Gravitation

Every object in the universe attracts other object by a force of attraction, called gravitation, which is directly proportional to the product of masses of the objects and inversely proportional to the square of distance between them. This is called Law of Gravitation or Universal Law of Gravitation.

The formula to find the magnitude of the gravitational force between the earth and an object on the surface of the earth is:

Gravitational force, F = G ×

Where,

G = Gravitational constant

M = Mass of the earth

m = Mass of the object

And R is Radius of the earth

When an object falls from any height under the influence of gravitational force only, it is known as free fall. In the case of free fall no change of direction takes place but the magnitude of velocity changes because of acceleration. This acceleration acts because of the force of gravitation and is denoted by ‘g’.

The acceleration gained by an object because of gravitational force is called its acceleration due to gravity. Its SI unit is m/s ². Acceleration due to gravity is a vector, which means it has both a magnitude and a direction. It has a standard value defined as 9.80665 m/s ².

Weight is the force with which a body is pulled towards the center of the earth or any other celestial body for that matter. The greater the force put by the celestial body, the greater the weight of an object on its surface.

The gravitational attraction depends upon the weight of the celestial body, since Earth has more weight compared to the moon. It is found that the gravitational acceleration is found to be 1.62 m-sec^-2 while for Earth the force is 9.8 m-sec^-2 which is 6 × gravitational acceleration of the moon.

As we know that:-

W(Weight)=M(mass) × g(gravitational acceleration)

Weight on the Earth W_E = M × 9.8

Weight on the Moon W_M = M ×1.62

Hence, the weight of an object on the moon is 1/6 the weight on the earth.

We know that lesser is the surface area of an object the more is the pressure it exerts. Hence, it is difficult to hold a school bag having a strap made of thin and strong string, because the thin string has very less contacting area in case of a school bag having a strap made of thin and strong string which increases pressure which is un comfortable to carry the school bag.

When an object is immersed partially or fully in a liquid, it experiences an upward force. This upward force is known as buoyant force and this phenomenon is known as buoyancy. Buoyancy is also known as up thrust.

An object will float on the surface of water when it has density less than that of water because it always displaces more weight of water than its own weight, and as the buoyant force is more than its own weight, therefore it floats. On the other hand, an object sinks in water when it has density more than that of water because it always displaces less weight of water than its own weight, and as the buoyant force is less than its own weight, it sinks in water.

Weight is the gravitational force acting on the object in the downward direction. According to the rule of buoyancy, all fluids apply an up thrust force in the upward direction. Hence, the air exerts an upward force (buoyant force or up thrust) on our body and makes us slightly lighter than we actually are. So the weight shown on the weighing scale would be less than 42 kg .Thus we can say that our actual weight / mass will be slightly more.

We know that,

Measured weight of the body = Actual weight of the body - Buoyant force

Therefore,

Actual weight = Measured weight + Buoyant force

As measured weight of cotton and iron is same, actual weight will only depend on the buoyant force.

Buoyant force is given by:

B = Volume of the body × g × Density of air.

Buoyant force is directly proportional to the volume of the body.

As volume of iron is less than the volume of cotton, therefore buoyant force acting on the iron will be less than that of iron.

Hence, actual weight of the cotton will be more than the iron.

The force between two objects of masses m₁ and m₂ separated by a distance "r" is given by:

Now according to question,

New distance R' = R/2
So, new force between them F'

The force has become 4× initial Force .. Ans

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 Acceleration= Force/Mass or Force = Acceleration x MassAs force is directly proportional to mass, acceleration is constant for a body of any mass.

Firstly, we will calculate the magnitude of gravitational force by using the formula:

F = G×_(i)

We know that,

Gravitational constant, G = 6.7 × 10^-11 Nm²/kg²

Given,

Mass of earth, m₁ = 6.4 × 10²⁴ kg

Mass of object, m₂ = 1kg

Distance between centre of earth and object=Radius of earth, r = 6.4× 10⁶ m

Putting all the values in (i), we get:

F = = 9.8144 N

Therefore, the magnitude of gravitational force is 9.8144 N

Yes the moon do attracts the earth but the earth also attracts the moon.In gravitational force earth and moon are attract with each other and but there masses are different so earth pull the moon but they have same gravitational force between them The gravitational pull of earth is higher than mass of the moon. Hence, the attraction exerted by the moon on the earth becomes negligible.

