8.1. Conservation lawsIn this chapter we will discuss conservation of energy. The conservation laws in physics can be expressed in very simple form:
" Consider a system of particles, completely isolated from outside influence. As the particles move about and interact with each other, there are certain properties of the system that do not change "
In short we can express this as
X = constantin which X is the conserved property.
8.2. Conservation of mechanical energyA mass hanging from the ceiling will have a kinetic energy equal to zero. If the cord breaks, the mass will rapidly increase its kinetic energy. This kinetic energy was somehow stored in the mass when it was hanging from the ceiling: the energy was hidden, but has the potential to reappear as kinetic energy. The stored energy is called potential energy. Conservation of energy tells us that the total energy of the system is conserved, and in this case, the sum of kinetic and potential energy must be constant. This means that every change in the kinetic energy of a system must be accompanied by an equal but opposite change in the potential energy:
[Delta]U + [Delta]K = 0and
E = U + K = constantThe work-energy theorem discussed in Chapter 7 relates the amount of work W to the change in the kinetic energy of the system
W = [Delta]KThe change in the potential energy of the system can now be related to the amount of work done on the system
[Delta]U = - [Delta]K = - Wwhich will be the definition of the potential energy. The unit of potential energy is the Joule (J).
The potential energy U can be obtained from the applied force F
Sometimes, the potential energy function U(x) is known. The force responsible for this potential can then be obtained
8.2.1. The spring forceThe force exerted by a spring on a mass m can be calculated using Hooke's law
F(x) = - k xwhere k is the spring constant, and x is the amount by which the spring is stretched (x > 0) or compressed (x < 0). When a moving object runs into a relaxed spring it will slow down, come to rest momentarily, before accelerating in a direction opposite to its original direction (see Figure 8.1). While the object is slowing down, it will compress the spring. As the spring is compressed, the kinetic energy of the block is gradually transferred to the spring where it is stored as potential energy. The potential energy of the spring in its relaxed position is defined to be zero. The potential energy of the spring in any other state can be obtained from Hooke's law
Figure 8.1. Conversion of kinetic energy into potential energy and vice-versa.Sample Problem 8-4
A spring of a spring gun is compressed a distance d from its relaxed state. A ball if mass m is put in the barrel. With what speed will the ball leave the barrel once the gun is fired ?
Suppose Ei is the mechanical energy of the system when the spring is compressed. Since the system is initially at rest, the total energy is just the potential energy of the compressed spring:
Suppose the ball in Figure 8.1 has an initial velocity v0 and a mass m. If the spring constant is k, what is the maximum compression of the spring ?
In the initial situation, the spring is in its relaxed position (U = 0). The total energy of the ball-spring system is given by
8.2.2. Gravitational forceA ball moving upwards in the gravitational field of the earth will lose its kinetic energy and come momentarily to rest at its highest point. The ball than reverses its direction, steadily regaining its kinetic energy that was lost on the way up. When the ball arrives at its starting point it will have a kinetic energy equal to its initial kinetic energy. The work done by the gravitational force on the ball is negative during the upwards motion while it is positive on the way down. The work done when the ball returns to its original position is zero.
The potential energy due to the gravitational force can be calculated
Sample Problem 8-3
A child with mass m is released from rest at the top of a curved water slide, a height h above the level of a pool. What is the velocity of the child when she is projected into the pool ? Assume that the slide is frictionless.
The initial energy consist only out of potential energy (since child is at rest the kinetic energy is zero)
Ei = m g hwhere we have taken the potential energy at pool level to be zero. At the bottom of the slide, the potential energy is zero, and the final energy consist only out of kinetic energy
Ei = EfThus
8.2.3. Friction forceA block of mass m projected onto a rough surface will be brought to rest by the kinetic friction force. There is no way to get back the original kinetic energy of the block after the friction force has brought it to rest. The directed long-scale motion of the block has been transformed into kinetic energy of the randomly directed moving atoms that make up the block and the plane. We can not associate a potential energy with the friction force.
