CBSE - Gravitation

The universal law of gravitation states that every object in the universe attracts every other object with a force called the gravitational force. The force acting between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
For two objects of masses m1and m2 and the distance between them r, the force (F) of attraction acting between them is given by the universal law of gravitation as:

Let M_E be the mass of the Earth and mbe the mass of an object on its surface. If R is the radius of the Earth, then according to the universal law of gravitation, the gravitational force (F) acting between the Earth and the object is given by the relation:

Gravity of earth attracts every object towards its center. When an object is dropped from a certain height , it begins to fall towards Earth’s surface under the influence of gravitational force. Such a motion of object is called free fall.

When an object falls freely towards the surface of earth from a certain height, then its velocity changes. This change in velocity produces acceleration in the object which is known as acceleration due to gravity denoted bu letter g . The value of acceleration due to gravity is g= 9.8 m/s².

Mass	Weight
Mass is the quantity of matter contained in the body.	Weight is the force of gravity acting on the body.
It is the measure of inertia of the body.	It is the measure of gravity
Mass is a constant quantity.	Weight is not a constant quantity. It is different at different places.
It only has magnitude.	It has magnitude as well as direction.
Its SI unit is kilogram (kg)	Its SI unit is the same as the SI unit of force, i.e., Newton (N)

The mass of moon is 1/100 times and its radius 1/4 times that of earth. As a result, the gravitional attraction on the moon is about one sixth when compared to earth. Hence, the the weight of an object on the moon 1/6th its weight on the earth.

It is difficult to hold a school bag having a thin strap because the pressure on the shoulders is quite large. This is because the pressure is inversely proportional to the surface area on which the force acts. The smaller is the surface area; the larger will be the pressure on the surface. In the case of a thin strap, the contact surface area is very small. Hence, the pressure exerted on the shoulder is very large.

The upward force exerted by a liquid on an object that is immersed in it is known as buoyancy.

→ An object sink in water if its density is greater than that of water.
→ An object floats in water if its density is less than that of water.

When we weigh our body, an upward force acts on it. This upward force is the buoyant force. As a result, the body gets pushed slightly upwards, causing the weighing machine to show a reading less than the actual value.

The cotton bag is heavier than the iron bar. The cotton bag experiences larger up thrust of air than the iron bar. So, the weighing machine indicates a smaller mass for cotton bag than its actual mass.

According to the universal law of gravitation, gravitational force (F) acting between two objects is inversely proportional to the square of the distance (r) between them, i.e.,

If distance r becomes r/2, then the gravitational force will be proportional to

Hence, if the distance is reduced to half, then the gravitational force becomes four times larger than the previous value./p>

All objects fall on ground with constant acceleration, called acceleration due to gravity (in the absence of air resistances). It is constant and does not depend upon the mass of an object. Hence, heavy objects do not fall faster than light objects.

According to the universal law of gravitation, gravitational force exerted on an object of mass m is given by:



Where,

Mass of Earth, M = 6 x 10²⁴kg

Mass of object, m = 1 kg

Universal gravitational constant, G = 6.7 x 10 ^{- 11}Nm2kg ^{- 2}

Since the object is on the surface of the Earth, r = radius of the Earth (R)

r= R = 6.4 x 10⁶ m

Gravitational force,

According to the universal law of gravitation, two objects attract each other with equal force, but in opposite directions. The Earth attracts the moon with an equal force with which the moon attracts the earth.

The Earth and the moon experience equal gravitational forces from each other. However, the mass of the Earth is much larger than the mass of the moon. Hence, it accelerates at a rate lesser than the acceleration rate of the moon towards the Earth. For this reason, the Earth does not move towards the moon.

What happens to the force between two objects, if
(i) the mass of one object is doubled?
(ii) the distance between the objects is doubled and tripled?
(iii) the masses of both objects are doubled?

Answer

From Universal law of , force exerted on an object of mass m by earth is given by

So as the mass of any one of the object is doubled the force is also doubled

(ii) The force F is inversely proportional to the distance between the objects. So if the distance between two objects is doubled then the gravitational force of attraction between them is reduced to one fourth of its original value. Similarly f the distance between two objects is tripled , then the gravitational force of attraction becomes one ninth of its original value.

(iii) Again fron Universal law of attraction from equation 1 force F is directly proportional to the product of both the masses. So if both the masses are doubled then the gravitational force of attraction becomes four times the original value.

Universal law of Gravitation is important because it it tells us about:
→ the force that is responsible for binding us to Earth.
→ the motion of moon around the earth
→ the motion of planets around the sun
→ the tides formed by rising and falling of water level in the ocean are due to the gravitational force exerted by both sun and moon on the earth.

Acceleration of free fall is the acceleration produced when a body falls under the influence of the force of gravitation of the earth alone. It is denoted by g and its value on the surface of the earth is 9.8 ms^-2.

