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Relation among energy types

The expression of mechanical energy of the “Earth – satellite” system is typical of two body system in which one body revolves around other along a circular path. Particularly note the expression of each of the energy in the equation,

E = K + U

G M m 2 r = G M m 2 r G M m r

Comparing above two equations, we see that magnitude of total mechanical energy is equal to kinetic energy, but different in sign. Hence,

E = - K

Also, we note that total mechanical energy is half of potential energy. Hence,

E = U 2

These relations are very significant. We shall find resemblance of forms of energies in the case of Bohr’s orbit as well. In that case, nucleus of hydrogen atom and electron form the two – body system and are held together by the electrostatic force.

Importantly, it provides an unique method to determine other energies, if we know any of them. For example, if the system has mechanical energy of - 200 X 10 6 J , then :

K = - E = 200 X 10 6 = 200 X 10 6 J

and

U = 2 E = - 400 X 10 6 J

Energy plots of a satellite

An inspection of the expression of energy forms reveals that that linear distance “r” is the only parameter that can be changed for a satellite of given mass, “m”. From these expressions, it is also easy to realize that they have similar structure apart from having different signs. The product “GMm” is divided by “r” or “2r”. This indicates that nature of variation in their values with linear distance “r” should be similar.

Energy plots

Plots of kinetic, potential and mechanical energy .vs. distance

Since kinetic energy is a positive quantity, a plot of kinetic energy .vs. linear distance, “r”, is a hyperbola in the first quadrant. The expression of mechanical energy is exactly same except for the negative sign. Its plot with linear distance, therefore, is an inverted replica of kinetic energy plot in fourth quadrant. Potential energy is also negative like mechanical energy. Its plot also falls in the fourth quadrant. However, magnitude of potential energy is greater than that of mechanical energy as such the plot is displaced further away from the origin as shown in the figure.

From plots, we can conclude one important aspect of zero potential reference at infinity. From the figure, it is clear that as the distance increases and becomes large, not only potential energy, but kinetic energy also tends to become zero. We can, therefore, conclude that an object at infinity possess zero potential and kinetic energy. In other words, mechanical energy of an object at infinity is considered zero.

Gravitational binding energy

A system is bounded when constituents of the system are held together. The “Earth-satellite” system is a bounded system as members of the system are held together by gravitational attraction. Subsequently, we shall study such other bounded systems, which exist in other contexts as well. Bounded system of nucleons in a nucleus is one such example.

The characterizing aspect of a bounded system is that mechanical energy of the system is negative. However, we need to qualify that it is guaranteed to be negative when zero reference potential energy is at infinity.

Questions & Answers

A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
Samson Reply
water boil at 100 and why
isaac Reply
what is upper limit of speed
Riya Reply
what temperature is 0 k
Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
Mustapha
How MKS system is the subset of SI system?
Clash Reply
which colour has the shortest wavelength in the white light spectrum
Mustapha Reply
how do we add
Jennifer Reply
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
Abhyanshu Reply
x=5.8-3.22 x=2.58
sajjad
what is the definition of resolution of forces
Atinuke Reply
what is energy?
James Reply
Ability of doing work is called energy energy neither be create nor destryoed but change in one form to an other form
Abdul
motion
Mustapha
highlights of atomic physics
Benjamin
can anyone tell who founded equations of motion !?
Ztechy Reply
n=a+b/T² find the linear express
Donsmart Reply
أوك
عباس
Quiklyyy
Sultan Reply
Moment of inertia of a bar in terms of perpendicular axis theorem
Sultan Reply
How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
Yara Reply
Centre of mass of two uniform rods of same length but made of different materials and kept at L-shape meeting point is origin of coordinate
Rama Reply

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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