Gravitation

We know,

F = G $\dfrac{\text{m}_1\text{m}_2}{\text{d}^2}$

Where,

F = force of attraction

G = constant of proportionality

m₁ = mass of first object

m₂ = mass of second object

d = distance between the two objects.

When distance is reduced to half then, d' = $\dfrac{\text{d}}{2}$

So, substituting we get,

F' = G $\dfrac{\text{m}_1\text{m}_2}{\Big(\dfrac{\text{d}}{2}\Big)^2}$ = G $\dfrac{\text{m}_1\text{m}_2}{\dfrac{\text{d}^2}{4}}$ = 4G $\dfrac{\text{m}_1\text{m}_2}{\text{d}^2}$ = 4F

Hence, when the distance between the objects is reduced to half, the gravitational force increases four times.

All objects fall on ground with constant acceleration, called acceleration due to gravity (in the absence of air resistances). This acceleration remains constant and is independent of an object's mass. Hence, heavy objects do not fall faster than light objects.

Mass of the body (m) = 1 kg

Mass of the earth (M) = 6 × 10²⁴ kg

Radius of the Earth (R) = 6.4 × 10⁶ m

G = 6.67 x 10^-11 Nm²kg^-2

According to the formula for gravitational force we know,

F = G $\dfrac{\text{M}\text{m}}{\text{R}^2}$

Substituting we get,

$\text{F} = \dfrac{6.67 \times 10^{-11}\times 6 \times 10^{24} \times 1}{(6.4 \times 10^6)^2} \\[1em] = \dfrac{6.67 \times 10^{13}}{6.4 \times 6.4 \times10^{12}} \\[1em] = 9.77 \text{ N} \approx 9.8 \text{ N}$

Hence, magnitude of the gravitational force = 9.8 N

According to universal law of gravitation, two objects with masses attract each other with equal gravitational force, but in opposite directions.

This force is given by: F = G $\dfrac{\text{M}\text{m}}{\text{R}^2}$

Hence, the earth attracts the moon with the same force with which the moon attracts the earth.

According to the universal law of gravitation, two objects with masses attract each other with equal gravitational force, but in opposite directions. Hence, moon and earth both attract each other with the same force. Therefore, the earth does not move towards the moon.

(i) We know,

F = G $\dfrac{\text{m}_1\text{m}_2}{\text{d}^2}$

Where,

F = force of attraction

G = constant of proportionality

m₁ = mass of first object

m₂ = mass of second object

d = distance between the two objects.

When mass of one object is doubled.

m' = 2m₁

So, substituting we get,

F' = G $\dfrac{\text{2m}_1\text{m}_2}{\text{d}^2}$ = 2G $\dfrac{\text{m}_1\text{m}_2}{\text{d}^2}$ = 2F

Hence, when the mass of one object is doubled, the gravitational force increases two times.

(ii) When distance is doubled then, d' = 2r

So, substituting we get,

F' = G $\dfrac{\text{m}_1\text{m}_2}{(2d)^2}$ = G $\dfrac{\text{m}_1\text{m}_2}{4d^2}$ = $\dfrac{\text{F}}{4}$

Hence, when the distance between the objects is doubled, the gravitational force becomes one fourth of its original force.

Now, if it's tripled

d' = 3r

So, substituting we get,

F' = G $\dfrac{\text{m}_1\text{m}_2}{(3d)^2}$ =G $\dfrac{\text{m}_1\text{m}_2}{9d^2}$ = $\dfrac{\text{F}}{9}$

Hence, when the distance between the objects is tripled, the gravitational force becomes one ninth of its original force.

(iii) If masses of both the objects are doubled, then

m₁' = 2m₁

and

m₂' = 2m₂

So, substituting we get,

F' = G $\dfrac{\text{2m}_1\text{2m}_2}{\text{d}^2}$ = 4G $\dfrac{\text{m}_1\text{m}_2}{\text{d}^2}$ = 4F

Hence, when the masses of both the objects are doubled, the gravitational force becomes four times the original force.

The universal law of gravitation successfully explained several phenomena that were believed to be unconnected:

1. The force that binds us to the earth
2. The motion of the moon around the earth
3. The motion of planets around the sun
4. The tides due to the moon and the sun.

The acceleration of free fall 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.

The weight of a body on the earth's surface (W) = mg (where m = mass of the body and g = acceleration due to gravity)

Given, value of g is larger at poles when compared to the equator. So, for the same mass of gold, weight will be less at the equator as compared to the poles. Therefore, Amit's friend will not agree with the weight of gold bought.

