Grade 11 Physics Note

Gravity and Gravitation

Gravity:

A force that attracts everything towards the centre of the earth is called as gravity. Its value is 9.8 m/s.  Its unit is m/s (meter per second). The radius in the equatorial plane is larger than the radius along the poles. The value of g is accordingly larger at poles and less at the equator, since g = GM / R2g will be larger where R is smaller. Therefore g at the equator is less than that of pole.

 

 

Escape velocity;

Escape velocity is defined as the minimum velocity that an object must have in order to escape the gravitational field of the earth, which is the escaping of the earth without ever falling back.

 

Weightlessness:

When a satellite is falling freely in space, everything within this freely falling system will appear to be weightless.  It does not matter where the object is, whether it is falling under the force of attraction of the earth, the sun, or some distant star.

                                        OR

The phenomenon of “weightlessness” occurs when there is no force of support on your body. When your body is effectively in “free fall” accelerating downward ate the acceleration of gravity, then you are not being supported.

Taking the example of roller-coaster ride which is constrained to follow a track, then the condition for “weightlessness is met when the downward acceleration of your seat is equal to the acceleration of gravity.

 =

 

Geo – stationary satellite:

Geo-stationary satellite is the one whose orbital motion is synchronized with the rotation of the earth.  In this way the synchronous satellite remains always over the same point on the equator as the earth spins on its axis.

Newton's law of gravitation

It states that, "the gravitational force of attraction between any two object is directly proportional to the product of the mass and inversely proportional to the square of distance between them."

i.e.  ------------- (i)

 ----------------- (ii)

Where, m1 and m2 be the masses of object A and B and 'r' be the distance between them

Combining equation (i) and (ii), we get

 ------- (iii)

Where G is a constant called universal gravitation constant and its value is  and its dimension is

If

 or

Hence the universal gravitational constant is defined as the gravitational force of attraction between any two objects of equal masses of 1 kg each separated by a distance of 1m.

 

The variation of acceleration due to gravity below the earth’s surface:

Bodies allowed to fall freely were found to fall at the same rate irrespective of their masses (air resistance being negligible). The velocity of a freely falling body increased at a steady rate, means the body had acceleration. This acceleration is called acceleration due to gravity or simply g.

 

If F is the force, m is the mass of the body, g is the acceleration due to gravity, M is the mass of the Earth, R is the radius of the Earth and G is the gravitational constant, then we know that

F = mg

By comparing above equations,

 

 

Variation of 'g' with depth

 

Consider a body of mass m, lying on the surface of the Earth of radius R and mass M. Let g be the acceleration due to gravity at that place.

 

Let the body be taken to a depth d from the surface of the Earth. Then, the force due to gravity acting on this body is only due to the sphere of radius R minus depth d that is (R - d).

 

If g is the acceleration due to gravity at depth ’d’ then

 

 

Let the Earth be of uniform density r and its shape is a perfect sphere.

 

Where r is the radius of the sphere

 

Mass of Earth = Volume of earth × Density of earth

 

M =

 

Where ρ is the density of the Earth, So

 

 

Acceleration due to gravity at the Surface,

 

 

Comparing g’ and g

 

 

 

 

Therefore the acceleration due to gravity decreases with increase in depth.

 

Gravitational potential Energy:

The work done in moving a unit mass from one point to another in a gravitational field is called gravitational potential difference.

The SI unit of gravitational potential energy is JKg-1.

 

Explaination

If a body of mass m is at a distance r from a body of mass M. The gravitational potential energy of the mass m is defined as the work done in bringing it from infinity to that point.

 

The work done in moving the mass m through a small distance dr in the gravitational field of the body or planet of mass M can be given as,

 

The total work done in bringing mass m from infinity to that point, is

 

 

 

 

 

 

Thus the Gravitational potential energy (U) can be given as,

 

In terms of Per Unit mass,

 

Gravitational Potential Energy on Earth’s Surface:

Gravitational Potential Energy per Unit mass on the surface of earth can be calculated by substituting following values in above equation.

