Gravitation

The force of attraction applied by earth is called its gravity.

The earth attracts (or pulls) all the objects towards its centre. The force with which it pulls objects towards itself is called gravitational force of earth or gravity.

Example:

A stone dropped from a height falls towards earth because the earth exerts a force of attraction (called gravity) on the stone and pulls it down.

According to Newton, every object in this universe attracts every other object with a certain force. The force with which two objects attract each other is called gravitation.

The gravitation acts between two objects even if the two objects are not connected by any means.

If the mass of the objects (or bodies) are small, then the gravitational force between them is also very small (which cannot be detected easily).

When an object is dropped from some height, a uniform acceleration is produced in it by the gravitational pull of earth and this acceleration does not depend upon the mass of falling body. This uniform acceleration is called acceleration due to gravity.

The acceleration due to gravity is represented by ‘g’ and its value is 9.8 ms^-2.

The value of g changes slightly from place to place.

Where, G = Gravitational Constant

M = Mass of Earth

R = Radius of Earth

Universal Law of Gravitation given by Newton is called Newton’s Law of Gravitation. This law states that, “Every object in the universe attracts every other object with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.”

Suppose two bodies A and B of mass m₁ and m₂ be at distance r from each other.

So, according to Newton’s law of gravitation,

F …..1

And F …..2

Thus, combining equations 1 and 2 we get,

Where, G = Universal Gravitational Constant

G = Nm²kg^-2

The earth attracts every object towards its centre with a certain force which depends on the mass of the body and the acceleration due to gravity at that place

The weight of the body is the force with which it is attracted towards the centre of the earth.

Force = Mass × Acceleration

Where, Force = Weight = W

Acceleration = Acceleration due to gravity = g

And, Mass = m

Thus, W = m × g

Also, weight is a vector quantity that is it has both magnitude as well as direction.

The weight of the body is the force with which it is attracted towards the centre of the earth.

W = m × g

Weight of the body changes from place to place, as the value of g changes from place to place.

Acceleration due to gravity g is zero at the centre of earth, as at the centre of earth we are surrounded by equal masses in all the direction, hence equivalent gravitational force acting on us due to earth is zero and thus making acceleration due to gravity to zero.

So, W = m × g = 0

Hence, weight at the centre of earth is zero.

The moon never actually falls on the surface of earth.

Now, to understand this, we first need to understand the nature of the force between the moon and earth and also the movement of moon under this force.

So, the gravitational force is the force that acts between the earth and the moon and the moon moves in the circular orbital due to the centripetal force generated by the gravitational force. Similarly, earth and sun also experience this gravitational force between themselves.

So, yes moon does fall towards earth, but earth also moves in its orbit and covers more distance away from the moon relative to the distance covered by the moon. Hence, moon never falls on the surface of the earth.

Suppose we have an object of mass = m also let the weight on moon be and weight on earth be.

Now, weight of object on moon will be:

…..1

Now the weight is actually the force with which moon attract the object.

M = Mass of Moon

r = radius of Moon

Now, we know

Mass of earth = 100 times mass of moon

Radius of earth = 4 times radius of moon

So, weight of object on earth will be given by:

…..2

Now, dividing equation 1 by 2, we get,

Thus we can say that weight of an object on the moon one-sixth its weight on the earth.

We know that weight of a body is the force with which the body is attracted towards the centre of earth.

W = m × g

Where, m = Mass of the body

And, g = Acceleration due to gravity

The mass of a body or object is constant and does not change from place to place.

But acceleration due to gravity is inversely proportional to the square of the distance from the centre of earth. So, as we go up from surface of the earth, the distance from the centre of earth increases and hence value of g decreases.

Since, W = m × g

So, as the g decreases, weight of the body also decreases with increase in height above the earth. Thus, as we go higher above the surface of the earth the weight of the body decreases.

Force on an object of mass m falling freely under gravitation by the earth is given by Universal Law of Gravitation:

F = or, F m

Now, clearly from above we can see that F depends on mass of the object.

Since, M = Mass of earth

r = Radius of earth

G = Nm²kg^-2 are constants

If g is the acceleration due to gravity on an object of mass m then,

F = m × g

Or, g =

Or, g =

Thus, g is independent of mass of the object.

