Module : Conservation of Energy

Kreshnik Angoni and Kevin Lenton
Worksheet.docx
Worksheet.pdf

Other Resources:

• Haliday & Resnick, Fundamentals of Physics 7.1-7,9
• Openstax
• Printable version

What is the total energy of the Universe?

Consider the universe as a closed, isolated system. Any force in the universe is internal to the universe and has an equal opposite reaction on some other object (Newton's Third Law). That is to say the amount of work that the force does on the universe has an equal and opposite work done somewhere else. In other words all you are doing is transferring work from one object in one part of the universe to another object in another part of the universe. You can see the concept of energy conservation beginning to develop.
The positive mass-energy and kinetic energy of the universe is canceled by the negative gravitational potential energy (defined below).

This implies that the total energy of the universe is perhaps zero, although that is debatable.
But it also implies that whatever the total energy is, it is constant unless inputs of work either come in or leave the universe system.

These concepts of system, and potential energy are crucial to the concept of Energy Conservation.

CONSERVATIVE FORCES, A precursor to Potential Energy

Some forces posses a specific and useful quality: the work done by these forces equals zero whenever the forces move along a closed path ( the endpoint of path is the same as its initial point). That is to say the work done is path independent, it depends on the final and initial positions, but not on the path taken between these points.
These are known as conservative forces.
Note that a conservative force does non-zero work whenever the final position is different from the initial position. For instance, in Fig. 4 the force moves from point “1” to point “2” and does work $W_{1-2}$. If the force then moves back from “2” to “1” $W_{1-2}$ (fig.4) then
$W_{2-1}=-W_{1-2}$
so that $W_{1-1}=W_{1-2}+W_{2-1}=0$.
If you do a certain amount of work in one direction, and get that work back exactly, going back in the opposite direction to the initial point, the force is essentially storing that energy for you. The energy is being conserved because the force is conservative.

Three main examples of such conservative forces are the gravitational, restorative (elastic, or spring) and static electric forces.
The work done by these forces is described as a Potential Energy, and is treated differently than non-conservative forces in conservation of energy problems, because you can get that energy back. Every conservative force will have Potential Energy.

Take Home Message

In the case of a conservative force applied on an object, it turns out that:
The work done by a conservative force depends only on its initial and final positions and not on the shape of the path from the initial to final location.

What about non-conservative forces?

Most forces are non-consevative. Any force that is directly pulling or pushing, e.g. friction, or the force of a hand is an example of a non-conservative force. The work done does depend on the path taken.
The net work done by a force of kinetic friction force when an object slides on the floor is negative even when object is moved along a closed path, because the friction removed energy from the object all along the path.
The normal force is always perpendicular to displacement 1 and its work is always zero, no matter what is the path. Figure 4 Forces that depend on the velocity (the drag force in fluids, the magnetic force,..) are not conservative forces.

 

INTRODUCTION TO POTENTIAL ENERGY

Consider an object at rest on the floor. As long as the application point of exerted forces ( FG , N ) does not move no work done.
If you move the object to a new height ‘h’ and release it , the object will fall, i.e. it will move vertically down back to h = 0.
The gravity force will do (positive) work on the block because the force (the weight) is moved by a distance s (=h) (Fig.1) is shifted along weight direction by s .
$W_{G}=F_{G}s=mgh\quad \quad \quad (1)$
You can place the object at the height ‘h’ by hand or by throwing it upward with the right initial velocity, or any other method you want. No matter how the object gets to the height ‘h’, once there, “it has the capacity to produce mechanical work”. This is entirely due to its height, because the object now possesses a potential mechanical energy. This type of energy is due to the weight force which is due to the gravitational attraction between the earth and object. When you move an object up, the earth also moves. So, when talking about potential energy, the system has to include both the earth and object.
In more general terms:

Any kind of energy which is due to the configuration of a system, i.e. its position, is called potential energy.

Elastic Potential Energy: The Block-spring system

If you extend a spring by a distance “x”, the restoring force produced by the spring will obey Hooke's Law $F_{elastic}=-kx$ and will

be directed towards the equilibrium position. When the block is

returned to the unstretched or equilibrium position, the elastic (or restoring) force has done positive work (see previous wiki on work):

$W_{elastic}=F_{elastic}\cdot s=(k/2)(x^{2}-0^{2})={\frac {1}{2}}kx^{2}\quad \quad \quad (2)$
If the spring is at rest, at its equilibrium(x = 0), it does not Figure 2

s

possess any capacity to produce work. But, if it is extended

or compressed (x.0), the spring “is able” to produce work “on the block”; this means that it does possess energy. Its capacity to produce work depending only on its configuration (extended or compressed); so this is a potential energy. The two main types of mechanical potential energy are: gravitational and elastic.