 

G = 6.67x10-11Nm2kg-2 

Mass of the Earth is 6.0x1024kg

Radius of the Earth is 6.4x106m

 

 

 

Expression for the total energy of a satellite orbiting round the earth:

When a satellite goes round the earth in any orbit, it possesses both kinetic energy as well as potential energy. The sum of these two

If  then, the total energy is,

 --------------- (i)

But,

 -------------- (ii)

Where,

m is the mass of the object

v is the velocity of the satellite in any orbit But, the satellite is moving far from earth. So,

or,  -----------(iii)

Putting the value of equation (ii) we get,

 ----------(iv)

But the potential energy is equal to the gravitational potential energy in that orbit.

i.e.

 ----------- (v)

Putting the value of

Or,  ------------- (vi)

The negative sign indicates that, this energy is due to attractive force between the earth and the satellite and hence, the satellite is bound to earth.

Gravitational potential energy is equal to gravitational potential* mass of the object:

The amount of work done in bringing an object from infinity to earth surface is called gravitational potential energy. It is denoted by u

So, gravitational potential energy (u) = W

Where, W is the work done to bring an object from infinity to earth's surface and it is found as follows:

Let O, M and R be the center, mass and radius of earth respectively. In which, the whole mass of earth is supposed to be concentrated at point O.

Let P be any point at distance 'x' from the center of earth in which an object of mass 'M' lies.

Then, gravitational force between the earth and the object is

 ---------- (ii)

If the same object is brought towards earth from P to Q through a small distance . Then, the small work done will be

 ---------- (iii)

Now, to bring the object from infinity to point A on earth's surface, we have to integrate equation (iii) to get the total work done. So,

Total work done(w)

or,

or,

or,

or,

or,

or,

or,

or,

or,  -------------(iv)

Putting the value of w in equation (i) we get,

or,  --------------(v)

The (-ve) sign indicates that the gravitational potential energy is due to the attractive force between the earth and the object.

Equation (v) gives the expression for gravitational potential energy on earth surface.

Gravitational Potential (UA): The gravitational potential energy per unit mass is called gravitational potential and it is denoted by UA at point A on earth surface.

So, gravitational potential (UA) =  ---------- (i)

Where, 'U' is the gravitational potential energy on earth's surface and it is found as follows:

Let O, M and R be the center, mass and radius of the earth in which the whole mass of earth is supposed to be concentrated at point O.

Let P be the any point on earth surface at a distance of 'x' from the center of earth in which an object of mass 'm' lies

Then, the gravitational force between them is given by,

 ----------- (ii)

If the same object is brought from P to Q at a distance of , then small work is done

 ---------- (iii)

Then, to bring the object from infinity to point A on earth's surface, we need to integrate the small work done in (iii) to get the total work done.

 -------------- (iv)

Putting the value of w in equation (i)

 ----------- (v)

Putting the value of U from equation (v) in (i), we get

 ------------ (vi)

The negative sign indicates that the gravitational potential is due to the attractive force between the earth and the object.

Equation (vi) gives the expression for gravitational potential on earth's surface.

Hence gravitational potential energy = gravitational potential* mass of the object.

Satellite, expression for the orbital velocity:

Satellite

It is an artificial body placed in orbit round the earth or another planet in order to collect information or for communication.

Or

It is a celestial body orbiting the earth or another planet.

Orbital Velocity

Orbital velocity is the velocity given to the body to keep it in orbit. This velocity is usually given to the artificial satellite so that it revolves round any particular planet.

 

Expression for Orbital Velocity

Orbital Velocity formula can be deduced from Equating Centripetal force of the Satellite revolving in the orbit with the gravitational force between planet and the Satellite. 
 

Let a satellite of mass m revolving around the Planet of mass M in the orbit of radius R with speed V then mathematically it can be expressed as,
               

 

Time Period

The Time period of a satellite is the time it takes it to make one full orbit around an object.

