Chapter 10 - Gravitation Page no :- 134
Solution
1
The universal law of gravitation states that every object in
the universe attracts every other object with a force called the gravitational
force. The gravitational 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 centres.
Consider two objects of masses m1 and m2 and
let the distance between their centres be r. The
gravitational force of attraction (F) acting between them is given by
the universal law of gravitation as:
Concept Insight -
where, G is the universal gravitation constant given by:
Solution
2
Let M
be the mass of the Earth and m be 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:
Concept Insight -
Chapter 10 - Gravitation Page no :- 136
Solution
1
Gravity of the Earth attracts every object towards its centre.
When an object is released from a height such that it falls towards the surface
of the Earth under the influence of gravitational force alone, the motion of
the object is called free fall.
Solution
2
During free fall of an object towards the earth, the
magnitude of velocity of the falling object goes on increasing. This changing
velocity produces acceleration in the object. This acceleration is known as
acceleration due to gravity (g). Its value is given by 9.8 m/s2.
Concept Insight -
The value
of acceleration due to gravity changes from place to place but for most of the
purposes it is taken to be 9.8 m/s2.
Chapter - Page no :-
Solution
1
Chapter 10 - Gravitation Page no :- 138
Solution
1
Let ME be the mass of the Earth and m be
the mass of an object on the surface of the Earth. Let RE be
the radius of the Earth. According to the universal law of gravitation, weight
WE of the object on the surface of the Earth is given by,
Let MM and RM be the mass and
radius of the Moon. Then, according to the universal law of gravitation, weight
WMof the same object on the surface of the Moon is given by:
Therefore, weight of an object on
the moon is 
th of its weight on the Earth.
th of its weight on the Earth.
Concept Insight Gravitational force of the
moon is th that of the earth.
th that of the earth.
Chapter 10 - Gravitation Page no :- 141
Solution
1
It is difficult to hold a school bag having a thin strap
because the pressure exerted 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 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 shoulders is very large.
Concept
Insight - Pressure =
Force/Area
Solution 2
Whenever an object is immersed in a fluid, either partially
or fully, it experiences an upward force by the fluid. This force is called
buoyant force or upthrust and the property due to which a fluid exerts this
upthrust on an object placed in it, is called buoyancy.
Solution
3
If the density of an object is more than the density of
water, then it sinks in water. This is because the buoyant force acting on the
object is less than the force of gravity acting on it. On the other hand, if
the density of an object is less than the density of water, then it floats on
the surface of water. This is because the buoyant force acting on the object is
greater than the force of gravity acting on it.
Concept Insight -Density of an object is a key factor to decide whether it'll float or sink in a liquid.
Chapter 10 - Gravitation Page no :- 142
Solution
1
When you weigh your body, an upward force (due to air) 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.
Concept Insight -The upthrust of air acting on the body reduces the weight of the body.
Solution 2
We know that true weight = apparent weight + upthrust. The
cotton bag is heavier than the iron bar. This is due to the reason, that the
bag of cotton which has more volume (as it has less density) than the iron bar
(which has more density), experiences more upthrust due to air.
Concept
Insight - Same mass does not
necessarily means same buoyant force.
Chapter 10 - Gravitation Page no :- 143
Solution
1
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.
Concept Insight - 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 would become four times.
Solution
2
It is true that gravitational force
acts on all objects in proportion to their masses. But a heavy object does not
fall faster than a light object. This is because of the reason that
As Force 
Mass, therefore,
acceleration is constant for a body of any mass.
Mass, therefore,
acceleration is constant for a body of any mass.
Concept
Insight - Acceleration due to
gravity is constant for a heavy object as well as a light object.
Solution 3
According
to the universal law of gravitation, gravitational force between two objects of
masses M and m at a distance r from each other is given by:
Let mass of Earth be represented
by M = 6 × 1024 kg
Let mass of
the object be represented by m = 1 kg
Universal
gravitational constant, G = 6.7 × 10-11 Nm2 kg-2
Since the
object is on the surface of the Earth, r = radius of the Earth (R)
r = R = 6.4
× 106 m
Concept Insight -
Gravitational force,
Solution
4
The Earth attracts the Moon with a
force which is same as the force with which the moon attracts the Earth because
according to Newton's third law of motion, force of action and reaction are
always equal and opposite. So, the force of attraction of Earth on Moon is
equal and opposite to the force of attraction of Moon on Earth.
