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The motion and energy of a mass attached to a horizontal spring, spring constant k, at various points in its motion. In figure (a) the mass is displaced to a position x = A to the right of x =0 and released from rest (v=0.) The spring is stretched. The force on the mass is to the left. The diagram is labeled with one half k A squared. (b) The mass is at x = 0 and moving in the negative x-direction with velocity – v sub max. The spring is relaxed. The Force on the mass is zero. The diagram is labeled with one half m quantity v sub max squared. (c) The mass is at minus A, to the left of x = 0 and is at rest (v =0.) The spring is compressed. The force F is to the right. The diagram is labeled with one half k quantity minus A squared. (d) The mass is at x = 0 and moving in the positive x-direction with velocity plus v sub max. The spring is relaxed. The Force on the mass is zero. The diagram is labeled with one half m v sub max squared. (e) the mass is again at x = A to the right of x =0. The diagram is labeled with one half k A squared.
The transformation of energy in SHM for an object attached to a spring on a frictionless surface. (a) When the mass is at the position x = + A , all the energy is stored as potential energy in the spring U = 1 2 k A 2 . The kinetic energy is equal to zero because the velocity of the mass is zero. (b) As the mass moves toward x = A , the mass crosses the position x = 0 . At this point, the spring is neither extended nor compressed, so the potential energy stored in the spring is zero. At x = 0 , the total energy is all kinetic energy where K = 1 2 m ( v max ) 2 . (c) The mass continues to move until it reaches x = A where the mass stops and starts moving toward x = + A . At the position x = A , the total energy is stored as potential energy in the compressed U = 1 2 k ( A ) 2 and the kinetic energy is zero. (d) As the mass passes through the position x = 0 , the kinetic energy is K = 1 2 m v max 2 and the potential energy stored in the spring is zero. (e) The mass returns to the position x = + A , where K = 0 and U = 1 2 k A 2 .

Consider [link] , which shows the energy at specific points on the periodic motion. While staying constant, the energy oscillates between the kinetic energy of the block and the potential energy stored in the spring:

E Total = U + K = 1 2 k x 2 + 1 2 m v 2 .

The motion of the block on a spring in SHM is defined by the position x ( t ) = A cos ( ω t + ϕ ) with a velocity of v ( t ) = A ω sin ( ω t + ϕ ) . Using these equations, the trigonometric identity cos 2 θ + sin 2 θ = 1 and ω = k m , we can find the total energy of the system:

E Total = 1 2 k A 2 cos 2 ( ω t + ϕ ) + 1 2 m A 2 ω 2 sin 2 ( ω t + ϕ ) = 1 2 k A 2 cos 2 ( ω t + ϕ ) + 1 2 m A 2 ( k m ) sin 2 ( ω t + ϕ ) = 1 2 k A 2 cos 2 ( ω t + ϕ ) + 1 2 k A 2 sin 2 ( ω t + ϕ ) = 1 2 k A 2 ( cos 2 ( ω t + ϕ ) + sin 2 ( ω t + ϕ ) ) = 1 2 k A 2 .

The total energy of the system of a block and a spring is equal to the sum of the potential energy stored in the spring plus the kinetic energy of the block and is proportional to the square of the amplitude E Total = ( 1 / 2 ) k A 2 . The total energy of the system is constant.

A closer look at the energy of the system shows that the kinetic energy oscillates like a sine-squared function, while the potential energy oscillates like a cosine-squared function. However, the total energy for the system is constant and is proportional to the amplitude squared. [link] shows a plot of the potential, kinetic, and total energies of the block and spring system as a function of time. Also plotted are the position and velocity as a function of time. Before time t = 0.0 s, the block is attached to the spring and placed at the equilibrium position. Work is done on the block by applying an external force, pulling it out to a position of x = + A . The system now has potential energy stored in the spring. At time t = 0.00 s, the position of the block is equal to the amplitude, the potential energy stored in the spring is equal to U = 1 2 k A 2 , and the force on the block is maximum and points in the negative x -direction ( F S = k A ) . The velocity and kinetic energy of the block are zero at time t = 0.00 s . At time t = 0.00 s, the block is released from rest.

