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.
| S. No. | Mass | Weight |
|---|---|---|
| 1 | Mass is the property of matter. The mass of an object is the same everywhere. | Weight depends on the effect of gravity. Weight increases or decreases with higher or lower gravity. |
| 2 | Mass can never be zero. | Weight can be zero if no gravity acts upon an object, as in space. |
| 3 | Mass can never be zero. | Weight is vector quantity. It is different at different places. |
| 4 | It only has magnitude. | It has magnitutde as well as direction. |
| 5 | Its SI unit is kilogram (kg). | Its SI unit is the same as the SI unit of force, i.e., Newton (N). |
WE = the object on the surface of the Earth
Let, MM = mass of the moon
RM = radius of the moon
WM = 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.
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
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
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
v = u + gt
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 = h
Initial velocity during the upward journey, u = 29.4 m/s
Final velocity, v = 0 m/s
Acceleration due to gravity, g = −9.8 m s−2
From the equation of motion,
(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 m/s
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 = 0 x 1 + ½ x 9.8 x 12
s = 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.
The buoyant force acts in the upward direction on the object which is immersed in a liquid.
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.
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.
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.
The motion of the planet around the earth is due to the centripetal force which is provided by the gravitational force between the planet and the Sun.
The force depends on the mass of the Sun, mass of the planet and the distance between the Sun and the planet.
The one which was simply dropped down will reach the ground first. If the height was large enough and measurements very accurate the earth continues to spin. Therefore, this stone will fall directly in a straight line joining the initial point on the centre of the earth but the point on the earth’s surface which was directly below the initial point will have moved. The vertical component of velocity of the other stone, which had been hurled parallel to the earth’s surface, is zero.
However, owing to the sphericity of the earth’s surface together with earth’s spin, it will have to fall greater distance in order to reach earth’s surface. Therefore, it will take longer.
If the gravity of earth suddenly becomes zero then the moon flies off along a straight line. This straight line will be a tangent to the circular path.
First of all one should know that earth’s gravity is stronger at poles than at equator. That’s why acceleration due to gravity is more at the poles than at the equator. So, if two identical packets are dropped from some height h, will accelerate more than the packet dropped at equator from the same height and hence will reach the surface of earth earlier.
The weight of any person on the moon is about 1/6 times of the earth (given)
If he can lift 15 kg on earth then he will lift 'x' kg or moon such that 1/6 of 'x' is equal to 15.
x/6 = 15
x = 90 kg.
We know that, g = GM / R^2 or M = gR^2 / G
Average density of the earth, D = Mass / Volume = gR^2 / G x Ve (where Ve is the volume of the earth)
D = gR^2 / G(4/3) πR^3 = 3g / 4πGR
Earth does not fall into the Sun because earth keeps revolving around the Sun. This creates a centripetal force that pulls earth away from the Sun. When this force is equal to the gravitational attraction, earth becomes stable. That is, it never falls into the Sun, nor she leaves the Sun.
The weight of a body is directly proportional to the mass of the earth and inversely proportional to the square of the radius of the earth.
Weight of an object of mass m,
When diameter of the earth becomes half, radius of earth also becomes half and mass becomes 4 times of its initial value, then
From Newton’s law of gravitation, force of attraction between two bodies is directly proportional to the product of their masses, and inversely proportional to the square of distance between their centres i.e.,
A student's 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 falling body.
For the object of mass m1 dropped from
height h1, u = 0, a = g, s = h,
Similarly, for the object of mass m2 dropped from height h2
The ratio will not change in either case because acceleration remains the same. In the case of free fall acceleration does not depend upon mass and size of the body.
(a) The density of the salt solution is greater than that of water so the cube will experience a greater buoyant force in the saturated salt solution.
The volume of the smaller cube is lesser than the initial cube so the smaller cube will experience less buoyant force.
(b) Here, Mass of ball, m = 4 kg
Density of ball, ρ = 4000 kg m-3
Density of water σ =103 kg m-3
Buoyant force = weight of the liquid displaced = density of water x volume of water displaced x g
= 1000 x (4 / 4000) x 10 = 10 N.