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I don't understand the wordings of this theorem. Can someone please help me in understanding this? Secondly on what basis are the sign conventions in this theorem applied? I get confused in plus and minus majorly while writing the the potential energy (due to earth or any other means) and the spring's work. When is it $-\frac{1}{2}kx^2$ and when $+$?

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In simple words, W-E theorem states that the net work done by forces on a body is equal to change in kinetic energy of the body. The kinetic energy includes sum of rotational and kinetic energies. That means simply sum up the work done by forces on the body: it is equal to change in KE of the body. Remember here that work done by force is important,not the force. It may be that force is present but it's work is zero. Now for the sign convention. Using a fixed sign convention for a question will give the same answer. But generally adopted convention is that if displacement is in same direction as force, work is positive, in opposite direction is taken to be negative. A point in which confusion occurs is that be careful about work done by body and work done on body. The sign convention is for work done by body. For example. In compression of spring,we apply a force to compress it. The displacement of spring is in same direction as we apply force , so work done by us is positive ,. Or we say work is done by us(work done on us is negative). But for spring it applies a force outwards and instead gets displaced inwards. So work done by spring is negative Or you say work is done on the spring. (Work done on spring is positive) Hope it clarifies your doubts

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  • $\begingroup$ Sorry , added by mistake $\endgroup$ – Tojrah Mar 31 at 9:28
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If you want to use the work-kinetic energy theorem you best stick to either a point like object or a particle, (or an object which can have both transnational kinetic energy and rotational kinetic energy) and then the theorem states that the work done by external forces on the particle (the system) is equal to the change in kinetic energy of the particle.

A particle (the system under consideration) is moving (ie has kinetic energy) along a rough surface so an external force due to the kinetic friction is acting on the particle.
Work done equals on the body is $\int \vec F_{\rm external} \cdot d\vec s$ where $F_{\rm external}$ is the frictional force and $d\vec s$ is the incremental displacement of the force.
In this case the direction of the friction force is opposite to the displacement of the particle so the work done on the particle is negative and so the change in kinetic energy of the particle is negative ie the kinetic energy of the particle is decreasing.
A particle (the system) is on a rough surface which is increasing in speed and the particle does not slip relative to the surface.
In this case the static frictional force is in the same direction as the displacement of the particle so the work done by the frictional force is positive and there is an increase in the kinetic energy of the particle.


Now suppose that you have a compressed horizontal spring fixed at one end and with a particle at rest at the other end.

Consider the particle as the system and so the spring exerts an external force on the particle.
Release the particle and the the force (external) on the particle due to the spring is in the sane direction as the motion of the spring so the spring does positive work on the particle and the kinetic energy of the particle increases.
If the compression of the spring was $x$ and the sping constant was $k$ then the work done by the spring on the particle is $\frac 12 k x^2$.
Please note that I have not used the term (elastic potential energy) as in the context of what is happening to the particle (the system) it is irrelevant as to the origin of the external force.

Reverse the process with a moving particle hitting an unextended spring and compressing it, then the force (external) is in the opposite direction to that of its displacement and so the work done on the particle by the spring is negative.
In this case the work done by the external force is $-\frac 12 k x^2$.


Let me explain why you cannot use the work - kinetic energy theorem for a spring.

Imagine an uncompressed spring (the system) fixed at one end.
You apply a force (external) to the spring and whilst you are compressing the spring you are doing work on the spring.
You compressing the spring and the spring has no kinetic energy so the work - kinetic energy theorem does not hold.
It does not hold because in doing the spring another form of energy is stored in the spring - elastic potential energy.
So in such a case you should be thinking of the work- energy theorem which states the the work done on a system by external forces is equal to the change in energy of the system.


Using the spring being compressed by a moving particle as an example of a system you could say that no work is done on the system and so the total energy of the system does not change.
The force on the spring due to the mass and the force on the mass due to the spring are internal forces so cannot be counted in the work done by external forces.
The external force exerted on the spring at the fixed end of the spring does not move and so does no work.

As there is no change in the total energy of the system (spring and particle),

$\rm (kinetic \:energy)_{\rm initial}+(elastic \:potential\: energy=0)_{\rm initial}=\rm (kinetic \:energy=0)_{\rm final}+(elastic \:potential\: energy)_{\rm final}$.

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  • $\begingroup$ Even after editing, the question promotes misleading or even wrong statements. $\endgroup$ – nasu Mar 31 at 13:51
  • $\begingroup$ 1. A key point which is omitted is that the change in kinetic energy equals the NET work. 2. The W-E theorem works for individual particles as well for systems of particles. Most mechanics books treat the case of systems of particles a little bit after the case of point particles. 3. There are no cases when the W-E does not work. It works very well for the compression of a spring or any other process. You just have to apply it properly. $\endgroup$ – nasu Mar 31 at 14:04
  • $\begingroup$ @Nasu please note that I have differentiated between the work - kinetic energy theorem and the work - energy theorem. The second works for all situations the first does not. I did this because of the title of this question. $\endgroup$ – Farcher Mar 31 at 14:08
  • $\begingroup$ The term "Work-energy" theorem refers to the connection between work and KINETIC energy. There is no separate theorem called "work-energy" theorem and having a different meaning. courses.lumenlearning.com/boundless-physics/chapter/… $\endgroup$ – nasu Mar 31 at 14:13
  • $\begingroup$ You cannot compress a spring without moving the parts of the spring and even the center of mass of the spring. If you compress an ideal spring for a while and then remove the compressing force the spring will oscillate so it does have kinetic energy. $\endgroup$ – nasu Mar 31 at 14:26

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