Chapter 10 Gravitation

Every object in the universe attracts every other object with a force which is proportional to the product of their masses and inversely proportional to the distance between them .

For two objects of masses m₁ and m₂ and the distance between them r, the force (F) of attraction acting between them is given by the universal law of gravitation as:

Where, G is the universal gravitation constant given by:

The mass of the earth = m₁

The mass of the object = m₂

Distance between the Earth’s centre and object ( radius of Earth ) = r

Therefore, Gravitational force is

A free fall motion means that the object is falling from a height under the influence of gravity only. It falls only due to its weight and not any other force. Provided these conditions the object experiences a free fall.

It is observed that in a free fall; free - falling objects do not encounter air resistance.

When a body falls towards the earth from a height, then its velocity changes during the fall. This changing velocity produces acceleration in the body. This is called acceleration due to gravity. Its value is given by 9.8 m/s².

Our weight on the moon is less than it would be on Earth due to a difference of the strength of gravity on the moon. The moon’s gravitational force is determined by the mass and the size of the moon. Since the moon has significantly less mass than the Earth, it will not pull objects toward itself as the strength that Earth will. This means that if you went to the moon you would weigh less, even though your mass stays the same; 

Let, M_E = the mass of the Earth 

m = an object on the surface of the Earth  

R_E = the radius of the Earth 

W_E = the object on the surface of the Earth

Let, M_M = mass of the moon 

 R_M = radius of the moon

 W_M = the object on the surface of the moon

It is difficult to hold a school bag having a strap made of thin and strong string, because the thin string has very less contact area in case of a school bag having a strap made of thin and strong string which increases pressure which is uncomfortable to carry the school bag.

The upward force exerted by a liquid on an object that is partly or wholly immersed in it is known as buoyancy. Buoyancy is caused by the differences in pressure acting on opposite sides of an object immersed in a static fluid. It is also known as the buoyant force.

An object sinks in water if its density is greater than that of water. This is because the buoyant force acting on the object is less than the force of gravity. On the other hand, an object floats in water if its density is less than that of water. This is because the buoyant force acting on the object is greater than the force of gravity.

The weighing machine reads slightly less than the actual value. This is because of the upthrust of air acting on our body. Hence, the body gets pushed slightly upwards, causing the weighing machine to show a reading less than the actual value.

The bag of cotton is heavier than the iron bar. This is because the density of cotton bag is less than that of the iron bar, so the volume of cotton bag is more compared to iron bar. So the cotton bag experiences more upthrust due to the presence of air.

Actual weight = Measured Weight + Buoyant Force

We know that the universal law of gravitation, 

gravitational force 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 

When the distance is reduced to half the gravitational force becomes four times larger than the previous value.

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 force is directly proportional to mass, acceleration is constant for a body of any mass. Hence, heavy objects do not fall faster than light objects.

Given; Mass of Earth, M = 6 × 10²⁴ kg

Mass of object, m = 1 kg

Universal gravitational constant, G = 6.7 × 10⁻¹¹ Nm² kg⁻²

radius of the Earth (R) = R = 6.4 × 10⁶ m

Gravitational force,

The Earth attracts the moon with an equal force with which the moon attracts the earth but these forces are in opposite directions.

According to universal law of gravitation,

Apply Newton’s third law which states that every action has an equal and opposite reaction. Therefore, the Earth and the moon experience the same amount of gravitational forces from each other. Both bodies revolve around their common centre of mass and centrifugal force balances the gravitational force. However, the mass of the Earth is much larger than the mass of the moon. 

For this reason, the Earth does not move towards the moon.

According to the universal law of gravitation,

(i) Force is directly proportional to the masses of the objects. If the mass of one object is doubled, then the gravitational force will also get doubled.

(ii) Force is inversely proportional to the square of the distances between the objects. If the distance is doubled, then the gravitational force becomes one-fourth.

Similarly, if the distance is tripled, then the gravitational force becomes one-ninth.

(iii) Force 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 importance of universal law of gravitation ;

1. It helps us bind together with the Earth.
2. A body which goes up will not come down if there is no gravitational force.

When the body falls due to Earth’s gravitational pull, its velocity changes and is said to be accelerated due to Earth’s gravity and it falls freely called free fall. Acceleration of free fall is 9.8 ms⁻², which is constant for all objects.

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

Weight of an object on the Earth is given by:

W = mg

Where,

m = Mass of the object

g = Acceleration due to gravity

Since the acceleration due to gravity is less at the equator as compared to that at poles, the weight of the gold will be less at the equator than at the poles. Hence, Amit’s friend will not agree with the weight of the gold bought.

A sheet of paper will fall slower than one that is crumpled into a ball because of the different drag force. Crumpled ball has a smaller surface area so the resistance offered by air is less than in the case of a sheet of paper which has a larger surface area.

Given; gravitational force on surface of the moon = 1/6 x gravitational force on surface of the Earth

Also,

Weight = Mass × Acceleration

Mass of the object = 10 kg

Acceleration due to gravity, g = 9.8 m/s²

Therefore, weight of  object on the Earth = 10 × 9.8 = 98 N

And, weight of the same object on the moon = 98 / 6 = 16.33 N

We know that 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

(i) Let; h = the maximum height attained by the ball.

Hence,

(ii) Let; t = time taken by the ball to reach the height 122.5 m

 then using 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

 Initial velocity of the stone = u = 0

 Final velocity of the stone = v

 Height of the stone = s = 19.6 m

 Acceleration due to gravity = g = 9.8 m s⁻²

According to the equation of motion under gravity:

v₂ − u₂ = 2 gs

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.

 Initial velocity of the stone = u = 40 m/s

 Final velocity of the stone = v = 0

 Height of the stone = s

 Acceleration due to gravity = g = −10 m s⁻²

Let h = maximum height attained by the stone.

According to the equation of motion under gravity:

v₂ − u₂ = 2 gs

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

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

G = Universal gravitational constant = 6.7 × 10⁻¹¹ Nm² kg⁻²

Force of gravitation = F =

Let the two stones meet after a time t.

(i) For the stone dropped from the tower:

Initial velocity, u = 0 m/s

Let the displacement = s

Acceleration due to gravity, g = 9.8 m s⁻²

From the equation of motion,

                  ...(1)

(ii) For the stone thrown upwards:

Initial velocity, u = 25 m/s

Let the displacement = s'.

Acceleration due to gravity, g = −9.8 m s⁻²

Equation of motion,

              ...(2)

The combined displacement is ;

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

Therefore, the stones will meet after 4 s and the distance is 80 m from the top.

The ball takes a total of 6 s for its upward and downward journey.

Time taken to reach maximum height = 6 / 2 = 3 s

Hence, it has taken 3 s to attain the maximum height

(a)

Let initial velocity = u m/s

Final velocity of the ball at the maximum height, v = 0 m/s

Acceleration due to gravity, g = −9.8 m s⁻²

Equation of motion

The buoyant force acts in the upward direction on the object which is immersed in a liquid.

When a plastic is immersed in water, there are two forces acting upon it. One is the gravitational force, which pulls the object downwards, and the other is the buoyant force, which pushes the object upwards. In this case, the buoyant force on plastic is more than the gravitational force. Thus, the plastic will come up to the surface of water.

An object sinks in liquid when its density is greater than the liquid. On the other hand, an object floats on liquid when its density is less than 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.

Density of the 500 g sealed packet 

Density of packet > density of water 

Hence, it will sink in water.

Mass of water displaced by packet = Volume of the packet = 350 g

The mass of water displaced is 350 g.