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.

We know that Gravitational force,

F = GMm/r ²

(i) When the mass of 1 object is doubled

F' = G (2M) m/r ²

F' = 2GMm/r ²

F' = 2F, the force will become 2 times.

(ii) (a)Distance between two objects is doubled

F' = GMm/(2r) ²

F' = GMm/4r ²

F' = 1/4 F, the force will become 1/4 times

(b) Distance between two objects is tripled

F' = GMm/(3r) ²

F' = GMm/9r ²

F' = 1/9F, the force will become 1/9 times

(iii) When masses of both objects is doubled

F' = G (2M)(2m)/r ²

F' = 4GMm/r ²

F' = 4F, the force will become 4 times.

The importance of universal law of gravitation is that it explains:

(i) The gravitational force of attraction of the Earth binds all terrestrial objects to the earth.

(ii) The gravitational force of earth keeps the atmosphere close to earth.

(iii) The moon revolves around the earth.

(iv) The gravitational pull of the sun on the planet keeps them revolving around the sun.

(v) The tides formed by the rising & falling of water level in the oceans are due to the gravitational force of sun & moon on the water.

As, the only force which is acting on body on free fall is Gravity.

Hence, its acceleration will also be relative to gravity.

If gravity is the only force acting on an object, then we find the object will accelerate at a rate of 9.8m/s ² down toward the centre of the Earth (this is often rounded to 10m/s)

In fact, all objects will fall at this rate if the air does not act to slow them down.

The Gravitational force between the earth and an object is called ‘earth’s gravity’.

Gravity is the force that attracts a body towards the centre of the earth, or towards any other physical body having mass.

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 g is 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.

Sheet of paper will fall slower because due to larger​ surface area the amount of upward Gravitational force causes resistance to its movement. The air resistance would make the actual force lesser than that imparted by the gravity of earth and hence slowing down the speed of sheet of paper.

We know that,

Acceleration due to gravity on the surface of earth = 9.8 m/s²

So, the acceleration due to gravity on the surface of moon will be of this value.

= 9.8 × m/s²

Firstly, we will calculate the weight on the moon:

Mass of objects, m = 10 kg

Acceleration due to gravity on moon, g = 9.8 × m/s²

Weight of object (on moon), W = m × g

= 10 × 9.8 ×

= 16.3 N

Now, we will calculate the weight of the object on the earth surface:

Mass of object, m = 10 kg

Acceleration due to gravity on earth, g = 9.8 m/s²

Weight of object (on earth), W = m × g

= 10 × 9.8

= 98 N

(i) The value of acceleration due to gravity g will be taken as negative as the ball is going upwards.

Given,

Initial velocity of ball, u = 49 m/s

Final velocity of ball, v = 0 (As the ball stops at top)

Acceleration due to gravity, g = - 9.8 m/s²

Height, h =?

We know that,

v² = u² + 2gh

(0)² = (49)² + 2 × (-9.8) × h

0 = 2401 – 19.6 h

19.6 h = 2401

h =

= 122.5 m

(ii) Now, we will calculate the time taken by the ball to reach at the highest point:

Given,

Final velocity, v = 0 (As the ball stops at top)

Initial velocity, u = 49 m/s

Acceleration due to gravity, g = - 9.8 m/s²

Time taken, t =?

0 = 49 + (-9.8) × t

0 = 49 – 9.8 t

9.8 t = 49

t =

t = 5 s

The ball will take equal time to return to the surface of earth.

Therefore, Total time = 5 + 5

= 10 seconds

Given,

Initial velocity, u = 0

Final velocity, v =?

Acceleration due to gravity, g = 9.8 m/s²

Height, h = 19.6 m

We know that,

v² = u² + 2gh

= (0)² + 2 × 9.8 × 19.6

= 19.6 × 19.6

= (19.6)2

= 19.6 m/s

Therefore, its final velocity just before touching the ground is 19.6 m/s.

