Gravitation

Gravitation is one of the four fundamental forces in the universe .

 Newton's Law of Gravitation

Newton's Law of Gravitation states that the gravitational force between two bodies is directly proportional to the product of their masses and inversely proportional to the square of distance between them .

**F = G m1m2 / r2**

; where G is the gravitational constant .

Gravitational Force is independent of the intervening medium .

## Gravitational Constant

G = 6.67 x 1011 Nm2/kg2

The Gravitational constant is universal , i.e. its value does not change in different medium . G is the same for all medium .

## Acceleration due to Gravity

Acceleration due to gravity , is the acceleration produced by gravitational force . It is also called as gravitational field intensity .

For Derivation , Click Here : Acceleration due to Gravity

= =**g = GM / R2**

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### **Variation with Height**

For Derivation , Click Here : Variation of g with height

g = GM / (R+h)2 |gh = g (R/(R+h)2) | gh = g (1 - 2h/R)

### **Variation with Depth**

For Derivation , Click Here : Variation of g with depth

g = gh(1 - d/R)

### **Variation with Latitude**

For Derivation , Click Here : Variation of g with latitude

g' = g - Rω2cos2Φ

### **Variation with height and depth**

   The acceleration due to gravitation varies linearly inside the Earth's surface and non-linearly above Earth's Surface . At the centre of Earth and at an infinite distance above Earth's surface the gravitational acceleration is zero .

## Concepts of Gravitation

Gravitation has concepts that are analogous to Electrostatics , like field intensity , potential and potential energy . These terms help us to understand Earth's gravitational field , so that we can derive expression for Binding Energy .

### **Gravitational Potential Energy**

Gravitational Potential Energy is the work done in bringing a charge of unit mass from infinity to a point against the gravitational field .

Potential Energy (U) = Force . Displacement

U = - GMm /r

... U = Vm

The Gravitational Potential Energy is used in the derivation for Binding Energy for a satellite at rest as well as in orbit .

### **Gravitational Potential**

Gravitational potential is the amount of work done per unit mass in a gravitational field .

Potential (V) = - GM / r

... V = U / m

**Note :** The direction of Gravitational potential and gravitational potential energy is same and both are opposite to acceleration due to gravitation .

## Rocket Science

### **Critical Velocity**

For derivation , Click Here : Critical Velocity

The velocity , that is required to maintain an object in a stationary circular orbit around a body is called as critical velocity .

vc = sqrt (GM/r)

vc = sqrt (ghr)

**Weightlessness in Space**

 When a satellite is orbiting in an orbit , the gravitational force is balanced by the centrifugal force . Thus , the two forces cancel out each other and there is no net force acting on the astronaut . Also , since the satellite and astronaut , both are attracted to the earth by the same force , there is no normal force on the astronaut by the spaceship . Thus , the astronaut feels weightless .

### **Time Period**

For Derivation , Click Here : Time Period of a Satellite

T = 2π sqrt (r3 / GM)

### **Binding Energy**

For Derivation , Click Here : Binding Energy

The Energy by which an object is bound to a body's gravitational field is called as Binding Energy .

The total energy binds the object to the Earth . Thus , an opposite energy has to be applied to escape Earth's gravitation . Hence , binding energy is negative of Total Energy .

**B.E. = - T.E.**

At Rest the Object has only Gravitational Potential Energy .

**B.E. = GMm / r** {At rest}

When in orbit , the object has both Gravitational Potential and Kinetic Energy (KE = 1/2 mv2 ; where v = critical velocity )

**B.E. = (GMm) / 2r** {In Orbit}

### **Escape Velocity**

For Derivation , Click Here : Escape Velocity

The velocity at which an object can escape the gravitational pull of a body is called as Escape Velocity .

**ve = sqrt (2ghr)**

**ve = sqrt (2GM/r)**

Escape Velocity for an object on earth is 11.2 km/s

## Kepler's Laws

### **Law of Orbits**

Kepler's Law of Orbits states that all planets revolve around the sun in elliptical orbits with sun as the focus .

### **Law of Areas**

Kepler's Law of Areas states that a planet sweeps equal Areas in equal intervals of time around the sun .

### **Law of Periods**

Kepler's Law of Periods states that the square of Time Period is directly proportional to the cube of Distance

... **T2 α r3**

## Test Yourself !

Appear for a short test on Gravitation here .