**Gravitational Force:**

** **The force of attraction between any two bodies in the universe is known as force.

**Newton’s Universal Law of Gravitation.**

It states that “Everybody in this universe attracts another body with force which is directly proportional to the product of their masses and inversely proportional to the square of distance between their centers.”

**Verification of Newton’s Universal Law of Gravitation.**

Consider two bodies of masses m_{1} and m_{2 }separated by the distance ‘d’ from their center. Let F be the force of attraction between them, then,

From Newton’s law of Gravitation,

Fa m_{1}m_{2} …………………….(i) (When d is constant)

Fa _{ }……………………….(ii) (When m_{1}m_{2} are constant) combining (i) and (ii) we get,

Fa

or, F=G [Where ‘G’ is gravitational constant whose value is 6.67×10^{-11} Nm^{2}/kg^{2}]

**Definition of Gravitational constant ‘G’:**

Gravitational constant is force of gravitation between two bodies of mass one kilogram each separated by a distance of 1 meter.

[G=6.67×10^{-11}Nm^{2}/kg^{2}]

**Gravity**

The force of attraction of massive bodies like earth which pulls an object lying on or near the surface towards its center.

. **Acceleration due to gravity ‘g’:**

Acceleration produced in a freely falling object towards the earth surface due to gravity of earth is called acceleration due

m |

**Relation between acceleration due to gravity and radius of planet:**

R |

Let ‘M’ is mass of earth and ‘R’ is radius, then force of attraction on object of mass ‘m’ lying at earth’s surface from Neton’s Law of gravitation will be,

M |

F=G …………………..(i)

Again if ‘g’ be the acceleration due to gravity on object, then from Newton’s second law of motion,

Force (f)=mg ………………(ii)

Here, in (i) and (ii) ‘F’ represents same force.

\mg=

or, g=

As mass of earth (M) is constant,

ga

The value of acceleration due to gravity is more (9.83m/s^{2}) at pole and less (9.788m/s^{2}) at equator as equatorial radius of earth is longer than polar radius.

**Factors affecting acceleration due to gravity ‘g’:**

i. Mass of earth (planet)

ii. Radius of earth (planet)

**Variation of acceleration due to gravity.**

i. Variation due to shape of earth:

The value of acceleration due to gravity is more (9.83m/s^{2}) at pole and less (9.788m/s^{2}) at equator as equatorial radius of earth is longer than polar radius. ga .

ii. Variation with change in height from earth’s surface: As ga value of ‘g’ decreases with increase in height from earth’s surface. Value of ‘g’ center of earth is zero.

**Acceleration due to gravity on surface of moon:**

For moon, mass (M) = 7.4×10^{22}kg

Radius (R)=1.74×10^{6}m

Then, g= =6.67×10^{-11}x =1.63m/s^{2}.

Hence, acceleration due to gravity of moon is about 1/6 of earth’s gravity. So, any object on moon’s surface is attracted by the force 6 times less than the force it is attracted in earth’s surface.

** Gravitational Field:**

It is defined as the space around the planet or satellite up to where it’s gravity can influence.

**Mass:**

Total amount of matter contained in a body is defined as mass. Its SI unit is kg. It is constant everywhere in the universe. It is scalar avantity and measured by using physical balance.

**Weight:**

The force of gravity acted on a body is called its gravity. Its SI unit is Newton. It is vector quantity. It is given as: w=mg.

As, wag, weight of an object changes with change value of acceleration due to gravity ‘g’.

**Free Fall:**

Freely falling of an object towards earth without external resistance is called free fall. During freefall, acceleration of an object is equal to acceleration due to gravity ‘g’.

** Weightlessness:**

The condition in which an object loses its weight apparently is called weightlessness. During freefall, body becomes weightlessness.

**Conclusion of coin and feather experiment:**

In the absence of external air resistance, acceleration due to gravity for all objects is same and doesn’t depend upon their masses.

**Escape velocity:** The minimum velocity required for the objects to escape from the gravitational pull of massive body such as earth is called escape velocity.

**Gravitational Potential:** The amount of work done by the gravitational force on bringing a unit mass from infinity to that point is called gravitational potential.

**Important formula**

**1. Gravitational force (F) = **

**2. Weight (W) =****, **

**3. Acceleration due to gravity (g) = ****, **

** g ^{1} = (**

**)**

^{2 }**×**

**g[At certain height (h)]**

** g ^{1}= (1-**

**)**

**×**

**g [At depth (d)]**

**4. Weight (W) = m ****×**** g**

**5. h=ut + ****gt ^{2}, v^{2}=u^{2}+2gh**

**6. Height jumped on earth ****×**** accel ^{n} due to gravity of earth= Height jumped on other heavenly body**

**×**

**acc**

^{n}due to gravity.**7. Mass lifted on earth ****×**** acc ^{n} due to gravity of earth = Mass lifted on other body **

**×**

**accelerate due to gravity of that body.**

**REASONABLE FACTS**

**1. Newton’s law of gravitation is called universal law.**

** Ans: **Since, the force of gravitation exists everywhere in the universe and holds true for all bodies whether microscopic or macroscopic so, Newton’s law of gravitation is called universal law.

