Princess, Betsy, Tik-Tok, Polychrome, the Shaggy Man, and the entire Army
of Oogaboo all fall through the straight Hollow Tube to the opposite side of
the earth. The retarding force of the air is evidently negligible during this
trip, since they all pop out neatly at the other end. For an object at a
distance r from the center of such a spherical mass distribution the
gravitational force has the:
Fg(r) = mgr/Re,and is directed toward the center of the earth. Use Re = 6.4
x 106 m for the radius of the earth:
(a) Do they undergo simple harmonic motion? How do you know?
(b) How long does their trip last?
Problem 11-7: An automobile with very bad shock absorbers behaves as
though it were simply mounted on a spring, as far as vertical oscillations are
concerned. When empty, the car's mass is 1000 kg, and the frequency of
oscillation is 2.00 Hz.
(a) What is the spring constant?
(b) How much energy does it take to set this car into oscillation with an
amplitude of 5.0 cm (assuming all damping can be neglected)?
(c) What is the maximum speed of the vertical motion in (b)?
(d) Suppose that four passengers with an average weight of 60 kg now enter the
car. What is the new frequency of oscillation?
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