A sheet of paper has a larger surface area in comparison to a crumpled paper ball. The larger surface area will suffer greater air resistance. Hence, a sheet of paper falls slower than the crumpled ball.

Given,

Acceleration due to earth's gravity g = 9.8 ms^-2

Object's mass, m = 10 kg

Acceleration due to moon gravity = g_m

Weight on the earth = W_e

Weight on the moon = W_m

Weight = mass x gravity

g_m = $\dfrac{1}{6}$ g_e (given)

So, W_m = m g_m = m x $\dfrac{1}{6}$ x g_e

W_m = 10 x $\dfrac{1}{6}$ x 9.8 = 16.34 N

W_e = m x g_e = 10 x 9.8

W_e = 98 N

Hence, weight on moon and earth = 16.34 and 98 N respectively.

Given,

Initial velocity (u) = 49 ms^-1

Final velocity v at maximum height = 0

Acceleration due to earth gravity g = -9.8 ms^-2 (negative as ball is thrown up).

Let H be the maximum height to which the ball rises.

By third equation of motion,

2gH = v² - u²

Substituting we get,

2 × (-9.8) × H = 0 - (49)²

-19.6 H = -2401

H = $\dfrac{-2401}{-19.6}$ = 122.5 m

Hence, the maximum height to which it rises = 122.5 m

(ii) Total time T = Time to ascend (T_a) + Time to descend (T_d)

According to the first equation of motion,

v = u + gt

Substituting we get,

0 = 49 + (-9.8) x T_a

T_a = $\dfrac{49}{9.8}$ = 5 s

Also, T_d = 5 s

∴ T = T_a + T_d = 5 + 5 = 10 s

Hence, total time it takes to return to the surface of the earth = 10 s

Given,

Initial velocity (u) = 0

Tower height = total distance = 19.6 m

g = 9.8 ms^-2

Final velocity (v) = ?

According to the third equation of motion,

v² = 2gs + u²

Substituting we get,

v² = (2 x 9.8 x 19.6) + 0

v = $\sqrt{384.16}$

v = 19.6 ms^-1

Hence, final velocity = 19.6 ms^-1

Given,

Initial velocity (u) = 40 ms^-1

Final velocity (v) = 0

g = 10 ms^-2

Max height = ?

According to the third equation of motion,

v² = u² - 2gs

Note: [negative g as the object goes up]

Substituting we get,

0 = (40)² - 2 x 10 x s

⇒ 1600 = 20s

⇒ s = $\dfrac{1600}{20}$ = 80 m

Total Distance = s_a + s_d = 80 + 80 = 160 m

where, s_a and s_d are distance going up and coming down, respectively.

Displacement = 0 [as the stone comes to the same point from where it was thrown ].

Hence, total distance covered = 160 m and displacement = 0, as the first point is the same as the last point

Given,

Mass of the sun (m) = 2 × 10³⁰ kg

Mass of the earth (M) = 6 × 10²⁴ kg

G = 6.67 x 10^-11 Nm²kg^-2

Average distance R = 1.5 × 10¹¹ m

According to the formula for gravitational force we know,

F = G $\dfrac{\text{M}\text{m}}{\text{R}^2}$

Substituting we get,

F = $\dfrac{6.67 \times 10^{-11}\times 6 \times 10^{24} \times 2 \times 10^{30} }{(1.5 \times 10^{11})^2}$ = 3.56 x 10²² N

Hence, magnitude of the gravitational force = 3.56 x 10²² N

Given,

(a) In the case, when the stone falls from the top of the tower,

Initial velocity u = 0

Distance travelled = x

Time taken = t

According to the second equation of motion,

S = ut + $\dfrac{1}{2}$gt²

⇒ S = 0 x t + $\dfrac{1}{2}$gt²

⇒ S = $\dfrac{1}{2}$gt²   [Eq 1]

(b) When another stone is projected vertically upwards,

Initial velocity u = 25 ms^-1

Distance travelled = (100 - x)

Time taken = t

Using S = ut + $\dfrac{1}{2}$gt²

we get,

S' = 25t - $\dfrac{1}{2}$gt²   [Eq 2]

From equations (1) and (2)

S + S' = $\dfrac{1}{2}$gt² + 25t - $\dfrac{1}{2}$gt²

100 = $\dfrac{1}{2}$gt² + 25t - $\dfrac{1}{2}$gt²

100 = 25t

t = $\dfrac{100}{25}$ = 4 sec.

Hence, after 4 secs, the two stones will meet.

From (1)

S = $\dfrac{1}{2}$ x 10 x 4² = 5 x 16 = 80m.

Hence, after 4 sec, 2 stones meet a distance of 80 m from the top.

