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Oscillations about an equilibrium position

We have just considered the energy of SHM as a function of time. Another interesting view of the simple harmonic oscillator is to consider the energy as a function of position. [link] shows a graph of the energy versus position of a system undergoing SHM.

Graph of energy E in Joules on the vertical axis versus position x in meters on the horizontal axis. The horizontal axis had x=0 labeled as the equilibrium position with F=0. Positions x=-A and x=+A are labeled as turning points. A concave down parabola in red, labeled as K, has its maximum value of E=E total at x=0 and is zero at x=-A and x=+A. A horizontal green line at a constant E value of E total is labeled as E total. A concave up parabola in blue, labeled as U, intersects the green line with a value of E=E total at x=-A and x=+A and is zero at x=0. The region of the graph to the left of x=0 is labeled with a red arrow pointing to the right and the equation F equals minus the derivative of U with respect to x. The region of the graph to the right of x=0 is labeled with a red arrow pointing to the left and the equation F equals minus the derivative of U with respect to x.
A graph of the kinetic energy (red), potential energy (blue), and total energy (green) of a simple harmonic oscillator. The force is equal to F = d U d x . The equilibrium position is shown as a black dot and is the point where the force is equal to zero. The force is positive when x < 0 , negative when x > 0 , and equal to zero when x = 0 .

The potential energy curve in [link] resembles a bowl. When a marble is placed in a bowl, it settles to the equilibrium position at the lowest point of the bowl ( x = 0 ) . This happens because a restoring force    points toward the equilibrium point. This equilibrium point is sometimes referred to as a fixed point . When the marble is disturbed to a different position ( x = + A ) , the marble oscillates around the equilibrium position. Looking back at the graph of potential energy, the force can be found by looking at the slope of the potential energy graph ( F = d U d x ) . Since the force on either side of the fixed point points back toward the equilibrium point, the equilibrium point is called a stable equilibrium point    . The points x = A and x = A are called the turning points . (See Potential Energy and Conservation of Energy .)

Stability is an important concept. If an equilibrium point is stable, a slight disturbance of an object that is initially at the stable equilibrium point will cause the object to oscillate around that point. The stable equilibrium point occurs because the force on either side is directed toward it. For an unstable equilibrium point, if the object is disturbed slightly, it does not return to the equilibrium point.

Consider the marble in the bowl example. If the bowl is right-side up, the marble, if disturbed slightly, will oscillate around the stable equilibrium point. If the bowl is turned upside down, the marble can be balanced on the top, at the equilibrium point where the net force is zero. However, if the marble is disturbed slightly, it will not return to the equilibrium point, but will instead roll off the bowl. The reason is that the force on either side of the equilibrium point is directed away from that point. This point is an unstable equilibrium point.

[link] shows three conditions. The first is a stable equilibrium point (a), the second is an unstable equilibrium point (b), and the last is also an unstable equilibrium point (c), because the force on only one side points toward the equilibrium point.

Three illustrations of a ball on a surface. In figure a, stable equilibrium point, the ball is inside a concave-up surface, at the bottom. A filled circle under the surface, below the ball, has two horizontal arrows labeled as F pointing toward it from either side. Gray arrows tangent to the surface are shown inside the surface, pointing down the slope, toward the ball’s position. In figure b, unstable equilibrium point, the ball is on top of a concave-down surface, at the top. An empty circle under the surface, below the ball, has two horizontal arrows labeled as F pointing away it from either side. Gray arrows tangent to the surface are shown inside the surface, pointing down the slope, away from the ball’s position. In figure c, unstable equilibrium point, the ball is on the inflection point of a surface. A half-filled circle under the surface, below the ball, has two horizontal arrows labeled as F, one on either side of the circle, both pointing to the left. Gray arrows tangent to the surface are shown inside the surface, pointing down the slope, one toward the ball and the other away from it.
Examples of equilibrium points. (a) Stable equilibrium point; (b) unstable equilibrium point; (c) unstable equilibrium point (sometimes referred to as a half-stable equilibrium point).