Given,

Initial velocity, u = 40 m/s

Final velocity, v = 0 (As the stone stops)

Acceleration due to gravity, g = - 10 m/s²

Height, h =?

v² = u² + 2gh

(0)² = (40)2 + 2 (-10) × h

0 = 1600 – 20 h

20 h = 1600

h =

h = 80 m

The maximum height to which stone is thrown up is 80 m.

As, the stone is thrown up from the ground and after reaching to a maximum height of 80m it falls back to the ground. Therefore, the net displacement of the stone is zero.

The distance covered by the stone in reaching the maximum height is 80 m.

The stone will cover the same distance of 80 m in coming back to ground.

So, the total distance covered by the stone = 80 + 80

= 160 m

We know that, Gravitational force acting between two object is given by:

F = G× ...........(1)

G is the Gravitational constant = 6.7 × 10^-11 Nm²/kg²

m₁ is the Mass of earth = 6 × 10²⁴ kg

m₂ is the Mass of sun, m₂ = 2 × 10³⁰ kg

r is the distance between the earth and the sun= 1.5 × 10¹¹ m

Now, putting all these values in equation 1, we get:

F =

F = 3.57 × 10²² N

The Force exerted on the Earth by the sun is 3.5 × 10²² N

According to Figure the height of tower is 100m, Assuming the two stones meet at point P which is at a height x from the ground, from this we get that distance between Point P and the top of tower is (100-x)

(i)Firstly, for the stone falling from top of tower:

Height, h = (100 - x)

Time, t =?

We know that,

(100 – x) = 0 × t +1/2× 10 × t²

(ii)Now, for stone projected vertically upwards:

Initial velocity, u = 25 m/s

g = -10 m/s² (As the stone is going up)

We know that,

x = 25t – 5t² (ii)

Now, by adding (i) and (ii), we get:

100 – x + x = 5t² + 25 t – 5t²

t =

Therefore, the 2 stones will meet after a time of 4 seconds.

100 – x = 5× (4)²

100 – x = 80

x = 20m

Therefore, the two stones will meet at a height of 20 m above the ground after 4 seconds

Since the ball thrown up vertically returns to the thrower in 6 seconds, this means that the ball will take half of this time.

= 3 s

(a) Given,

Final velocity, v = 0

Initial velocity, u =?

Acceleration due to gravity, g = -9.8 m/s² (As the ball goes up)

Time taken, t = 3 s

We know that,

v = u + g × t

0 = u + (-9.8) × 3

0 = u – 29.4

u = 29.4 m/s

(b) Now, calculation of maximum height:

We know that, at maximum height velocity of object will be Zero

v² = u² + 2gh

(0)² = (29.4)² + 2 × (-9.8) × h

0 = 864.36 – 19.6 h

19.6 h = 864.36

h =

h = 44.1 m

(c) Finally, we have to calculate the position of ball after 4 seconds:

We know that, the ball will at max height when t=3 sec at t = 4 sec the ball has been falling for 1 sec from the max height

h = 0×1+ ×9.8 × (1)² (Because t = 1 s)

= 0 + 4.9 ×1

= 4.9 m
Hence, ball has been fallen 4.9 m from the max height

The height from the ground= maximum height - height object has fallen in 1 sec= 44.1- 4.9= 39.2 m above the ground

The buoyant force on an object immersed in a liquid acts in upward direction (i.e., opposite to weight of the object).

Or in other words we can say that the buoyant force on an object submerged in a fluid is caused by the pressure difference between the top and bottom of the object. To overcome the gravitational force, the buoyant force acts in the upward direction. The larger pressure at greater depth pushes upward on the object.

The density of plastic is less than that of water, so the force of buoyancy on plastic block will be greater than the weight of plastic block displaced. Hence, the acceleration of plastic block will be in upward direction, and comes up to the surface of water.

Given,

Mass of substance = 50 g

Volume of substance = 20 cm³

We know that,

Density of substance =

=

= 2.5 g cm^-3



The density of the substance is more than that of water( 1g cm^-3) as a result of which the substance will sink in water.