As we have studied the gravitational
potential due to a solid sphere at various points in surrounding, and inside of
the sphere we can use the same results for the gravitational potential due to
Earth. Using these expressions, we can find the gravitational potential energy
of any mass located at a point anywhere on Earth. The position can be at some
height on surface or at a depth inside the surface.
See the video given below for its analysis.
https://youtu.be/dR7291ZXlCE
As an application of potential energy of a
body on Earth, we now study an application on calculation-the work done on a
body in displacing it from one height to another, in Earth's gravitational field.
This can be calculated by difference of potential energy of body at the two
locations.
See the example videos below to understand
the concept by an example -
https://youtu.be/8lWuA_0wDI8
https://youtu.be/jOa0kv_wv7o
In the previous articles, we studied about
the gravitational potential due to a uniform ring on its axis. Using the
concept we can also find the gravitational potential energy of a body. With a
uniform disc as due potential to a disc, can be obtained by integrating the
results due to elemental rings considered in the disc.
For detailed explanation of the process see
the example video below -
A Tunnel in a Solid Sphere : If a body is released from rest at some distance outside a solid sphere, having a diametrical tunnel along its length, then the body will accelerate due to gravitational force by the sphere. It enters into the tunnel, and inside the sphere the gravity decreases and becomes zero at its centre. To analyse the speed of a body during motion we can use its potential energy and apply work energy theorem to solve the case. See the example video below on same situation for understanding this application -
https://youtu.be/le2P6s8kNSA
Motion of a Body along the Axis of a Ring : When a body is released from a point anywhere on the axis of a uniform fixed ring, it starts moving towards the centre of the ring, and due to gravity its speed increases. Similarly if a body is projected from the centre of a ring with some initial speed along the axis, its speed decreases as it moves away from the ring. The analysis of speed of a moving body along the axis of a uniform ring in its gravitational field can be done by using work energy theorem. To see the same analysis in a case, see the example video below -
When we assemble a system of particles in a
given configuration and shape, the work done against the gravitational forces
of the system will be considered as the interaction potential energy of the
system. Similarly for a single body the interaction energy of all the particles
of the body is called gravitational self-energy of the body.
Same can also be defined as the work done
in assembling the body by bringing all its particles from infinity to the given
mass and shape of body. To understand the detailed explanation watch the video
below:
https://youtu.be/JLu4Vnq175U
Gravitational
Self Energy of a Uniform Hollow Sphere: This can be
calculated by finding the work done in bringing an elemental mass dm from
infinity to the surface of a hollow sphere and integrating the same over the
whole mass of the shell. The process is explained in the video below:
https://youtu.be/MWxYIyZ9jpc
Gravitational
Self Energy of a Uniform Solid Sphere: This can be
calculated by considering a solid sphere of radius x and we find the work done
in bringing an elemental mass dm from infinity to the surface of this sphere in
increasing the radius of sphere by dx, then we integrate the work done in
making the sphere from its radius limits 0 to R for the whole mass of sphere.
The process is explained in the video below:
Spaceships can be launched from earth to space in different
ways. To launch a satellite it has to be launched at specific speeds at
different altitudes where the speed depends upon the shape of orbit or the
trajectory in which it has to be set.
Watch the below video for understanding of trajectory of
spaceships and satellites in motion around a planet in its gravitational field.
https://youtu.be/wDQqSMjvgm0
Depending upon the speed of projection at an altitude the
spaceship can be setup in circular orbit, inner ellipse, outer ellipse,
parabolic or hyperbolic trajectory. All such cases are useful in different
types of space missions.
Kepler's Laws can be used in different types of problems
based on cases of satellite motion. See the Example videos below for understanding
of applications of Kepler's Laws.
Communication
Satellites: These are used in broadcasting of information over the Earth
surface at various regions. For a given satellite, if we draw the tangential
conical surface of the Earth, with the satellite located at the vertex of the cone,
then the area on the Earth's surface is called the Broadcasting region of the
satellite at an instant. To cover the basic understanding and analysis of
communication satellites, watch the video below:
https://youtu.be/gpKAVq5_VOA
Geostationary
Satellites: These are satellites which are deployed on the equatorial plane
of Earth, which is the plane containing the Earth's Equator, and which always
appears to be at rest with respect to a point on the Earth's surface. That
happens because the angular speed of geostationary satellites is equal to that
of the angular speed of rotation of Earth so this satellite will always
maintain a position vertically above a point on the equator on Earth's surface.