Gravitational force between the earth and an object is known as the weight of the object.

Weight of a body on the Earth is given by:
W= mg
Where,
m= Mass of the body
g= Acceleration due to gravity
The value of gis greater at poles than at the equator. Therefore, gold at the equator weighs less than at the poles. Hence, Amit's friend will not agree with the weight of the gold bought.

When a sheet of paper is crumbled into a ball, then its density increases. Hence, resistance to its motion through the air decreases and it falls faster than the sheet of paper.

Weight of an object on the moon = 1/6 x Weight of an object on the Earth
Also,
Weight = Mass x Acceleration
Acceleration due to gravity, g = 9.8 m/s²
Therefore, weight of a 10 kg object on the Earth = 10 x 9.8 = 98 N
And, weight of the same object on the moon= 1.6 x 9.8 = 16.3 N.

A ball is thrown vertically upwards with a velocity of 49 m/s. Calculate
(i) the maximum height to which it rises.
(ii) the total time it takes to return to the surface of the earth.

Answer :



According to the equation of motion under gravity:

v² - u²= 2 gs

Where,

u= Initial velocity of the ball

v= Final velocity of the ball

s= Height achieved by the ball

g= Acceleration due to gravity

At maximum height, final velocity of the ball is zero, i.e., v= 0

u = 49 m/s

During upward motion, g = - 9.8 m s^{- 2}

Let h be the maximum height attained by the ball.

Hence,



Let t be the time taken by the ball to reach the height 122.5 m, then according to the equation of motion:

v= u + gt

We get,



But,

Time of ascent = Time of descent

Therefore, total time taken by the ball to return = 5 + 5 = 10 s

(i) 122.5 m (ii) 10 s

According to the equation of motion under gravity:
v² − u² = 2 gs
Where,
u = Initial velocity of the stone = 0
v = Final velocity of the stone
s = Height of the stone = 19.6 m
g = Acceleration due to gravity = 9.8 m s⁻²
∴ v² − 0² = 2 � 9.8 � 19.6
v2 = 2 � 9.8 � 19.6 = (19.6)²
v = 19.6 m s^{− 1}
Hence, the velocity of the stone just before touching the ground is 19.6 m s⁻¹.

According to the equation of motion under gravity:
v² − u² = 2 gs
Where,
u = Initial velocity of the stone = 40 m/s
v = Final velocity of the stone = 0
s = Height of the stone 
g = Acceleration due to gravity = −10 m s⁻²
Let h be the maximum height attained by the stone. 
Therefore, 
0 - (40)² = 2 x h x (-10)
h= 40 x 40 / 20 = 80 m

Therefore, total distance covered by the stone during its upward and downward journey = 80 + 80 = 160 m
Net displacement of the stone during its upward and downward journey
= 80 + (−80) = 0

According to question,

M_Sun = Mass of the Sun = 2 � 10³⁰ kg
M_Earth = Mass of the Earth = 6 � 10²⁴ kg
R = Average distance between the Earth and the Sun = 1.5 � 1011 m
From Universal law of gravitation,

Let the two stones meet after a time t.

(i) For the stone dropped from the tower:

Initial velocity, u= 0

Let the displacement of the stone in time t from the top of the tower be s.

Acceleration due to gravity, g = 9.8 m s^{- 2}

From the equation of motion,

(ii) For the stone thrown upwards:

Initial velocity, u= 25 m s^{- 1}

Let the displacement of the stone from the ground in time tbe s'.

Acceleration due to gravity, g = - 9.8 m s ^{- 2}

Equation of motion,



The combined displacement of both the stones at the meeting point is equal to the height of the tower 100 m.



In 4 s, the falling stone has covered a distance given by equation (1) as



Therefore, the stones will meet after 4 s

A ball thrown up vertically returns to the thrower after 6 s. Find
(a) the velocity with which it was thrown up,
(b) the maximum height it reaches, and
(c) its position after 4 s.

Answer

(a) Time of ascent is equal to the time of descent. The ball takes a total of 6 s for its upward and downward journey.
Hence, it has taken 3 s to attain the maximum height.
Final velocity of the ball at the maximum height, v = 0 
Acceleration due to gravity, g = −9.8 m s⁻²
Equation of motion, v = u + gt will give,
0 = u + (−9.8 × 3)
u = 9.8 × 3 = 29.4 ms⁻¹
Hence, the ball was thrown upwards with a velocity of 29.4 m s⁻¹.