8.3. Conservative and non-conservative forcesIf a potential energy can be associated with a force, we call the force conservative. Examples of conservative forces are the spring force and the gravitational force. If a potential energy can not be associated with a force, we call that force non-conservative. An example is the friction force. Alternative tests of the conservative nature of a force are:
1. A force is conservative if the work it does on a particle that moves through a round trip is zero; otherwise, the force is non conservative. The requirement of zero work for a round trip is not met by the friction force.
2. A force is conservative if the work done by it on a particle that moves between two points is the same for all paths connecting these points; otherwise, it is non-conservative.
Test 1 and test 2 are equivalent. For example, assume that the work done for the round trip from A to B and back to A (see Figure 8.2) is zero. This means that
Figure 8.2. Particle on a round trip from A to B back to A, and from A to B via 2 different routes.
WAB,1 + WBA,2 = 0or
WAB,1 = - WBA,2The work done by the force on each segment reverses sign if we revert the direction
WAB,2 = - WBA,2This relation than can be used to show that
WAB,1 = WAB,2which is exactly what test 2 states (the work done by the force on the object depends only on the initial and final position of the object and not on the path taken).
Figure 8.3 shows two possible trajectories to get from A to B. What is the work done on the object by the gravitational force for trajectory 1 and for trajectory 2 ? The work done if the mass is moved along route 1 is equal to
Figure 8.3. Two possible trajectories to get from A to B.
8.4. Potential energy curveA plot of the potential energy as function of the x-coordinate tells us a lot about the motion of the object (see for example Figure 8.12 in Halliday, Resnick and Walker). By differentiating U(x) we can obtain the force acting on the object
U(x) + K = ESince the kinetic energy can not be negative, the particle can only be in those regions for which E - U is zero or positive. The points at which E - U = K = 0 are called the turning points. The potential energy curve (Figure 8.12 in Halliday, Resnick and Walker) shows several local maxima and minima. The force at each of these maxima and minima is zero. A point is a position of stable equilibrium if the potential energy has a minimum at that point (in this case, small displacements in either direction will result in a force that pushes the particle back towards the position of stable equilibrium). Points of unstable equilibrium appear as maxima in the potential energy curve (if the particle is displaced slightly from the position of unstable equilibrium, the forces acting on it will tend to push the particle even further away).
8.5. Non-conservative forcesIf we look at a block-spring system, oscillating on a rough surface, we will see that the amplitude of the motion decreases continuously. Because of the frictional force, the mechanical energy is no longer conserved. If we look at a system on which several conservative forces act, in addition to the friction force. The total work done on the system is
8.6. Conservation of energyIn the presence of non-conservative forces, mechanical energy is converted into internal energy Uint (or thermal energy):
[Delta]Uint = - WfWith this definition of the internal energy, the work-energy theorem can be rewritten as
" Energy may be transformed from one kind into another in an isolated system but it can not be created or destroyed; the total energy of a system always remains constant. "
Sample Problem 8-8
A ball bearing whose mass is m is fired vertically downward from a height h with an initial velocity v0 (see Figure 8.4). It buries itself in the sand at a depth d. What average upward resistive force f does the sand exert on the ball as it comes to rest ?
Figure 8.4. Sample Problem 5.The work done by the friction force f is given by
Ef = Uf = m g (- d) = - m g dThe change in mechanical energy is
A block whose mass is m is fired up an inclined plane (see Figure 8.5) with an initial velocity v0. It travels a distance d up the plane, comes momentarily to rest, and then slides back down to the bottom of the plane. What is the magnitude of the kinetic friction force that acts on the block while it is moving ? What will the velocity be when the block returns at its original position.
The work done by the friction force is equal to the change in the mechanical energy of the system. The potential energy at the origin is taken to be zero. Therefore, the initial mechanical energy of the system is just the kinetic energy of the block
Figure 8.5. Example Problem 2.The final mechanical energy (at maximum height) is just the potential energy of the block at height h:
Ef = m g h = m g d sin([theta])The change in mechanical energy is
Wf = - f dand must be equal to [Delta]E. Thus
Wf = - 2 f dThis must be equal to the change in mechanical energy of the system. When the block returns at the origin, there is no change in its potential energy. The change in the mechanical energy of the system is due to a change in the velocity of the block: