- Why should we study the concept of energy?
- From Force to Energy
- Newton's Laws
- Back to Energy
- Conservation of Energy
- Kinetic Energy
- Potential Energy
- Gravitational Potential Energy
- Elastic Potential Energy
- Mass as a Form of Potential Energy
- Gravitational and Electrostatic Potential Energy
- Negative Energy:Bound States

(3.1) |

(3.2) | |||

(3.3) |

The dimension of momentum is given by . If there are many particles, the total momentum of the system is the sum of the individual momenta. Most of us have an intuitive idea of force. We know that for instance, to move a heavy object, we have to exert a lot of ``force''. Newton took this experience of force, and gave it a precise mathematical formulation. Let a force act on a particle from time to time , and during this time interval let the momentum change from to . Then the change in momentum, in time interval , is given by .

(3.4) | |||

(3.5) |

Given that in most circumstances mass is a constant, the second law is equivalent to saying that for the object to experience acceleration, which is the rate of change in velocity - be it in its magnitude or direction - there must be a force acting on it. Hence, for acceleration given by , we have

(3.6) | |||

(3.7) |

The force equation looks very reasonable. The more the force, the greater the change in the amount of motion of the particle. The dimension of force is given by . Let us look at the units for above equation. Mass is measured in , acceleration is measured in ; hence unit of force, called a Newton, is given by .

We can now state Newton's three laws of motion.

**First Law**A body continues in uniform motion if no force acts on it.**Second Law**A force acting on a body causes it to accelerate (or decelerate).**Third Law**The total momentum of an isolated system is constant.

- Consider a bullet fired from a gun. According to Newton's third law, momentum is conserved, so that the recoil of the gun must have the same momentum as that of the bullet, and hence their combined momentum before and after remains zero. But common sense tells us that something very significant and irreversible has happened; something was taken from gunpowder and transformed into motion. Physics must be able to capture the transformation of gunpowder into motion, and Newton's laws are silent on what this connection is.
- Car manufacturers need to know how to build cars which can accelerate, coast at a constant speed, and, of course, decelerate as well. Newton's laws are useful in telling us how much force is required if we want to accelerate, and also how much force the brakes must exert to decelerate. But Newton is silent on the great difference between the force that accelerates the car, and the brakes which decelerate.

- A non-directional conservation law, that unlike conservation of momentum, does not cancel out for motion in opposite direction.
- A quantity that takes into account the direction of force relative to the motion, so that a force along the direction of motion has a positive effect, force against the direction of motion has a negative effect, and force perpendicular to motion has no effect.
- And most importantly, the concept must connect apparently unrelated things, such as the explosion of gunpowder with the motion of the bullet, the burning of gasoline with the motion of a car.

(3.8) |

After a few days, she finds that the blocks no longer add up to 28, and she finds that the bathtub has water full of soap, submerged in which, she suspects, there are some more blocks. Given that the original height of the bath-water was 0.5m, and that each block raises the height of water by 0.1m, she now has a new formula for computing the number of blocks, namely

(3.9) |

We see that the calculation for the number of blocks is becoming more and more abstract, and has less and less to do with counting the blocks themselves. Measuring the weight of a toy box or the height of bath water has no direct correlation with the blocks themselves. The application of this example is straightforward to energy. In other words, there are many, many forms of energy, that taken together are conserved, but there is no analogy of what is energy

(3.10) |

There are various forms of energy such as solar energy, chemical energy, nuclear energy, wind energy, and so on.

**All the forms of energy
can be classified under two great headings: kinetic and
potential.**

Energy due to motion
in space is called kinetic energy , and energy due to position
or the internal
configuration of the material body is called potential energy .
For example a stone on a mountain has potential energy due to its
position, and a stretched spring has potential energy due to its internal
composition.
Denoting total energy by we have the fundamental relation

Suppose the system has energy at time and energy at a later time , then, the change in energy is . Conservation of energy implies that

(3.12) | |||

(3.13) |

Note an important fact that since all we know is that , the absolute value of has not been fixed. Hence, energy is only defined upto a constant, since and constant would both be equally conserved. In both the example of the bullet being fired or the car being accelerated, kinetic energy due to motion was created by expending energy that was present as potential energy in the gunpowder and in the gasoline, respectively.

