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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

what is a wave?
DAVID Reply
show that coefficient of friction of solid block inclined at an angle is equivalent to trignometric tangent of angle
DAVID
thanks for that definition.
Dodou Reply
Hi everyone please can dere be motion without force?
Lafon
no...
Enyia
Thanks
Lafon
hi
Omomaro
whats is schrodinger equation
Omomaro
l went spiral spring
Xalat
what is position?
Adhar Reply
position is simply where you are or where you were
Shii
position is the location of an object with respect to a two or three dimensional axes or space.
Bamidele
Can dere be motion without force?
Lafon
what is the law of homogeinity?
auson Reply
two electric lines of force never interested each other. why?
Sujit Reply
proof that for BBC lattice structure 4r\root 5 and find Apf for the BBC structure
Eric Reply
what is physics?
Abdulaziz Reply
physics is deine as the specific measrument of of volume, area,nd distances...
Olakojo
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia
yes
Alex
proof that for a BBC lattice structure a= 4r/ root 5 find the APF for the BBC structure
Eric
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia Reply
tell me conceptual quetions of mechanics
Syeda Reply
I want to solve a physical question
ahmed
ok
PUBG
a displacement vector has a magnitude of 1.62km and point due north . another displacement vector B has a magnitude of 2.48 km and points due east.determine the magnitude and direction of (a) a+ b and (b) a_ b
Kou Reply
quantum
George
a+b=2.9
SUNJO
a+b
Yekeen
use Pythogorous
Dhritwan
A student opens a 12kgs door by applying a constant force of 40N at a perpendicular distance of 0.9m from the hinges. if the door is 2.0m high and 1.0m wide determine the magnitude of the angular acceleration of the door. ( assume that the door rotates freely on its hinges.) please assist me to d
Mike
what is conditions met to produce shm
Enocy Reply
what is shm
Manzoor
shm?
Grant
Why is Maxwell saying that light is an electromagnetic wave?
Bong
1st condition; It(th e BBC's system) must have some inertia which will enable it to possess Kinetic energy 2. must be able to store potential energy
Calleb
I meant "the system" not the BBC'S....."
Calleb
what a answer bro
Manzoor
kindly tell us the name of your university
Manzoor
GUlam Ishaq Khan INSTITUTE of engineering science
ali
Department of Environment Ionian University Zante Greece
why light wave travel faster than sounds
ALI Reply
Why light travel faster than sounds?
ALI
Light travel faster than sound because it does not need any medium to travel through.
alhassan
when an aeroplane flies....why it does not fall on the earth?
Frazali
As an aeroplane moves, it hits a wind,we have the wind flowing at the upper and lower zone of the aeroplane, the one that is moving on the upper zone moves at a greater speed than that of the lower zone, this creates a low pressure on the upper zone and a greater pressure at the lower zone.
Kipkoech
which thing of aeroplane moves it upward?
Frazali
good question
Manzoor
about force
Barataa
am pleased to join the group
Nesru
yea
caleb
It a privilege to be here
olajire
hi
Awode
hello
Manzoor
Light speed is more than sound speed. C=3×10*8m/s V=320-340 m/s
siva
A body of mass 2kg slides down a rough plane inclined to horizontal at 30degrees. find the energy that is wasted as a result of friction if the co-efficient of kinetic f
official Reply
ten applications of Newton's second law of motion
Alale Reply
Calculate the volume at S.T.P of a gas whose volume at -5° and 746 mmHg
Mlungisi Reply
Practice Key Terms 3

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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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