These satellites have a fixed broadcasting region as these appear to be stationary
from Earth. For understanding geostationary satellites, watch the video given
below:
Broadcasting Region of a Satellite and Applications of Satellite Motion
From the satellite the surface area of earth to which directly signals can be sent is called the
broadcasting region of the satellite. For a geostationary satellite the area on earth surface to which it
can broadcast signals will remain fixed as the satellite appear stationary with respect to earth at a fix
altitude. If a satellite is revolving around the earth surface will be able to cover more area on earth
surface. Analysis of which can be done by using the concept of solid angle. Watch the video below for
this.
https://youtu.be/wV_uY79ny90
Time taken by a Satellite in moving from one point to another : When a satellite is not in parking orbit,
it appears travelling from any point on earth surface. Watch the Example Video below for a situation on
time calculation of a satellite in moving from position above one point to another on earth surface.
https://youtu.be/VThjr100URI
Maximum Broadcasting latitude by a Geostationary Satellite : A geostationary satellite is always in
equatorial plane of earth and if we draw the tangents on earth surface from this satellite the boundary
of a region is obtained on earth surface in which the satellite can broadcast its signals. Watch the
Example Video below for analysis of such a situation.
https://youtu.be/e8x4QJ9yihs
Time calculation of two satellites at nearest positions : When two satellites are orbiting around earth in
same plane there are periodic instances when the two satellites gets closest to each other. To find the
time duration and the analysis of such a situation watch the example video given below.
https://youtu.be/h0QKmNIhDLE
Change in Orbit of a
Satellite: When a satellite is orbiting around the earth in an orbit of a given
radius, and for some reasons we need to shift the satellite from one orbit to another,
work is required to be done in such a situation. For an analysis watch the
Illustration Video below.
https://youtu.be/mo9eFDZJLrw
Concept of
Degenerated Ellipse: When a body is moving in the influence of
gravitational field of a planet its path in general is regarded as elliptical
and when a body moves directed in radial direction the trajectory can be
approximated as ellipse with minor axis zero. The analysis of this situation is
explained in the Example Video below.
https://youtu.be/cAfBzGgtqH0
Satellite hitting the
Earth Surface: A satellite orbiting in a circular orbit around earth when
changes its speed due to ejection of a rocket and finally hits the earth
surface, the situation can be analyzed by using conservation laws. Watch the
Example Video below for the detailed analysis.
https://youtu.be/bKPsTLfW3jc
Maximum height
attained by a body projected from Earth Surface: When a body is projected
from earth surface, its kinetic energy decreases and potential energy increases
as it moves away from the centre of earth. Analysis of such a situation is
explained in the below Example Video.
Physics
for JEE 2016: Gravitational Field and Satellite Motion
Transporting
a body between two stars: When a body is shifted
from surface of one planet or a start to the surface of another, the direction
of gravitational field changes during motion from one to another. We need to be
careful about the null point between the two while transporting. Watch the
Illustration video below for analyzing such a case.
https://youtu.be/Wf38jnwypFQ
Analysis
of a Double Start System: When two stars only under their mutual
gravitational force revolve around their common center of mass then such a
system is called 'Double Star' system. Watch the Illustration video below for
analysis on how to calculate the time period of revolution of a double start
system.
https://youtu.be/oxdg_ARt4is
Collision
of a body with a moving Satellite: When a satellite is revolving around a planet
and another moving body hits the satellite and stick to it than the resulting
velocity of the satellite after collision can be calculated by using law of
conservation of liner momentum and the resulting path of motion of satellite
around the planet changes. To analyze such a case watch the illustration video
below.
https://youtu.be/6AO1SROC4Fg
Third
Cosmic Velocity on Earth Surface: We already
discussed first and second cosmic velocities of a projected body from Earth
surface. Third cosmic velocity is defined as the projection speed at which a
body when projected will leave the solar system and come out of the Sun's gravity.
Watch the illustration video below to understand the same.
Advance Illustrations set III of Gravitational Field and Satellite Motion
A body projected from a satellite : When a body is projected from a satellite at some speed with respect to it, its net projection velocity will be the vector sum of its velocity with respect to the satellite and the velocity of the satellite. Depending upon the net velocity gained, the path of body can be analysed. Watch the Illustration video given below for understanding of such a case.
https://youtu.be/v3yFmghba2A
Effect of shrinking earth on gravity : If we consider that the earth starts shrinking due to some reason, it will affect the gravity on the earth's surface. Watch the Illustration video below to analyse the similar situation.
https://youtu.be/L5FnhJsCFnI
Escape velocity due to gravity of two planets : We have already discussed about the escape velocity of a body when it is thrown from the surface of a planet. To understand the concept of escaping a body from a point where a body is moving under the influence of two planets, watch the illustration video below for this. Similar case can be extended further for more than two planets also.
https://youtu.be/XpVk9TI4yHE
A particle moving inside a spherical cavity : We have already analysed the gravitational field due to a solid sphere. If inside the sphere we make a cavity and a particle's motion is to be analysed in this cavity, we first need to understand the gravitational field inside the cavity. Watch the Illustration video below to understand such a case.
JEE 2016 office sending text messages
to applicants whose documents sport errors
If you have committed any errors
while filling the JEE 2016 application form, be ready to receive a message from Joint Entrance Examination (JEE) authorities asking for changes.