(3.14) |

This doesn't tell us much, since we are not able to translate this information into how much energy must be expended by the gunpowder. However, the total energy expended in shooting the bullet is given by

(3.16) | |||

(3.17) |

Using eqn.(3.23) we have the result that to design a gun which specifies the velocity of the bullet to be , we need to spend an amount of energy given by

(3.18) |

**Rotational Motion**

There are only two kinds of kinetic motion. Namely, rectilinear, or linear, motion,
that is, motion in a straight line, and kinetic
motion associated with rotation. Rotational motion, in general, is also associated
with periodic motion.

(3.19) |

We have , similar to kinetic energy for linear motion,
the angular component of kinetic energy given by

(3.20) |

is the

(3.21) |

Similar to linear momentum, the total angular momentum of a physical system is conserved. In general, for a solid body that is moving with velocity and spinning with angular velocity , the total kinetic energy is given by

(3.23) |

(3.24) |

(3.25) | |||

(3.26) |

The dimension of work , which is the same as the dimension of energy. Hence we can in principle add work to energy, showing that there is a deep relationship between energy and work. This is an instance where dimensional analysis yields new insights into dimensionally related quantities. Work is a measure of the action of force acting over a distance. Think of pushing a car; the heavier the car, the more the force required to move it. Also, the longer the distance the car needs to be pushed, the larger the amount of work that needs to be done. So the definition seems reasonable. However, many students have an intuitive problem with this definition of work. We all know that if we hold up a big piece of stone, even though the stone is not moving, and consequently no work is being done, we will soon break out into a sweat from the exertion that we are undergoing. So what's going on? How can there be no work done, even though we have had to exert ourselves? The answer to this counter-intuitive result lies in the physiology of human muscle. There are two kinds of muscle cells, one which change over a long period of time and the other which change over a short period.The clam, for example, has a muscle cell which relaxes over a very long interval, and hence a clam can be in an open position and support a large weight without expending any energy. In contrast, human muscle needs constant electrical impulses to hold its position, and consequently to hold a piece of stone requires a large expenditure of biological energy. Hence we are doing internal biological work, and not work on the piece of stone, when we hold it in air. To confirm that this is true, one can just place the stone on a table, and it will sit on the table without any work being done, and with no energy being used up for that purpose. The fact that no work is done on a load put on a table is the same reason why high rise buildings can stand without any work being done. All the high floors are stationary, and hence do not require any expenditure of work (energy) to hold them up. Of course overloading may cause the floor to break, and this then becomes a problem of material science rather than that of mechanics. One cannot store work, since once the body ceases to move, no more work is done. This is a reflection of the dynamic nature of force. However, unlike work, we can store energy. When work is done on, or by, a particle, the result is to increase, or to decrease, its energy. In other words, the deep connection between work and energy is that force results in work , which in turn increases or decreases the energy of a body. Of course, energy conservation tells us that we are simply transforming or transferring energy from one form to another. Consider a moving particle with mass that at initial time has a position of and speed of . Let a constant force act on it from time to , during which time it travels, along the direction of the force, to the final position of . At the end of time , it has increased its velocity to . The particle has only kinetic energy, and the conservation of energy then tells us that the increase in the (kinetic) energy of the particle must be due to the work done on it by the action of the constant force . Distance , given by

(3.28) | |||

(3.29) |

was covered during time given by

(3.30) |

(3.33) |

(3.34) | |||

(3.35) |

The change in comes about by constant force acting over a distance s. Power, denoted by , is then defined as

(3.36) | |||

(3.37) |

Since the time interval over which the kinetic energy has changed is , we have , and hence

This is an important equation, since it tells us that the rate of change of kinetic energy of a particle, that is the power being expended on the particle, is equal to the force times the velocity of the particle. The dimension of power is . One unit of power is defined to be 1 Watt (W) defined to be 1 Joule per second, and W has units of .

(3.40) |

From eq.(3.51) we see that we have recovered the result of mechanics of how velocity and acceleration are related to distance covered. Since acceleration is a constant, we also have

Hence, combining eqs.(3.51) and (3.53), for , we have

We see, from the equations (3.51),(3.53) and (3.54), that all the equations of motion for constant acceleration can be derived from the principle of energy conservation.