In both cases, initially, an external force (its source is not part of system; like hand’s force) does a positive work Wext on the system and shift it from an initial configuration ‘i’ to a final configuration ‘f ’. Then, just because of being at “f ” configuration the system can provide work and the amount of work it can produce (its capacity for work production) depends only on system configuration. This means that we may calculate this capacity by use of a configuration function1; let’s call it U = Uconf. If we refer to values of this function at initial-final configurations Ui and Uf and remember that the positive external work Wext brings the system from state “i” to state “f ”, the simplest logical relation between U and Wext would be W =Uf -U =.U (3) where Uf >Ui

ext i

E

This relation fits perfectly with the logic that the energy of the system increases when an external force achieves positive external work (Wext > 0) on it (see figure 3). Actually, this definition requires that all the exterior work go only for configuration changes and not for the change of kinetic energy of system parts. So, the system must be moved from the initial configuration (Ui ) to the final configuration Uf by a constant speed (in practice

Figure 3

1 Mathematical function that depends only on the location (coordinates) not on velocities or accelerations

Uf Ui 0 Wext   - To avoid the ambiguity related to the external force, one prefers to tie the definition of potential energy to the work by internal forces. The third law tells that during system transfer from Ui to Uf , the work by internal forces Wint = - Wext. From relation (3), one gets to the basic definition for potential energy as -W =.U or .U = (U - U ) =-W (4)

intfi in

-Note that the equation (4) is based on the difference .U = Uf - Ui and not on the U- values. This means that only .U has physical meaning (not U values). This definition for U leaves “free choice” for the selection of configuration where U = 0. In practice, one fixes Ui = 0 to an initial configuration which makes easier the solution of the considered problem. Then, considering the system shifted from Ui = 0 to Uf , it comes out that (Uf - 0) =-Win . So, in practice, one starts by fixing u=0 to a given system configuration and the calculate the potential energy function as Uf = U = - Wint (5)

Example: One selects Ui = Ufloor = 0 when studying the displacement by “h” of an object from the floor up and Ui = Uground = 0 when shifting it from the ground level up. In both cases, the system is the same “earth

object”, the internal force is the weight and W = W = F .

. s =-mgh . So, UG = - (-mgh) = mgh

int GG

In case of the “spring- block” system, one selects Ui = 0 for spring at equilibrium (x = 0) and the Hook’s

2

22

fx x

force is the internal force. Then Wint =Wel =-kx

-k and Uelastic =-Wint

k 22 2

To define a potential energy U(x); define the system; chose a coordinative frame; note the location where U = 0; -calculate the work (Wi) by the internal force from there till a location “x” and use relation (5).

NOTES: A potential energy is due to the interaction between system’s constituents.

-So, it does not make sense to talk about potential energy of a single object or particle alone. When one says that the potential energy of an object with mass ‘m’ at height ‘h‘ is U = mgh , actually, this means the energy UG of the ‘system object – earth due to their gravitational interaction.’ -The mechanical potential energy is due to gravitational and restoring (elastic) forces. - From a general point of view, one may define a potential energy only if the internal force at origin of this energy is conservative. In the following we explain the meaning of a conservative force.

b) One can define a potential U(x,y,z) (or potential energy) which is a function of space coordinates. The physics history showed that this function is very useful for solving difficult problems.

-Remember that a mechanic force is produced by an object “source” and is applied on another object. So, we may talk always about the system of two objects. Meanwhile, only if this force is conservative, one can use efficiently the concept of system and define the potential energy of the system (due to this force).

Related Notes:

a) Any two or more objects undergo their gravitational attraction forces. As gravitational forces are conservative, one may always define a system and a gravitational potential energy for this system. If an elastic (restoring) force acts over one of these objects, this is an additional conservative force. So, one includes the source of this force into the system and adds the corresponding PE term to potential energy of system.

b) If one of contributing forces is much bigger than others, at a first approximation, one may consider only the major term of the potential and neglect the smaller ones. c) As non-conservative forces cannot provide a potential function their source is not considered as part of a system and non conservative forces are considered as external forces to the system.