JEE authorities have started checking
applications and begun to send text messages to those which seem like errors.
The errors are being primarily found in the Photograph, Thumb Impression, and
Signature areas.
JK Yadav, Deputy Secretary, JEE Unit confirmed
this development to PaGaLGuY. He said:
"We are verifying all the 2016 JEE applications. Those with discrepancies are being
sent text messages by the JEE office." Those who have received such messages
will need their 'Application Number' and 'Password' to login to make the necessary
changes.
Please note that this facility has been extended only to those who
receive messages from the JEE office and not all applicants. Those not
sure of their documents can visit the JEE website anyway and cross check.
If an applicant is welcomed with this
message -
'No discrepancy found in your uploaded Photograph, Thumb Impression and
Signature. You are not allowed to Login through this window,' - it means that
their uploaded documents have been accepted by the JEE Unit.
JEE is conducted by the Central Board
of Secondary Education (CBSE) every year and is the gateway to IITs, NITs,
IIITs, government and private engineering colleges in the country. For 2016,
JEE Main offline examination will be conducted on April 3, and online
examination on April 9 and 10.
When a substance is heated, the internal energy as well as vibrations of its molecules increases due to absorption of heat. Due to vibrations, the molecules in the substance also collide continuously with their neighbouring molecules and exert pressure on each other. Due to internal collisions, the whole substance experiences an outward pressure, and the separation between molecules of the substance starts increasing. This is called thermal expansion of body. For basic understanding of thermal expansion, watch the video below.
https://youtu.be/iaDmOBYLwyk
The internal energy of molecules is measured by temperature of the substance on different scales like Celsius, Kelvin, Fahrenheit, Reaumur, etc. For measurement of temperature on a scale, two fixed points are defined which are corresponding to freezing and boiling point of water. To understand the scales and conversions watch the video below.
Types of Thermal Expansion and Application for expansion of Cavities
There are three ways in which we categorise thermal expansion - Linear Expansion, Superficial Expansion and Cubical Expansion. These correspond to expansion in length, area and volume of a substance. For all types of thermal expansion, we define coefficients of expansion, which give an idea about the expansion tendency of materials. Higher coefficient of expansion indicates more expansion tendency of material. For step by step understanding of thermal expansion and its types, watch the video given below.
https://youtu.be/7ZKHc5J6R5Q
Expansion of Bounded Lengths & Cavities : When a material is heated and its temperature increases, the separation between the substance's molecules increases uniformly in its volume if the lattice structure of the substance is isotropic. If we consider a length between two molecules of a substance, then this length will expand/contract with the same coefficient of substance, no matter if the material is solid or hollow. Same concept is used for expansion of cavities in solid bodies. To understand this concept watch the video below.
Force on a Rod and Effect of temperature on Pendulum Clocks
Force on a Rod between rigid walls : If a rod is supported between two rigid walls of exactly same size as that of separation between the walls, a normal force is developed between ends of the rod and walls if temperature of system increases. Due to rise in temperature the rod expands and it pushes the walls outside and experiences same force inward by walls. This inward force compresses the rod elastically as walls are rigid. The normal force can be calculated by using the Young's modulus of the rod. Watch the video below for complete analysis of this situation.
https://youtu.be/viQ_E2kcEtU
Effect of Temperature on Pendulum Clocks : The time period of a pendulum depends on its length and as in summers temperature increases, time period of pendulum clocks also increase and due to this clocks slow down and start loosing time. Similarly in winters due to low temperature pendulum clocks gain time and become faster. Watch the analysis video below to understand the concept.
https://youtu.be/0RcpqFSGtII
To understand the applications of the concept see the example videos given below -
The admit card for JEE Main 2016 will be released before March 12, 2016, JEE authorities confirmed the development to PaGaLGuY. Applicants are advised to download their admit cards from JEE Main website only. Candidates will require their Application Number, and Class 12 board exam details, to download the admit card.
Apparent Expansion of a Liquid and Tension in a Clamped Wire
Concept of Apparent Expansion : When a liquid is filled in a container and it is heated, due to rise in temperature it expands and liquid always expands more than solid so if it is completely filled in container, some liquid will overflow. But due to expansion of container whole expansion volume of the liquid will not overflow. This is called apparent expansion of the liquid with respect to container. This concept helps in solving numerical problems based on this concept. To understand how to quickly solve such numerical problems, watch the below given video.
https://youtu.be/-8EIN5_u5v0
Tension in a Clamped Wire : When a metal wire is clamped between two rigid supports in almost taut state without any tension, on raising the temperature it expands and a sag is developed in the wire but if temperature is decreased, it tends to contract but as clamps are rigid it elastically expands to compensate the thermal contraction which develops a tension in the wire. To understand the concept, watch the below given video.