Now, let two objects A of mass and B of mass fall from height h such that

Now, using the following equation of motion,

Here, u = 0, s = h and a = g

So,

Or,

Now, from above, since time depends on both h and g, which are both independent of masses, so both A and B come down at same time.

Weight of a body is nothing but the force with which it is attracted towards the centre of earth.

Force = Mass × Acceleration

W = m × g

Since, we have Force = Weight = W and acceleration is acceleration due to gravity or g.

A body becomes weightless when g becomes zero and this happens when acceleration due to gravity becomes zero and thus body becomes weightless.

Example:

At centre of earth, far away from earth’s surface, freely falling body

As we go deeper inside the earth weight of object changes and mass remains invariant.

Since, mass of an object is the amount of matter contained in the object and as we know that going deep inside the earth will not change the amount of matter contained in the object, so mass remains constant. In fact mass of an object always remains constant and does not change from one place to another.

On the other hand, weight of an object is the force with which earth pulls an object towards its centre.

W = m × g

From above, as we know that g decreases as we go deep inside the earth, so the value of the weight of the object also decreases.

When a body is taken from earth to the moon, then its weight will decrease to one – sixth its weight on earth.

Explanation:

Suppose we have an object of mass = m also let the weight on moon be and weight on earth be.

Now, weight of object on moon will be:

…..1

Now the weight is actually the force with which moon attract the object.

M = Mass of Moon

r = radius of Moon

Now, we know

Mass of earth = 100 times mass of moon

Radius of earth = 4 times radius of moon

So, weight of object on earth will be given by:

…..2

Now, dividing equation 1 by 2, we get,

Thus we can say that weight of an object on the moon one-sixth its weight on the earth.

Universal Law of Gravitation given by Newton is called Newton’s Law of Gravitation. This law states that, “Every object in the universe attracts every other object with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.”

Suppose two bodies A and B of mass m₁ and m₂ be at distance r from each other.

So, according to Newton’s law of gravitation,

F …..1

And F …..2

Thus, combining equations 1 and 2 we get,

F =

Where, G = Universal Gravitational Constant

G = Nm²kg^-2

Newton’s law of gravitation is called universal law of gravitation because it is applicable to all bodies having mass: whether big or small; whether the bodies are terrestrial (which are on earth) or celestial (which are in outer space) such as stones, planets and satellites.

The acceleration produced in a freely falling body by the gravitational pull of the earth is called the acceleration due to gravitation.

We know that,

Force = Mass × Acceleration

F = m × a

a = …..1

Where F is the force on the object of mass m dropped from a distance r from the centre of earth of mass M.

So, force exerted by the earth on the object is

F = …..2

M = Mass of Earth

m = mass of object

r = distance of object from centre of earth

Now, from equation 1 and 2,

a =

a =

Now, from above,

a = g = Acceleration due to gravity

We also see that, although force is depending on the mass of the object,

F =

But acceleration due to gravity is independent of the mass.

g =

Factors on which g depends are:

(i) Value of gravitational constant (G)

(ii) Mass of Earth (M)

(iii) Radius of Earth (r)

As gravitational constant G and mass of earth M are always constant, so the value of acceleration due to gravity g is constant as long as the radius of earth remains constant.

At the surface of earth also, the value of g is not constant. ()

At the poles, radius of earth is minimum, hence the g is maximum. Similarly at the equator, the radius of earth is maximum, and hence the value of g is minimum. Also as we go up from the surface of the earth, distance from the centre of the earth increases and hence value of g decreases.

The value of g also decreases as we go inside the surface of earth and g is zero at the centre of earth, as at the centre of the earth the object has mass around it, so net force cancels and thus net acceleration becomes zero.

Let H be the height of the cliff.