Graphs of the energy, position, and velocity as functions of time for a mass on a spring. On the left is the graph of energy in Joules (J) versus time in seconds. The vertical axis range is zero to one half k A squared. The horizontal axis range is zero to T. Three curves are shown. The total energy E sub total is shown as a green line. The total energy is a constant at a value of one half k A squared. The kinetic energy K equals one half m v squared is shown as a red curve. K starts at zero energy at t=0, and rises to a maximum value of one half k A squared at time 1/4 T, then decreases to zero at 1/2 T, rises to one half k A squared at 3/4 T, and is zero again at T. Potential energy U equals one half k x squared is shown as a blue curve. U starts at maximum energy of one half k A squared at t=0, decreases to zero at 1/4 T, rises to one half k A squared at 1/2 T, is zero again at 3/4 T and is at the maximum of one half k A squared again at t=T. On the right is a graph of position versus time above a graph of velocity versus time. The position graph has x in meters, ranging from –A to +A, versus time in seconds. The position is at +A and decreasing at t=0, reaches a minimum of –A, then rises to +A. The velocity graph has v in m/s, ranging from minus v sub max to plus v sub max, versus time in seconds. The velocity is zero and decreasing at t=0, and reaches a minimum of minus v sub max at the same time that the position graph is zero. The velocity is zero again when the position is at x=-A, rises to plus v sub max when the position is zero, and v=0 at the end of the graph, where the position Is again maximum.
Graph of the kinetic energy, potential energy, and total energy of a block oscillating on a spring in SHM. Also shown are the graphs of position versus time and velocity versus time. The total energy remains constant, but the energy oscillates between kinetic energy and potential energy. When the kinetic energy is maximum, the potential energy is zero. This occurs when the velocity is maximum and the mass is at the equilibrium position. The potential energy is maximum when the speed is zero. The total energy is the sum of the kinetic energy plus the potential energy and it is constant.

Questions & Answers

definition of inertia
philip Reply
the reluctance of a body to start moving when it is at rest and to stop moving when it is in motion
charles
An inherent property by virtue of which the body remains in its pure state or initial state
Kushal
why current is not a vector quantity , whereas it have magnitude as well as direction.
Aniket Reply
why
daniel
the flow of current is not current
fitzgerald
bcoz it doesn't satisfy the algabric laws of vectors
Shiekh
The Electric current can be defined as the dot product of the current density and the differential cross-sectional area vector : ... So the electric current is a scalar quantity . Scalars are related to tensors by the fact that a scalar is a tensor of order or rank zero .
Kushal
what is binomial theorem
Tollum Reply
hello are you ready to ask aquestion?
Saadaq Reply
what is binary operations
Tollum
What is the formula to calculat parallel forces that acts in opposite direction?
Martan Reply
position, velocity and acceleration of vector
Manuel Reply
hi
peter
hi
daniel
hi
Vedisha
*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
imam
hello
Lydia
hello Lydia.
Sackson
What is momentum
isijola
hello
Saadaq
A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
Ekuri Reply
I need the solving for this question
philip
is the eye the same like the camera
EDWIN Reply
I can't understand
Suraia
same here please
Josh
I think the question is that ,,, the working principal of eye and camera same or not?
Sardar
yes i think is same as the camera
muhammad
what are the dimensions of surface tension
samsfavor
why is the "_" sign used for a wave to the right instead of to the left?
MUNGWA Reply
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
Victor
principle of superposition?
Naveen Reply
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
MB
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
Shubhrant
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
MB
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Shubhrant
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
Daniel Reply
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
Goodness Reply
(10/6) ÷0.4=4.167 per sec
Shubhrant
what is the formula for pressure?
Goodness Reply
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus
who is Newton?
John Reply
scientist
Jeevan
a scientist
Peter
that discovered law of motion
Peter
ok
John
but who is Isaac newton?
John
a postmodernist would say that he did not discover them, he made them up and they're not actually a reality in itself, but a mere construct by which we decided to observe the word around us
elo
how?
Qhoshe
Besides his work on universal gravitation (gravity), Newton developed the 3 laws of motion which form the basic principles of modern physics. His discovery of calculus led the way to more powerful methods of solving mathematical problems. His work in optics included the study of white light and
Daniel
and the color spectrum
Daniel
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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