Given,

Ball returns to thrower in 6s. So, time_up + time_down = 6 s hence, time_up = 3 s.

g = 9.8 ms^-2

(a) Final velocity (v) = 0

From first equation of motion

v = u – gt_up

⇒ u = v + gt_up

⇒ u = 0 + (9.8 x 3)

⇒ u = 29.4 ms^-1

Hence, the velocity with which it was thrown up = 29.4 ms^-1.

(b) According to the second equation of motion,

S = ut - $\dfrac{1}{2}$gt²

Substituting we get,

S = (29.4 x 3) - ($\dfrac{1}{2}$ x 9.8 x 3 x 3)
= 88.2 - 44.1 = 44.1 m

Hence, maximum height stone reaches is 44.1 m

(c) In 3 sec, it reaches the maximum height.

Distance travelled in another 1 sec = s'

According to the second equation of motion,

S = ut + $\dfrac{1}{2}$gt²

g is +ve as ball is going down

Substituting we get,

S = 0 + ($\dfrac{1}{2}$ x 9.8 x 1 x 1)
= 4.9 m

Hence, in 4 sec the ball will be 4.9 m from the top or 39.2 m (i.e., 44.1 - 4.9) from the bottom.

The buoyant force acts vertically upwards on the object that is immersed in a liquid.

When plastic is immersed in water, two forces act on it:

1. Gravitational force or its weight in vertically downward direction
2. Buoyant force in vertically upward direction.

As the density of plastic is less than that of water, the buoyant force is greater than its weight. Hence, the plastic comes up to the surface.

Given,

Volume of substance = 20 cm³

Mass of substance = 50 g

Density of water = 1 gcm^-3

Density of the substance = ?

Density = $\dfrac{\text{Mass}}{\text{Volume}}$ = $\dfrac{50}{20}$ = 2.5 gcm^-3

As density of the substance is greater than density of water. Hence, the substance will sink.

Given,

Volume of sealed packet = 350 cm³

Mass of sealed packet = 500 g

Density of water = 1 gcm^-3

Density of the sealed packet = ?

Density = $\dfrac{\text{Mass}}{\text{Volume}}$ = $\dfrac{500}{350}$ = 1.42 gcm^-3

As density of the packet is greater than density of water. Hence, the sealed packet will sink.

According to Archimedes Principle,

Displaced water volume = Volume of packet

Volume of water displaced = 350 cm³

Mass of water displaced = ρ x V = 1 × 350 = 350 g

Hence, mass of displaced water = 350 g

The universal law of gravitation states that 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 square of the distance between them. The force is along the line joining the centres of two objects.

F = G $\dfrac{\text{Mm}}{\text{d}^2}$

where,

F = force of attraction

G = constant of proportionality

M = mass of earth

m = mass of the object

d = distance between the earth's centre and object's centre

Earth attracts objects towards itself due to the gravitational force. Whenever objects fall towards the earth under this force alone, we say that the objects are in free fall.

Whenever an object falls towards the earth, an acceleration is involved due to the earth's gravitational force. So, this acceleration is called the acceleration due to the gravitational force of the earth (or acceleration due to gravity). It is denoted by g and its value is 9.8 ms^-2.

Differences between the mass of an object and its weight:

The moon's gravitation force is determined by the mass and the size of the moon.

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

Pressure is inversely proportional to the surface area on which the force acts. Incase of a strap made of a thin and strong string, the surface area of the strap in contact with the shoulder is less, hence, pressure on the shoulders is more which makes it difficult to hold the school bag.

The property of liquid to exert an upward force on a body immersed in it, is called buoyancy.

An object floats or sinks in water because of the difference in densities of the substance and water. When the density of the substance is less than the density of water, then the upthrust of water is greater than the weight of the substance. Hence it floats.

When the density of substance is more than water, then the upthrust of water on the substance is less than the weight of the substance. Hence it sinks.

When we stand on a weighing machine, the weight acts downwards while the upthrust due to air acts upwards.

As, true weight = (apparent weight + up thrust)

So our apparent weight becomes less than the true weight. This apparent weight is measured by the weighing machine and hence, the mass indicated is less than the actual mass. So our actual mass will be more than 42 kg.

The cotton bag is heavier than an iron bar as it experiences a larger air thrust than the iron bar.

As, true weight = (apparent weight + up thrust)

The cotton bag's density is less than that of the iron bar, so, the volume of the cotton bag is more compared to the iron bar. Hence, the cotton bag experience more upthrust due to the presence of air. So, in the presence of air, the cotton bag's true weight is more compared to the true weight of the iron bar.