Now, ball is initially at rest

So, u = initial velocity = 0

a = acceleration = g

v = final velocity

t = B seconds

(i) From the equations of motion, we know that,

(ii) Now, we also know that,

v = u + at

v = 0 + gB

v = gB

Let,

u = Initial velocity of stone

H = Maximum height attained by the stone

v = final velocity at maximum height = 0

t = time taken to attain maximum height = 3 s

a = -g = acceleration due to gravity

(i) We know that,

v = u + at

0 = u – g × 3

u = 3g = 3 × 9.8 ms^-2

u = 29.4 ms^-1

(ii) From the equations of motion, we know that,

H = 132.3 m

Let,

u = Initial velocity

v = Final velocity

h = Height attained

a = acceleration

t = time

Ascent:

Ball thrown upward with velocity u. So, v = 0 at maximum height h. Also a = -g

Now, using v = u + at

0 = u - g

…..1

Where = time of ascent

Also, by using

u = …..2

From equations 1 and 2, we get,

…..A

Descent:

Ball comes down from height h. So, u = 0 and a = g.

So, using v = u + at

v = g …..3

Where = Time of descent

Now, using

v = …..4

From equations 3 and 4, we get,

…..B

From equations A and B, we see that,

Given:

u = Initial velocity

Ball is thrown up

Ball goes up and attains a maximum height H and then again comes down.

Let

u = Initial Speed

v = Final Speed

t = time

a = acceleration

s = distance

Case – I

Ball is going upward

a = -g

u = u

v = 0

s = H

Since,

H = …..1

Case – II

Ball comes downward

So, u = 0

v = Final Speed

a = g

Since,

H = …..2

From equations 1 and 2, we see that,

v = u

Let the mass of the object be m.

Now, it is given that weight of the object = 98 N.

Also, we know that,

Weight = Mass × Acceleration

W = m × g

m =

m = = 10 kg

A sheet of paper falls slower than the one crumpled into a ball since the sheet of paper has the larger surface area. The air introduces a resistance to the falling paper. Since the sheet of paper has more surface area than the crumpled paper, so sheet of paper experiences more air resistance than that of the crumpled paper.

If there was no air resistance (friction), then both the sheet of paper and crumpled ball of paper would have fallen at the same time.

Force of gravity or the gravitational force of earth is the force that we experience towards the centre of the earth.

The gravitational force (pull) of earth is responsible for holding everything near earth’s surface, it keeps us firmly on the ground.

If the force of gravity is somehow vanished away, then we would no longer be bound to the earth’s surface and would be sent flying into the space.

I will weigh more at poles and weigh less at the equator since weight is the amount of force with which an object is pulled towards the centre of earth.

W = m × g …..1

Where, W = Weight

m = mass

g = Acceleration due to gravity

Since, mass is constant all the time, so,

W g …..2

And we also know that,

g =

Where, G = Gravitational Constant

R = Radius of earth

M = Mass of earth

So, g …..3

From equations 2 and 3, we get,

W

Since R is minimum at poles, so weight is maximum at poles and similarly since R is maximum at equator, so weight is minimum at equator.

Hence, we weigh more at poles than at equator.

We know that earth is not a perfect sphere. So the value of R is not the same at all places.

So, the value of acceleration due to gravity g is not constant at all places on the surface of earth.

Where, G = Gravitational Constant

M = Mass of Earth

R = Radius of Earth

Since, G and M are constant, so g depends on the radius of earth.

So, from the above we can say that,

g

Now, radius of earth is minimum at poles. So, acceleration due to gravity is maximum at poles.

But radius at equator is maximum, so acceleration due to gravity is minimum at equator.

No, acceleration due to gravity do not depend on the mass of the body.

Explanation:

The acceleration produced in a freely falling body by the gravitational pull of the earth is called the acceleration due to gravitation.

We know that,

Force = Mass × Acceleration

F = m × a

a = …..1

Where F is the force on the object of mass m dropped from a distance r from the centre of earth of mass M.

So, force exerted by the earth on the object is

F = …..2

M = Mass of Earth

m = mass of object

r = distance of object from centre of earth

Now, from equation 1 and 2,

a =

a =

Now, from above,

a = g = Acceleration due to gravity

We also see that, although force is depending on the mass of the object,

F =

But acceleration due to gravity is independent of the mass.

g =

Newton’s law of gravitation states that: Every object in the universe attracts every other object with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Hence, option D is correct.

We know that the value of g is inversely proportional to the square of radius or g .

Radius of earth is minimum at poles, so acceleration due to gravity is maximum at poles. Similarly, radius of earth is maximum at equator, so acceleration due to gravity is minimum at equator. Also acceleration due to gravity decreases as we go higher from the earth surface.