The process of determining whether an equilibrium point is stable or unstable can be formalized. Consider the potential energy curves shown in [link] . The force can be found by analyzing the slope of the graph. The force is F = d U d x . In (a), the fixed point is at x = 0.00 m . When x < 0.00 m, the force is positive. When x > 0.00 m, the force is negative. This is a stable point. In (b), the fixed point is at x = 0.00 m . When x < 0.00 m, the force is negative. When x > 0.00 m, the force is also negative. This is an unstable point.

Questions & Answers

two particles rotate in a rigid body then acceleration will be ?
kinza Reply
same acceleration for all particles because all prticles will be moving with same angular velocity.so at any time interval u find same acceleration of all the prticles
Zaheer
what is electromagnetism
David Reply
It is the study of the electromagnetic force, one of the four fundamental forces of nature. ... It includes the electric force, which pushes all charged particles, and the magnetic force, which only pushes moving charges.
Energy
what is units?
Subhajit Reply
units as in how
praise
What is th formular for force
Joseph Reply
F = m x a
Santos
State newton's second law of motion
Seth Reply
can u tell me I cant remember
Indigo
force is equal to mass times acceleration
Santos
The acceleration of a system is directly proportional to the and in the same direction as the external force acting on the system and inversely proportional to its mass that is f=ma
David
The uniform seesaw shown below is balanced on a fulcrum located 3.0 m from the left end. The smaller boy on the right has a mass of 40 kg and the bigger boy on the left has a mass 80 kg. What is the mass of the board?
Asad Reply
Consider a wave produced on a stretched spring by holding one end and shaking it up and down. Does the wavelength depend on the distance you move your hand up and down?
Sohail Reply
no, only the frequency and the material of the spring
Chun
how to read physics ncert?
Tech
beat line read important. line under line
Rahul
how can one calculate the value of a given quantity
Helen Reply
means?
Manorama
To determine the exact value of a percent of a given quantity we need to express the given percent as fraction and multiply it by the given number.
AMIT
meaning
Winford
briefly discuss rocket in physics
Ibrahim Reply
ok let's discuss
Jay
What is physics
Nura Reply
physics is the study of natural phenomena with concern with matter and energy and relationships between them
Ibrahim
a potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inductance of the inductor that should be connected to the capacitor for the circuit to oscillate at 1125Hza potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inducta
Royalty Reply
L= 0.002H
NNAEMEKA
how did you get it?
Favour
is the magnetic field of earth changing
tibebeab Reply
what is thought to be the energy density of multiverse and is the space between universes really space
tibebeab
can you explain it
Guhan
Energy can not either created nor destroyed .therefore who created? and how did it come to existence?
Suzana Reply
this greatly depend on the kind of energy. for gravitational energy, it is result of the shattering effect violent collision of two black holes on the space-time which caused space time to be disturbed. this is according to recent study on gravitons and gravitational ripple. and many other studies
tibebeab
and not every thing have to pop into existence. and it could have always been there . and some scientists think that energy might have been the only entity in the euclidean(imaginary time T=it) which is time undergone wick rotation.
tibebeab
What is projectile?
Nana Reply
An object that is launched from a device
Grant
2 dimensional motion under constant acceleration due to gravity
Awais
Not always 2D Awais
Grant
no comments
Awais
why not? a bullet is a projectile, so is a rock I throw
Grant
bullet travel in x and y comment same as rock which is 2 dimensional
Awais
components
Awais
no all pf you are wrong. projectile is any object propelled through space by excretion of a force which cease after launch
tibebeab
for awais, there is no such thing as constant acceleration due to gravity, because gravity change from place to place and from different height
tibebeab
it is the object not the motion or its components
tibebeab
where are body center of mass on present.
Balwant Reply
on the mid point
Suzana
is the magnetic field of the earth changing?
tibebeab
does shock waves come to effect when in earth's inner atmosphere or can it have an effect on the thermosphere or ionosphere?
tibebeab
and for the question from bal want do you mean human body or just any object in space
tibebeab
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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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