Speed of a satellite around a planet can be given as,

The satellite travels around the entire circumference of the circle, which is

If R is the radius of the orbit in the period T, Then the orbital speed must be,

Giving you

If you solve this for the period of the satellite, you get

 

Escape velocity:

It is the speed at which the sum of an object's kinetic energy and its gravitational potential energy is equal to zero. It is the speed needed to "break free" from the gravitational attraction of a massive body, without further propulsion, i.e., without spending more fuel.

 

Escape velocity is actually a speed (not a velocity) because it does not specify a direction: no matter what the direction of travel is, the object can escape the gravitational field (provided its path does not intersect the planet).

Factors involved in gravitational escape velocity                                                                          

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Expression for the Escape Velocity:

The simplest way of deriving the formula for escape velocity is to use conservation of energy. Imagine that a spaceship of mass m is at a distance R from the center of mass of the planet whose mass is M. Its initial speed is equal to its escape velocity,v_e

 

If Kinetic energy of the Body launched from earth is equal to its Gravitational potential energy, then it could escape safely form the gravitational field.

 

That is,

 

 

By Solving above we get

 

Black holes:

Black holes are some of the strangest and most fascinating objects found in outer space. They are objects of extreme density with such strong gravitational attraction that even light cannot escape from their grasp if it comes near enough.

Albert Einstein first predicted black holes in 1916 with his general theory of relativity. The term "black hole" was coined in 1967 by American astronomer John Wheeler, and the first one was discovered in 1971.

There are three types:

1. Stellar black holes

2. Super-massive black holes

3. Intermediate black holes.

 

Stellar black holes — small but deadly

When a star burns through the last of its fuel, it may find itself collapsing. For smaller stars, up to about three times the sun's mass, the new core will be a neutron star or a white dwarf. But when a larger star collapses, it continues to fall in on itself to create a stellar black hole.

 

Super-massive black holes — the birth of giants

Small black holes populate the universe, but their cousins, super-massive black holes, dominate. Super-massive black holes are millions or even billions of times as massive as the sun, but have a radius similar to that of Earth's closest star. Such black holes are thought to lie at the center of pretty much every galaxy, including the Milky Way.

 

Intermediate black holes – stuck in the middle

Scientists once thought black holes came in only small and large sizes, but recent research has revealed the possibility for the existence of midsize, or intermediate, black holes. Such bodies could form when stars in a cluster collide in a chain reaction. Several of these forming in the same region could eventually fall together in the center of a galaxy and create a super-massive black hole.

 

Black hole theory — how they tick

Black holes are incredibly massive, but cover only a small region. Because of the relationship between mass and gravity, this means they have an extremely powerful gravitational force. Virtually nothing can escape from them — under classical physics, even light is trapped by a black hole.

Such a strong pull creates an observational problem when it comes to black holes — scientists can't "see" them the way they can see stars and other objects in space. Instead, scientists must rely on the radiation that is emitted as dust and gas are drawn into the dense creatures. Super-massive black holes, lying in the center of a galaxy, may find themselves shrouded by the dust and gas thick around them, which can block the tell-tale emissions.

Sometimes as matter is drawn toward a black hole, it ricochets off of the event horizon and is hurled outward, rather than being tugged into the maw. Bright jets of material traveling at near-relativistic speeds are created. Although the black hole itself remains unseen, these powerful jets can be viewed from great distances.

Black holes have three "layers" — the outer and inner event horizon and the singularity.

The event horizon of a black hole is the boundary around the mouth of the black hole where light loses its ability to escape. Once a particle crosses the event horizon, it cannot leave. Gravity is constant across the event horizon.

The inner region of a black hole, where its mass lies, is known as its singularity, the single point in space-time where the mass of the black hole is concentrated.

Under the classical mechanics of physics, nothing can escape from a black hole. However, things shift slightly when quantum mechanics are added to the equation. Under quantum mechanics, for every particle, there is an antiparticle, a particle with the same mass and opposite electric charge. When they meet, particle-antiparticle pairs can annihilate one another.

If a particle-antiparticle pair is created just beyond the reach of the event horizon of a black hole, it is possible to have one drawn into the black hole itself while the other is ejected. The result is that the event horizon of the black hole has been reduced and black holes can decay a process that is rejected under classical mechanics.


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