Concept
Insight - Action and reaction
forces are always equal and opposite.
Solution 5
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 much lesser than the acceleration
rate of the Moon towards the Earth. For this reason, we do not see the Earth
moving towards the Moon.
Concept Insight - For the same force, an object with greater mass obtains less acceleration as compared to an object with lesser mass.
Chapter 10 - Gravitation Page no :- 144
Solution
1
According
to the universal law of gravitation, the force of gravitation between two
objects is given by:
(i)
F is directly proportional to the product of masses of the objects. If the mass
of one object is doubled, then the gravitational force will also get doubled.
(ii) F is inversely proportional to the square of the
distance between the objects. If the distance is doubled, then the
gravitational force becomes one-fourth of its original value.
Similarly, if the distance is tripled, then the gravitational
force becomes one-ninth of its original value.
(iii) F is directly proportional to the product of masses of
the objects. If the masses of both the objects are doubled, then the
gravitational force becomes four times the original value.
Solution
2
The universal law of gravitation is important because it
accounts for the force that binds us to the Earth, motion of planets around the
Sun, motion of the Moon and other artificial satellites around the Earth, tides
due to the Moon and the Sun and many other phenomena.
Solution
3
When objects fall towards the Earth under the effect of
gravitational force alone, then they are said to be in free fall. This
force produces a uniform acceleration in the objects, which is called
acceleration of free fall or acceleration due to gravity. Its value is 9.8 m/s2.
Concept Insight - The value of acceleration of free fall is 9.8
m/s2, which is constant for all objects (irrespective of their
masses).
Solution
4
Gravitational force between the Earth and an object is known
as Earth's gravity or weight of the object.
Solution
5
Weight of a body on the earth is given by:
W = m g
where,
m = Mass of the body
g = Acceleration due to gravity
The value of g is greater at poles than at
the equator. Therefore, the same mass of gold weighs lesser at the equator than
at the poles. Hence, Amit's friend will not agree with the weight of the gold
bought.
Concept
Insight - Acceleration due to
gravity changes from place to place.
Solution 6
A sheet of paper has more surface
area than a crumpled ball of paper. So, the resistance offered by air to a
sheet of paper falling through it is more than the resistance offered to a
falling crumpled ball of paper. This decreases the speed of the sheet of paper
and hence it falls slower than the crumpled ball.
Concept Insight - Amount of air resistance depends on the
cross-sectional area of the object.
Solution 7
Weight = Mass × Acceleration
Acceleration due to gravity on earth, ge =
9.8 m/s2
Therefore, weight of a 10 kg object on the Earth = 10
× 9.8 = 98 N
Acceleration due to gravity on Moon, 
Therefore, weight of the same object on the Moon = 
Concept Insight Weight of an object on Moon
= 
of the weight of the object
on Earth
of the weight of the object
on Earth
Solution
8
(i) For
the upward motion of the ball, we use the equation:
v2 -
u2 = 2 g h
where,
u = Initial
velocity of the ball = 49 m/s (Given)
v = Final
velocity of the ball = 0 (At the highest point)
h = Maximum
height attained by the ball
g =
Acceleration due to gravity = -9.8 m/s2 (Ball goes up)
Putting the
values, we get
0 - (49)2 =
2 × (-9.8) × h
(ii) Let t be the time taken by
the ball to reach the height 122.5 m, then to calculate t we use the following
equation of motion:
v = u + g t
Putting the
values, we get
0 = 49 +
(-9.8) × t
0 = 49 -
9.8 t
49 = 9.8 t
But, Time of ascent = Time of
descent
Therefore, the total time taken
by the ball to return = 5 + 5 = 10 s
Concept
Insight - When a body is thrown vertically upwards, its velocity
decreases, so the acceleration due to gravity g is taken as negative.
Solution
9
Initial
velocity of the stone, u = 0
Final
velocity of the stone, v = ?
Height of
the tower, h = 19.6 m
Acceleration
due to gravity, g = 9.8 ms-2
For a
freely falling body:
v2 -
u2 = 2 g h
v2 -
02 = 2 × 9.8 × 19.6
v2 =
19.6 × 19.6 = (19.6)2
v2 =
384.16
v = 19.6 ms-1
Hence, the
velocity of the stone just before touching the ground is 19.6 ms-1.