Potential Energy

One may correctly object that since energy is always conserved, it should always be possible to express force in terms of a potential. In principle, this observation is true. However, there are many

A body can come to rest only if there is no force acting on it.
Since force is equal to the negative gradient of the potential,
that is

(3.47) |

**A body reaches equilibrium when its position is at the
minimum value of the potential**

There are almost unlimited forms of potential energy. We discuss a few
of these, in particular, the
gravitational and electrical potential energy, mass as potential energy as well as
elastic potential energy that is
stored in springs and other elastic objects. These examples are
chosen to give a flavor of great variety of forms that potential
energy can take.
Other forms of potential energy such as chemical energy, radiant energy, nuclear energy
and so on are more subtle, and require deeper study.

We have called the proportionality constant ; clearly is linked to gravity, since if there was no gravitational force, the particle would not fall towards the earth to start with. By dimensional analysis, since is energy, it has dimensions of , and hence the dimension of is , which is the dimension for acceleration. From dimensional analysis, must be proportional to acceleration due to gravity, and in fact is precisely equal to acceleration arising from earth's gravity, and is equal to . The gravitational potential given in (3.64) is an approximation, valid only for bodies that are close to the earth's surface. At distances from the earth's surface that are comparable to the earth's radius, the correct expression for is given by the ``inverse law''. Consider a particle falling under the force of gravity. At a height of , it has velocity , and hence its total energy that is given by

(3.50) | |||

(3.51) |

As the particle falls, its velocity increases as its height decreases. Conservation of energy then requires that energy always be a constant, that is, the change in be zero. Hence

(3.52) |

To understand the content of above equation, at height let the velocity of the particle be and its energy , and at height let it be and respectively. Energy conservation requires that . We have with height that the particle has fallen through. Hence, from (3.68) we have

We see that we have recovered (3.51), with the additional information that the constant acceleration is due to gravity. Note the significant fact that, unlike the case for (3.51), we did not use the idea of force at any stage in deriving (3.69). This is a reflection of a the general procedure of replacing conservative force by potential .

(3.55) |

Let the ball be at some position . We can for example ask the following question: What is the force required to displace it by a small distance ? Let the ball have zero velocity. To change the energy of the ball by displacing it, an external force has to do work on it. Hence we have

(3.58) | |||

(3.59) | |||

(3.60) | |||

(3.61) |

where in obtaining the last equation we have taken to be so small that it can be completely ignored. For equilibrium, the restoring force due to the spring has to exactly cancel the external force. Hence, if the ball is moved to a position , we have a restoring force acting on the ball that is given by

(3.62) | |||

(3.63) |

In other words, the force required to stretch the spring is
proportional to the amount of stretching, namely . Note that
the spring is always a restoring entity since it acts against any change of
position of the ball. For the force is negative,
that is, it acts to pull back to , whereas if then
the force also acts to push the ball back to .

We solve for the motion of
the particle that is moving in the elastic potential. From
(3.72) we have for a particle moving with energy the
following.

One can easily verify, by differentiating (3.87), that to satisfy the energy equation given in (3.72), we have the following

Note has dimension of , has dimension of and the phase is a dimensionless angle measured in radians. Recall is the position of equilibrium of the particle. Suppose the particle starts its oscillations at time from the position . Then is fixed by

One can see from the behaviour of that if the ball is disturbed from its equilibrium position at , it will undergo oscillations about this position; note that if one increases time by the amount , that is, for , the particle returns to the position that it occupied at time . In other words, it is undergoing

(3.70) |

(3.71) |

Gravitational and Electrostatic Potential Energy

where is Newton's gravitational constant with dimension of , and its numerical value is given by

(3.74) |

The Coulomb constant is also defined in terms of the so called permittivity constant by the relation