(b) Let the maximum height attained by the ball be h.
Initial velocity during the upward journey, u = 29.4 m s⁻¹
Final velocity, v = 0
Acceleration due to gravity, g = −9.8 m s⁻²
From the equation of motion, s= ut + 1/2 at²
h= 29.4 x 3 + 1/2 x -9.8 x (3)² = 44.1 m

(c) Ball attains the maximum height after 3 s. After attaining this height, it will start falling downwards. 
In this case,
Initial velocity, u = 0
Position of the ball after 4 s of the throw is given by the distance travelled by it during its downward journey in 4 s − 3 s = 1 s.
Equation of motion, s= ut + 1/2 gt² will give,
s= 0 x t + 1/2 x 9.8 x 1² = 4.9 m
Total height = 44.1 m
This means that the ball is 39.2 m (44.1 m − 4.9 m) above the ground after 4 seconds.

An object immersed in a liquid experiences buoyant force in the upward direction.

For an object immersed in water two force acts on it
→ gravitational force which tends to pull object in downward direction
→ buoyant force that pushes the object in upward direction
here in this case buoyant force is greater than the gravitational pull on the plastic block. This is the reason the plastic block comes up to the surface of the water as soon as it is released under water.

If the density of an object is more than the density of a liquid, then it sinks in the liquid. On the other hand, if the density of an object is less than the density of a liquid, then it floats on the surface of the liquid.
Here, density of the substance = Mass of the substance / Volume of the substance
= 50 / 20
= 2.5 g cm^-3
The density of the substance is more than the density of water (1 g cm⁻³). Hence, the substance will sink in water.

Density of the 500 g sealed packet= Mass of the Packet / Volume of the Packet
= 500 / 350
= 1.428 g cm⁻³
The density of the substance is more than the density of water (1 g cm⁻³). Hence, it will sink in water.
The mass of water displaced by the packet is equal to the volume of the packet, i.e., 350 g.

1. Two objects of different masses falling freely near the surface of moon would
(a) have same velocities at any instant
(b) have different accelerations
(c) experience forces of same magnitude
(d) undergo a change in their inertia
Ans. (a) have same velocities at any instant
Explanation: When an object is under free fall, its acceleration depends only on the acceleration due to gravity and mass has no role to play. Due to this; irrespective of their masses, the object under free fall will have same velocity at any instant.

2. The value of acceleration due to gravity
(a) is same on equator and poles
(b) is least on poles
(c) is least on equator
(d) increases from pole to equator
Ans. (c) is least on equator
Explanation: The distance between surface of earth and its centre is more on equator than on the poles. Moreover, g varies inversely as square of radius. Due to this, acceleration due to gravity is least on equator.

3. The gravitational force between two objects is F. If masses of both objects are halved without changing distance between them, then the gravitational force would become
(a) F/4
(b) F/2
(c) F
(d) 2 F
Ans. (a) F/4
Explanation: Gravitational force between two object varies directly as their masses and inversely as the square of distance between them. So, when the masses of both objects are halved without changing the distance, the gravitational force between them would become one-fourth of the original value.

4. A boy is whirling a stone tied with a string in an horizontal circular path. If the string breaks, the stone
(a) will continue to move in the circular path
(b) will move along a straight line towards the centre of the circular path
(c) will move along a straight line tangential to the circular path
(d) will move along a straight line perpendicular to the circular path away from the boy
Ans. (c) will move along a straight line tangential to the circular path

Explanation: When an object is in circular motion, it tends to be in rectilinear motion at any instance of time. This is due to the centripetal force that the object remains moving on the circular path but when the string breaks, the stone will move along a straight line tangential to the circular path.

5. An object is put one by one in three liquids having different densities. The object floats with 1/9, 2/11, and 3/7 parts of their volumes of outside the liquid surface in liquids of densities d1, d2 and d3 respectively. Which of the following statement is correct?
(a) d1> d2> d3
(b) d1> d2< d3
(c) d1< d2> d3
(d) d1< d2< d3
Ans. (d) d1< d2< d3


Explanation: Let us find LCM of fractions of volumes outside the liquid surface in each other.

LCM of denominator is 693 and using this, the fractions can be written as follows:

This shows that these fractions are arranged in ascending order. Hence, densities of liquids are in ascending order.


6. In the relation F= G M m/d2, the quantity G
(a) depends on the value of g at the place of observation
(b) is used only when the earth is one of the two masses
(c) is greatest at the surface of the earth
(d) is universal constant of nature
Ans. (d) is universal constant of nature

7. Law of gravitation gives the gravitational force between
(a) the earth and a point mass only
(b) the earth and Sun only
(c) any two bodies having some mass
(d) two charged bodies only
Ans. (c) any two bodies having some mass

8. The value of quantity G in the law of gravitation
(a) depends on mass of earth only
(b) depends on radius of earth only
(c) depends on both mass and radius of earth
(d) is independent of mass and radius of the earth
Ans. (d) is independent of mass and radius of the earth

Explanation: Since G is a universal constant, hence its value does not depend on mass or radius of the earth.
9. Two particles are placed at some distance. If the mass of each of the two particles is doubled, keeping the distance between them unchanged, the value of gravitational force between them will be
(a) � times
(b) 4 times
(c) � times
(d) unchanged
Ans. (b) 4 times

Explanation: Gravitational force between two objects varies directly as their masses and inversely as the square of distance between them. So, when the masses of both objects are doubled without changing the distance, the gravitational force between them would become four times of the original value.