**2. The value of ‘g’ at the poles is greater than at the equator.**

** Ans: **The value of ‘g’ at a place on the earth is inversely proportional to the square of the radius of the earth at the place (i.e.g¥1/R^{2}). The radius of the earth at the poles is less than that at the equator. So, the value of ‘g’ at the poles is greater (i.e.g=9.83m/s^{2}) than at the equator (i.e.g=9.78m/s^{2}) (or the value of g varies from place to place on the earth surface)

**3. Acceleration due to gravity decreases beyond the earth surface.**

** Ans-** Acceleration due to gravity decreases beyond the earth surface because it is inversely proportional to height from the earth surface.

**4. Weight of body in the earth is greater at the bottom of mountain than at its top.**

** Ans-** Weight is directly proportional to the acceleration due to gravity (W ¥ g) and the value of g at a place on the earth is inversely proportional to the square of the radius of the earth at the place (i.e.g¥1/R^{2}). The radius is shorter at the base than at its top. Hence weight of a body is greater at the bottom of mountain than at its top.

**5. Weight of body in the earth is greater than that in the moon.**

** Ans-** Weight is directly proportional to the acceleration due to gravity (W ¥ g). The acceleration due to gravity of the earth is six times greater than that of the moon. So the weight of body in the earth is six times greater than that in the moon.

**6. There is no use of parachute on the moon.**

** Ans-**The parachute falls freely with acceleration due to gravity on the moon due to absence of air resistance (Atmosphere). Hence there is no use of parachute on the moon.

**7. A feather and a iron ball simultaneously fall from the same height in a vacuum (on the moon surface).**

** Ans-** A feather and a iron ball simultaneously fall from the same height in a vacuum (or on the moon surface) due to free fall as the acceleration produced on the bodies is same irrespective of their masses if there is no air resistance so their speeds increase with the same rate. As a result, they reach the ground together.

**OR**

The time of fall for a freely falling body does not depend on the mass of the falling body. So both the bodies reach the ground simultaneously when released from the same height in a vacuum.

**8. A coin and a feather fall together at the condition of free fall.**

** Ans – **At the time of free fall, both coin and feather gain same acceleration of 9.8m/s^{2 }because the acceleration produced in the freely falling bodies is same for all the bodies and does not depend on the mass of falling bodies in the absence of external resistance. So, a coin and a feather fall together at the condition of free fall.

**9. A coin drops fast if the coin and feather are dropped simultaneously from the same height in air.**

** Ans – **A coin drops fast if the coin and feather are dropped simultaneously from the same height in air because the surface of the feather is very large compared to its mass (i.e.air resistance ¥ 1/acceleration of falling body).

**10. Weight of a body at the centre of the earth is zero.**

** Ans- **Since the value of acceleration due to gravity is zero at the centre of the earth, the weight of a body is zero at the centre of the earth.

**11. It is easier to lift a small stone than a large stone.**

** Ans- **Since the force of gravity is directly proportional to the mass of body so the force of gravity is more in the large stone as it has more mass (load) than in small stone. Hence, it is easier to lift a small stone than a large stone.

**12. The weight of a body is more in Biratnagar than in Kathmandu valley.**

** Ans- **As Biratnagar is near to the centre than from Kathmandu and we know that the acceleration due to gravity is inversely proportional with the square of the distance from the centre of the earth and the mass of a body always remains constant. Hence, the weight of a body is more Biratnagar than in Kathmandu valley.

**13. Parachutists are not hurt when they jump out of a plane from a very high place.**

** Ans- **When parachutists jump from a very high place, parachute falls down at the uniform velocity due to air resistance so parachutists can balance their body and are not hurt.

**14. Acceleration due to gravity of the Jupiter is large than that of earth.**

** Ans- **Acceleration due to gravity of the planet is directly proportional to its mass. The mass of the Jupiter is about 319 times greater than that of earth. So, acceleration due to gravity of the Jupiter is larger than that of the earth.

**15. A body appears weightless during free fall.**

** Ans- **The weight of a body is equal to the reaction which the body gets from supporting medium. A body is said to have free fall if it does not experience any kind of resistance or reaction is zero. Therefore, the weight of a body is zero during free fall.

**16. The fall of a parachute towards earth’s surface is not a free fall.**

** Ans-** The fall of a parachute towards earth’s surface is not a free fall because there is a presence of atmosphere around the earth’s surface that creates external resistance and reduces the acceleration of the parachute.