3] INTERNAL FORCES. TOTAL MECHANIAL ENERGY OF AN ISOLATED SYSTEM

-If one has defined a simple system (one object & one source) in the upper terms, the source of the considered conservative force is part of the system and the force is an internal force of this system . Then, one may fix the frame origin at “source”, select Ox axe along the direction “source - object” and find out U-expression. This potential energy function U(x) depends only on x-coordinate and, one may show that, the internal force acting on the “object”, can be derived from the potential function U(x) as

dU

f =- (6)intdx The U-function (potential energy) is a physical parameter of the system; very often it is noted PE. (If the “system” contains several “sources” and “objects”, the potential function will depend on a set of coordinates and there will be several internal forces.)

Exemple: An aeroplane with mass m is flying at height y from earth surface. The gravitational force acting on the plane (weight) is a conservative force. The source of the weight is the earth. We define the system earth-plane. We select Oy-axe along vertical and the origin O earth surface (or its center). The potential energy of this system (we are used to call it aeroplane potential energy) is U = E = m * g * y

p

dU

Then, the plane weight (internal force of system) can be calculated by use of formula f =-=-mg

dy

The “–“ shows the weight direction opposite to selected Oy positive direction. 

-Assume that only a single force is applied on an object and it is conservative. So, we can define a system, tie a reference frame Ox to the source of force and calculate its potential energy U(x). Next, we calculate the kinetic energy of the object K (with respect to this frame) and apply the work-energy theorem over the object. Wnet = Kfin - Kin = .K and referring to a small shift “dx” we get the differential form dWnet = dK (7)

As dWnet = dWint + dWext = dWint (because dWext = 0) and dWint = fint*dx we get

dU

te (8)

dWnet = dWint = fint * dx =-

• dx =-dU =

dK . dK + dU = d(K + U ) = 0 . K +U = c

dx

  So “the sum of kinetic energy K and potential energy U of the “object” remains constant in time” The potential energy concerns the object and the “source” of force. As the kinetic energy of the “source” is zero (reference frame is tied to it), it comes out that the kinetic energy of the system is equal to that of the object under study. So, the sum K + U presents actually the total mechanical energy Emech (or ME) of the system and the expression (8) tells that

ME=KE+PE= const or Emech = K + U = const (9)

Remember that we assumed no action from outside, i.e. an isolated system. So, the expression (9) shows that the total mechanical energy of an isolated system is conserved.

4] CONSERVATION OF MECANICAL ENERGY FOR NON ISOLATED SYSTEMS

How to deal with situations where any kind of forces (conservative and non-conservative) act on the same object while it is shifted from location “1” to location “2” of space ?

Step_1 Identify the conservative forces(weight, restoring) acting on the objects under study.

Step_2 Define a potential function for each conservative force. Chose the reference frame and define clearly(write) where each potential is zero. Take their sum as common U potential for system.

Step_4 Divide the set of all acting forces on the object into internal (that do contribute to total U) and external (do not contribute to U).

Step_5 Note that if there are no external forces the total mechanical energy of the system remains constant; Emech-fin = Emech-in.

So, if the external forces are doing a positive work Wext_net on the system (actually on “the objects” which are part of the system) the total energy of the system will be increased by this amount and

Emech-fin = Emech-in + Wext_net or .Emech = Emech_fin – Emech_in = Wext_net (10)

Notes: - The relation (10) is valid even for the case of a negative work by external forces. In this case it shows a decrease of total mechanical energy of the system.

-Expression (10) presents the general form of the principle of mechanical energy conservation. It is valid for any kind of system (isolated Wext_net = 0 or not isolated Wext_net . 0). REMEMBER: The kinetic, potential and total mechanical energies of a system are mathematical functions defined at a reference frame.

-While the definition of kinetic energy does not need the concept of system, the potential and total energy functions cannot be defined without going through system concept. The values of KE, PE, ME functions do not have any precise physical meaning by themselves because they depend on the choice of reference frame (the change of the reference frame changes the value of those functions). But, the change of those functions has a very precise physical meaning. It is related to the external work achieved on the system and this quantity does not depend on the selected frame for calculations. This is the basic issue one must not forget when dealing with energy related problems.

-The zero value of U-function (PE) is chosen in such a way that makes easier the solution of problem.