10. The atmosphere is held to the earth by
(a) gravity
(b) wind
(c) clouds
(d) earth�s magnetic field
Ans. (a) gravity

11. The force of attraction between two unit point masses separated by a unit distance is called
(a) gravitational potential
(b) acceleration due to gravity
(c) gravitational field
(d) universal gravitational constant
Ans. (d) universal gravitational constant

12. The weight of an object at the centre of the earth of radius R is
(a) zero
(b) infinite
(c) R times the weight at the surface of the earth
(d) 1/R2 times the weight at surface of the earth
Ans. (a) zero
Explanation: The acceleration due to gravity is zero at the centre of the earth. We know that weight is a product of mass and g. Hence, when g = 0, weight would be zero.

13. An object weighs 10 N in air. When immersed fully in water, it weighs only 8 N. The weight of the liquid displaced by the object will be
(a) 2 N
(b) 8 N
(c) 10 N

(d) 12 N
Ans. (a) 2 N
Explanation: The weight of liquid displaced by object = weight in air � weight in liquid
=10 N � 8 N = 2 N. Moreover, according to Archimedes� theory; when an object is immersed in water, there is apparent reduction in its weight and the reduction is equal to the weight of liquid displaced by the object.

14. A girl stands on a box having 60 cm length, 40 cm breadth and 20 cm width in three ways. In which of the following cases, pressure exerted by the brick will be
(a) maximum when length and breadth form the base
(b) maximum when breadth and width form the base
(c) maximum when width and length form the base
(d) the same in all the above three cases
Ans. (b) maximum when breadth and width form the base

Explanation: Pressure is inversely proportional to surface area. When surface area is minimum then pressure will be maximum. Hence, option (b) is the correct answer.

15. An apple falls from a tree because of gravitational attraction between the earth and apple. If F1 is the magnitude of force exerted by the earth on the apple and F2 is the magnitude of force exerted by apple on earth, then

(a) F1 is very much greater than F2
(b) F2 is very much greater than F1
(c) F1 is only a little greater than F2
(d) F1 and F2 are equal
Ans. (d) F1 and F2 are equal
Explanation: This is in accordance to Newton�s Third Law of Motion which states that for every action there is an equal and opposite reaction.

Gravitational force. This force depends on the product of the masses of the planet and sun and the distance between them.

Both stones will take the same time to reach the ground because the two stones fall from the same height.

The moon will begin to move in a straight line in the direction in which it was moving at that instant because the circular motion of moon is due to centripetal force provided by the gravitational force of earth.

The value of ‘g’ at the equator of the earth is less than that at poles. Therefore, the packet falls slowly at equator in comparison to the poles. Thus, the packet will remain in air for longer time interval, when it is dropped at the equator.

Since the weight of any person on moon is about 1/6 times that on the earth, hence acceleration due to gravity at moon is 1/6 of that on earth. This means that by applying the same force a person can lift six times heavier object on the moon than what he could lift on the earth. So, the maximum mass which can be lifted by the same force applied by
the person on the moon is 6 × 15 kg = 90 kg. 

The gravitational of the sun provides the necessary centripetal force to keep the earth in its orbit. This counters the centrifugal force which arises due to circular motion of the earth. Thus, earth remains on its orbit without either falling into the sun or going away tangentially to its orbit. 

Weight of a body α M /R²

The weight will become 16 times.

F α m1 m2

F α 1/d²
This hypothesis is not correct. The two bricks, like a single body, fall with the same speed to reach the ground at the same time in case of free-fall. This is because acceleration due to gravity is independent of the mass of the falling body

Ratio will not change in either case because acceleration remains the same. In case of free-fall acceleration does not depend upon mass and size

(a) A cube of side 5 cm is immersed in water and then in saturated salt solution. In which case, will it experience a greater buoyant force. If each side of the cube is reduced to 4 cm and then immersed in water, what will be the effect on the buoyant force experienced by the cube as compared to the first case for water. Give reason for each case.
(b) A ball weight 4 kg of density 4000 kg m^-3 is completely immersed in water of density 10³ kg m^-3. Find the force of buoyancy on it. (Given g = 10 ms^-2.)

Ans. (a) (i) The cube will experience a greater buoyant force in the saturated salt solution because the density of the salt solution is greater than that of water.
(ii) The smaller cube will experience lesser buoyant force as its volume is lesser than the initial cube.
(b) Buoyant force = weight of the liquid displaced
= density of water � volume of water displaced �g