**17. The probability of getting hurt is more when a man jumps from greatest height, why?**

** Ans- **Acceleration is produced in falling body towards earth’s surface due to the influence of gravity. The velocity of the falling body also increases in every second of time at the rate of 9.8m/s. (i.e.g=9.8m/s^{2}). The force of the body increase as the increase in velocity. Therefore the probability of getting hurt is more as a man jump from a greatest or significant height.

**18. Parachutists are not hurt when they jump out of a plane from very high place. Why?**

** Ans- **Parachutists opens and expands when parachutists jump out of a plane. Larger the area of the parachute more will be the resistance of the air. As a result the acceleration of parachute decreases. Due to decreased acceleration due to gravity, parachutists fall down slowly. As a result of the slow motion, parachutists can balance their body and are not hurt when they jump out of a plane.

**19. Explain how an astronaut orbiting the earth in a spaceship is said to be weightless though the force of gravity at that distance may not be zero.**

** Ans- **When an astronaut in the spaceship is orbiting the earth, the spaceship and the astronaut are in the state of free fall towards the earth with the same acceleration due to gravity. The astronaut does not exert any force on the sides of the spaceship because the down ward acceleration of the astronaut is the same as that of the spaceship. As a result the astronaut feels to be floating weightlessly. Thus the astronaut becomes weightless with respect to the spaceship though the force of gravity at that distance may not zero.

**20. Gravitational constant (G) is also called universal constant.**

** Ans-** Gravitational constant is independent of nature and size of the bodies as well as nature of intervening medium. So, Gravitational constant also called universal constant is also called universal constant.

**21. The satellites do not need energy to revolve round the earth.**

** Ans- **The centripetal force necessary to revolve round the earth is provided by the gravitational force of earth. So, the satellites do not need any energy to revolve round the earth.

** **Note:

Satellites are launched from the earth’s surface to circle the earth. They are kept in their orbit by gravitational attraction of earth. Consider a satellite of mass ‘m’ which just circles the earth of mass ‘M’ close to its surface.

Centripetal force == = mg

V^{2}=gr

V=

=8000m/s (approx)

In Practice, the satellite is carried by rocket to the height of orbit and then given an impulse by firing jets to set it in a direction parallel to the tangent of the orbit. Its velocity is boosted to 8000m/s so that it stays in the orbit.

**22. The earth does not fall towards the sun due to its attraction.**

** Ans-** The gravitational force of attraction between the sun and the earth provides necessary centripetal force for the earth to revolve around the sun. The direction of velocity of the earth at any point on its orbit is tangential i.e. perpendicular to the force. Velocity does not have any component along the direction of force. Therefore, the earth does not collapse in to the sun.

**23. The space rockets are launched from west to east, usually in the equatorial plane.**

** Ans- **The earth rotates about its axis from west to east. So any point on the surface of the earth has linear velocity from west to east. When the rocket is launched from west to east, the linear velocity of the earth is added to the launching velocity of the rocket. At the equator, the linear velocity of rotation of earth is maximum. Therefore the space rockets are launched from west to east, usually in the equatorial plane.

**24. It is difficult to accelerate a heavy body as compared to a light one.**

** Ans-** The inertial mass of a body given by m=f/a. and a=f/m. The heavy body has greater inertial mass then the lighter one. To produce the same acceleration more force is required for heavy body.

## Additional Study

Gravitational force is the most prevalent force in the universe, pulling together on any two objects with mass in the universe. Further define gravitational force, along with the Universal Law of Gravitation, inverse square law, and learn how to write the equation used to calculate the gravitational force between two objects with mass.

## What is the Gravitational Force?

The universe has a lot of forces, a lot of pushes and pulls. We’re always pushing or pulling something, even if only the ground. But it turns out that in physics, there are really only four fundamental forces from which everything else is derived: the strong force, the weak force, the electromagnetic force, and the gravitational force.

The **gravitational force** is a force that attracts any two objects with mass. We call the gravitational force *attractive* because it always tries to pull masses together, it never pushes them apart. In fact, every object, including you, is pulling on every other object in the entire universe! This is called Newton’s **Universal Law of Gravitation**. Admittedly, you don’t have a very large mass and so, you’re not pulling on those other objects much. And objects that are really far apart from each other don’t pull on each other noticeably either. But the force is there and we can calculate it

This equation describes the force between any two objects in the universe:

In the equation:

*F*is the force of gravity (measured in Newtons, N)*G*is the gravitational constant of the universe and is always the same number*M*is the mass of one object (measured in kilograms, kg)*m*is the mass of the other object (measured in kilograms, kg)*r*is the distance those objects are apart (measured in meters, m)

So if you know how massive two objects are and how far they are apart, you can figure out the force between them.