Hence, option D is correct.

Here,

u = Initial velocity = 0

v = Final velocity

s = Distance travelled = 20 m

a = acceleration = g = 10 ms^-2

Now, we know from the equations of motion that,

ms^-1

v = 20 ms^-1

Hence, option D is correct.

When a ball is thrown vertically upward, then acceleration due to gravity is directed opposite to the direction of motion since the ball experiences a force of attraction directed towards the centre of earth due to earth’s gravity.

Direction of acceleration is in the direction of force as,

So, from above, we can conclude that acceleration is also directed towards the centre of earth which is opposite to the vertically upward motion of the ball.

Hence, option A is correct.

At the top of its path in a projectile, the acceleration is in downward direction.

This acceleration is the acceleration due to gravity and is directed towards the centre of earth.

Hence, option C is correct.

The acceleration due to gravity goes on decreasing for both the scientists A and B.

For scientist B,

Since,

So, as the distance from the centre of earth increases, the value of g decreases.

For scientist A,

Let scientist A be at a depth d. Then only attraction is due to the sphere (R – d), where R is the radius of earth. As the attraction due to rest of the earth cancels out, so the value of g decreases.

Hence, option D is correct.

The revolution of earth around the sun shows that there must be a force acting on the earth and directed towards the sun.

Since revolution requires a centripetal force and this centripetal force has provided the gravitational force between the earth and sun.

Hence, option B is correct.

If the earth stops rotating, then the value of g at the equator increases and becomes equal to the value at poles. This is because, if the earth will stop rotating, then the equatorial bulge will vanish and earth will have a uniform radius and since, so g is same through the surface of earth now.

Hence, option A is correct.

Since weight of object is more at poles than at the equator, so it is beneficial to purchase goods at equator and sell at poles provided a spring balance is used as a spring balance measures the weight of an object whereas equal beam balance measures mass and mass is constant and does not change from place to place.

Hence option B is correct.

The gravitational constant is a universal constant. It remains as it is in any condition. It does not depend on any of the factors like the temperature, mass, distance between the masses.

Value of G is Nm²kg^-2.

Hence, option D is correct.

We know that, where M is mass of earth and R is radius of earth.

Now, from the question,

M’ =

R’ =

So, the g’ will be:

Hence, option B is correct.

In vacuum all falling objects will have same acceleration.

The constant acceleration is due to gravitational force of earth and is known as acceleration due to gravity given by:

Where,

G = Gravitational Constant = Nm²kg^-2

M = Mass of Earth

R = Radius of Earth

From above, we see that all variables in the equation are constant. Thus g dosen’t depend on any other factors, and hence is constant.

Hence, option C is correct.

We know that,

Where,

G = Gravitational Constant = Nm²kg^-2

M = Mass of Earth

R = Radius of Earth

Now, the spaceship is at a distance R’ = 2R

So, g will be:

ms^-2

Hence, option D is correct.

We know that,

For earth

Now, for the new planet we have,

M’ = and R’ =

So, g’ will be calculated as:

ms^-2

Hence, option A is correct.

The stone is dropped from the cliff. So,

Initial Speed = u = 0

Distance = s = 100 m

Final Speed = v

Acceleration = a = g = 9.8 ms^-2

Now,

v = 44.27 ms^-1

Hence, option B is correct.

A ball is thrown upward. So,

Initial Speed = u

Distance = s = 100 m

Final Speed = v = 0

Acceleration = a = g = -9.8 ms^-2 (negative, since acceleration is opposite the motion)

Now,

u = 44.27 ms^-1

Hence, option B is correct.

Let the height be H

Time taken = t = 4 seconds

Initial Velocity = u = 0

Acceleration = a = g = 9.8 ms^-2

Now,

m

Hence, option D is correct.

Weight of an object is the force with which it is attracted towards the earth.

The quantity of the matter present in an object is called the mass and it is related with the object’s inertia whereas weight is written as W = m × g, where g is acceleration due to gravity.

Hence, option D is correct.

The equation is valid for spherical bodies.