Concept
Insight When a body is falling vertically downwards, its
velocity increases, so the acceleration due to gravity g is taken as positive.
Solution
10
Given:
Initial
velocity of the stone, u = 40 m/s
Final
velocity of the stone, v = 0 (At the highest point)
Maximum
height reached by the stone, h = ?
Acceleration
due to gravity, g = -10 ms-2 (Stone goes up)
Using the
equation of motion:
v2 -
u2 = 2 g h
0 - (40)2 =
2 × (-10) × h
0 - 1600 =
- 20 h
-1600 = -20
h
1600 = 20 h
Thus, 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 = 0 (since
final position coincides with the initial position)
Concept
Insight Distance is the length of the actual path covered by
an object, while displacement is the shortest distance between the initial and
final position of the object.
Solution
11
According to the universal
law of gravitation, the force of attraction between the Earth and the Sun is
given by:
Concept Insight- 
where,
MSun =
Mass of the Sun = 2 × 1030 kg
MEarth =
Mass of the Earth = 6 × 1024 kg
R = Average
distance between the Earth and the Sun = 1.5 × 1011 m
G =
Universal gravitational constant = 6.7 × 10-11 Nm2 kg-2
Substituting
the values, we get
Solution
12
Let the
two stones meet after a time t.
(i) For the
stone dropped from the top of 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 ms-2
From the
equation of motion,
(ii) For
the stone thrown upwards:
Initial
velocity, u = 25 ms-1
Let the
displacement of the stone from the ground in time t be s'.
Acceleration
due to gravity, g = -9.8 ms-2
From the
equation of motion,
The combined displacement (s +
s') of both the stones at the meeting point is equal to the height of the tower
100 m.
From eqs. (1) and (2), we get,
s + s' = 4.9 t2 +
25 t - 4.9 t2
100 = 25 t
In 4 s, the falling stone has
covered a distance given by equation (1) as
s = 4.9 t2 =
4.9 × (4)2 = 78.4 m
Therefore, the stones will meet
after 4 s at a height (100 - 78.4) = 21.6 m from the ground.
Concept
Insight Choose the equation of motion wisely to minimize the
number of steps in calculations.
Solution
13
(a) Concept Insight - 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.
Let the
initial velocity of the ball be u.
Final
velocity of the ball at the maximum height, v = 0
Acceleration
due to gravity, g = -9.8 ms-2
From the
equation of motion, v = u + gt, we get,
0 = u +
(-9.8) × 3
0 = u - 29.4
u = 29.4 ms-1
Hence, the
ball was thrown upwards with a velocity of 29.4 ms -1.
(b) Let the
maximum height attained by the ball be h.
Initial
velocity during the upward journey, u = 29.4 ms -1
Final
velocity, v = 0
Acceleration
due to gravity, g = -9.8 m s -2
Concept
Insight Choose the equation of motion wisely to minimize the
number of steps in calculations.
From the equation of
motion, 
h = 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
In this case,
Initial velocity, u = 0
Position of the ball after 4 s of
the throw is given by its position during the downward journey in 4 s - 3 s = 1
s.
Concept Insight - Choose
the equation of motion wisely to minimize the number of steps in calculations.
From the equation of
motion, , we get,
, we get,
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.
Chapter 10 - Gravitation Page no :- 145
Solution
1
An
object immersed in a liquid experiences buoyant force in the vertically upward
direction, i.e., in a direction opposite to the weight of the object.
Solution
2
In case of a block of plastic, the upward buoyant force is
greater than the weight of the object. The large buoyant force on the block is
due to its density being smaller than that of water. Due to the larger buoyant
force, the block of plastic comes up when released under water.
Concept Insight - A body floats when its weight
is lesser than the upward buoyant force acting on it.
Solution
3
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 
The
density of the substance is more than the density of water (1 g cm-3).
Hence, the substance will sink in water.
Concept Insight - Density of an object is a key factor in deciding
whether the object will sink or float.
Solution
4
The density of the substance is
more than the density of water (1 g cm-3). Hence, it will sink in
water.
As the
packet is fully submerged in water,
Mass of
water displaced by the packet = volume of the packet x density of water
= 350 cm3 × 1 g/cm3 =
350 g
Concept
Insight - Density of an object is a key factor in deciding
whether the object will sink or float.
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