- Mass is always a positive quantity, so gravitational potential
energy
always gives an
**attractive**force of attraction between any two masses. This is the reason that one can never 'shield' a system from gravity, since there is no way of cancelling it out. So although gravity is an extremely weak force, as we will soon show, its effects keep piling up. Hence for celestial bodies, solar systems, stars, galaxies and so on, the effects of gravity are the most important. Due to the fact that gravity is always attractive, one can imagine that if one gathers together a large enough mass, there could be**no**force which could be strong enough to stop the inward pull of gravity. This is precisely what happens when we have a star with mass three times bigger than that of our Sun : for such stars, the force of gravitational attraction is so strong, that the star - under the inward pull of gravity - undergoes gravitational collapse, and results in the formation of a black hole. - Another difference between electrical and gravitational potential energy is that every physical entity feels the force of gravity - there are no gravitationally ``neutral'' entities. On the other hand, there are many fundamental entities like the neutron that are electrically neutral, and do not feel the effect of electrical forces.
- On the other hand, electrical
charge can be positive or negative. The gravitational and Coulomb
potential also differ by a
**minus**sign, which implies that although masses attract gravitationally, two particles with like charges**repel**whereas as opposite charges**attract**. Unlike gravitational forces, we can completely shield a system from electrical forces by using negative charge to cancel positive charge, since having a net charge of zero means we have effectively cancelled out electrical forces. Recall all elements are made out of electrically neutral atoms. The atoms of each element are made out of a nucleus consisting of an equal number of electrically neutral neutrons and positively charged protons, which is surrounded by a 'cloud' of an equal number of electrons. The Coulomb potential holds the electrons in a bound state with the nucleus. The repulsive Coulomb potential - inside the nucleus - between like charges is overcome by strong nuclear forces which glue together the protons inside the nucleus. The nuclear force acts over very short distances, it can hence cancel Coulomb repulsion only over short distances. However, since the Coulomb potential acts over long distances, it finally wins over strong nuclear forces. Namely, if the number of protons exceeds a certain value close to 140, the Coulomb repulsion becomes so large that the nuclear force can no longer overcome it, and the nucleus falls apart (disintegrates). This repulsion due to the Coulomb potential is the reason that there are only a finite number of stable elements in the universe, numbering about 100 only.**Problem**Suppose the strong attractive force between protons acts over a distance of , and that the electrostatic potential acts over an infinite distance. Estimate how many protons would make the nucleus unstable. - The greatest difference in the gravitational and electrical
potential energy lies in their relative
**strengths**, characterized by the empirical values of the two constants and . To determine the relative strengths of the two potential energies, we evaluate the ratio of their values for the case of an electron and proton separated by a distance . Consider an electron, having mass and a charge , and a proton having mass and a charge . We hence have

We see that, for any distance of separation of two charged bodies, the gravitational potential energy is almost smaller in magnitude than the electrical potential energy. Even if we consider bodies with much bigger masses than an electron or a proton, the effect of gravity is extremely small. So for all cases of interest, we can ignore the effects due to gravity compared to other forces. One may object to the thought of ignoring gravity, considering that we always feel the force of gravity in our daily lives. This objection is correct if the mass involved is immense; for example, what we experience in daily life is the gravitational effect of the earth, which is an immense mass indeed. Compared to the gravitational effect of the earth's mass, the gravitational effect of any other massive body on the earth's surface, on say an object undergoing experiments in the laboratory, are negligible. The weakness of the gravitational force is ultimately the reason why the idea of an isolated system is possible.

(3.79) | |||

(3.80) |

We have to first remove the arbitrariness in energy, which is only defined upto a constant. Only then can we talk of positive and negative energy. Since there is no arbitrariness in , we have to first define what we mean by zero potential energy . Take the case of the gravitational potential. We consider the approximate expression given in eq.(3.64), namely

(3.81) |

(3.82) |

Hence, for a bound state

(3.83) | |||

(3.84) |

What is the physics of this result? Negative energy, , means that the particle cannot reach the surface, since on the surface its energy is at least zero (and higher if it has non-zero velocity on the surface). Hence, as long as , the particle can move inside the well, but cannot escape from it. The particle is consequently

(3.85) |

For a bound state, we have

(3.86) |

(3.89) | |||

(3.90) |

From eq.(3.111), we see that the escape velocity depends

(3.91) | |||

(3.92) | |||

(3.93) |

The escape velocity from a body with the earth's mass but smaller radius is given, from (3.111), by

(3.94) | |||

(3.95) |

Note that we can keep on increasing the value of by reducing the value of . What is the maximum value that can have? We know that the maximum velocity that any object in the universe can have is the velocity of light, namely . When the escape velocity approaches , this is an indication that nothing, not even light, can escape from the gravitational pull of a highly dense object. Such an object is called a black hole. The radius into which the mass of the earth has to be squeezed so that it forms a black hole is called the Schwarzchild radius of the earth, called , and is given by

Surprisingly, a much more complicated calculation based on General Relativity yields exactly the same result for the Schwarzchild radius as given in eqn.(3.122). So the Schwarzchild radius of the earth is really tiny. For the Sun the Schwarzchild radius is km. As mentioned earlier, it is know from astrophysics that stars with masses greater than 3 times the mass of our Sun all end their stellar evolution by undergoing gravitational collapse and forming black holes.