The gravitational equations are valid for point masses. For spherical bodies the centre of mass can be conveniently chosen at the centre and mass is evenly distributed about the centre.

Hence, option B is correct.

If the distance between two objects is doubled the gravitational force becomes one fourth since,

Where R is the distance between the bodies.

So, if R’ = 2R, then

Hence, option D is correct.

Let the aircraft be moving horizontally with speed u. Now, when the bomb is released, it experiences gravitational force on it and thus acceleration due to gravity in vertically downward direction.

Initially:

Finally:

Substituting equation 1 in 2, we get,

The above equation is an equation of parabola.

Hence, option B is correct.

Given:

Height of Tower = 100 m

Time taken to reach ground = T seconds

Acceleration = a = g = 9.8 ms^-2

Initial Speed = u = 0

Final Speed = v

Now, we know that,

At t = T and s = 100 m …..1

Let t = and s = x

Then, …..2

So, dividing equation 1 by 2, we get,

x = 25 m

So, 25 m from the top at t = seconds or (100 – 25) m, that is 75 m from the bottom at seconds.

Hence, option B is correct.

Now, initial separation = 1 m

Body 1:

Initial Speed = 0

Acceleration = 9.8 ms^-2

Time = 2 seconds

Now, we know that,

So,

m

Body 2:

Initial Speed = 0

Acceleration = 9.8 ms^-2

Time = 2 seconds

Now, we know that,

So,

m

Since, so final separation is also 1 m.

Hence, option D is correct.

False

Explanation:

Force of attraction between two bodies is called gravitation.

Gravity is the force of attraction on any object by the earth that is a gravitational pull towards the centre of earth.

False

Explanation:

G is also known as Universal Gravitational Constant. Value of G is Nm²kg^-2. The value of G is independent of mass of object and the distance between them.

True

Explanation:

Spring Balance gives the weight of the object.

W = m × g

If the heavy object is released, then it will read zero, since no mass implies no weight.

False

Explanation:

G is universal gravitational constant and is always constant.

G = Nm²kg^-2

It does not change with radius of the body and thus is independent of it.

True

Explanation:

Centre of mass means the mean position of all the matter in the body.

Centre of Gravity is a point at which the weight of body is considered.

If the object is small then gravitational field is uniform and both lie at the same point.

False

Explanation:

Gravitational Force between two bodies does not change if a material body is placed between them, since according to Newton’s law,

Where, G = Universal gravitational Constant

= Masses of bodies

R = Distance between two bodies

Here, F does not depend on the material medium and hence it will not change due to change in medium.

True

Explanation:

When a body is thrown up and when a body comes down, in both the cases the only force acting on the body is the gravitational force attraction that acts towards the centre of the earth.

F = m × a

Since, F and m both are same in both the cases, hence acceleration is also same.

They are opposite in sign since, when a body comes down, then the acceleration is in direction of motion and hence positive, whereas when body is thrown up, then the acceleration is in opposite direction of motion and hence negative.

True

Explanation:

The value of g is zero at the centre of earth. This is because at the centre of earth the object is surrounded from masses from all sides. The total force acting on the object is thus zero and thus acceleration due to gravity is also zero.

True

Explanation:

Yes, the inertia of the system depends on its mass.

Inertia is the property to resist the change in motion. The more the mass of the object, the more is the resistance to the change in velocity.

F = m × a

For same force, if mass is more, then acceleration is less and thus mass is the measure of inertia.

True

Explanation:

Yes, all objects attract each other along the line joining their centre of mass.

The force of attraction is the gravitational force of attraction and is given as,

Where, G = Universal gravitational Constant

= Masses of bodies

R = Distance between two bodies

Gravitational force acts only along the line joining the two masses.

True

Explanation:

The acceleration produced in a freely falling body by the gravitational pull of the earth is called the acceleration due to gravitation.

We know that,

Force = Mass × Acceleration

F = m × a

a = …..1

Where, F is the force on the object of mass m dropped from a distance r from the centre of earth of mass M.

So, force exerted by the earth on the object is

F = …..2

M = Mass of Earth

m = mass of object

r = distance of object from centre of earth

Now, from equation 1 and 2,

a =

a =

Now, from above,

a = g = Acceleration due to gravity

Hence,

g =