What is Gravitation?
Gravitation is the universal phenomenon where every object in the universe attracts every other object. This attractive force between any two objects is referred to as the "force of gravitation." Although the gravitational force is weak and typically requires substantial masses to be observed, it acts over long distances.
Gravitation and Gravity
Universal Law of Gravitation
According to Newton’s law of gravitation, the force of gravitational attraction between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Question for Chapter Notes: Gravitation
Try yourself:What is the formula for the force of gravitational attraction between two objects?
Explanation
F_{g} = Gm_{1}m_{2}/r^{2}
G is the gravitational constant with the value of 6.67 ×10^{−}^{11} Nm^{2}/kg^{2}, m_{1} is the mass of a body, m_{2} is the mass of other body, r is the distance between the two bodies.
If M and m be masses of two objects separated by a distance d, the gravitational force of attraction between them is given by: F = G x (m^{1 }x m^{2}) / d^{2}where G is a constant, known as the universal constant of gravitation.
- The universal constant of gravitation G is numerically equal to the force of attraction between two objects of unit mass each separated by unit distance.
- The value of G is 6.673 x 10^{-11 }N m^{2} kg^{-2}.
- G is called a universal constant because its value does not depend on the nature of the intervening medium or temperature or any other physical variable.
- As the value of G is extremely small, the gravitational force between ordinary terrestrial objects is so small that it cannot be detected. However, the force of attraction acting on an object due to Earth, the force of attraction between Earth and the moon, and the force experienced by planets due to the gravitational attraction of Sun can be easily felt and measured.
Example 1: Suppose we have two objects: Object A with a mass of 5 kilograms and Object B with a mass of 10 kilograms. The distance between the centers of these objects is 2 meters. We'll assume the gravitational constant, G, to be approximately 6.674 × 10^{-11 }N m^{2}/kg^{2}.
Solution:
Using the Universal Law of Gravitation, we can calculate the gravitational force between these objects:
F = (G * (m^{1} * m^{2})) / r^{2}
F = (6.674 × 10^{-11} N m^{2}/kg^{2} * (5 kg * 10 kg)) / (2 m)^{2}
F = (6.674 × 10^{-11} N m^{2}/kg^{2 }* 50 kg^{2}) / 4 m^{2}
F = (6.674 × 10^{-11} N m^{2}) / 4
F ≈ 1.6685 × 10^{-11} N
Therefore, the gravitational force between Object A and Object B is approximately 1.6685 × 10^{-11} Newtons.
Question for Chapter Notes: Gravitation
Try yourself:Which of the following pairs of objects will experience the greatest gravitational force?
Explanation
Let's calculate the gravitational forces for each option and determine which pair experiences the greatest force:
A) Object X with a mass of 100 kilograms and Object Y with a mass of 200 kilograms, separated by a distance of 1 meter.
Using the Universal Law of Gravitation:
F_A = (G * (m_X * m_Y)) / r^2
F_A = (6.674 × 10^-11 N m^2/kg^2 * (100 kg * 200 kg)) / (1 m)^2
F_A = (6.674 × 10^-11 N m^2/kg^2 * 20000 kg^2) / 1 m^2
F_A = 1.3348 × 10^-6 N
B) Object P with a mass of 50 kilograms and Object Q with a mass of 100 kilograms, separated by a distance of 2 meters.
F_B = (G * (m_P * m_Q)) / r^2
F_B = (6.674 × 10^-11 N m^2/kg^2 * (50 kg * 100 kg)) / (2 m)^2
F_B = (6.674 × 10^-11 N m^2/kg^2 * 5000 kg^2) / 4 m^2
F_B = 8.3435 × 10^-7 N
C) Object M with a mass of 200 kilograms and Object N with a mass of 400 kilograms, separated by a distance of 3 meters.
F_C = (G * (m_M * m_N)) / r^2
F_C = (6.674 × 10^-11 N m^2/kg^2 * (200 kg * 400 kg)) / (3 m)^2
F_C = (6.674 × 10^-11 N m^2/kg^2 * 80000 kg^2) / 9 m^2
F_C = 5.9111 × 10^-6 N
D) Object R with a mass of 150 kilograms and Object S with a mass of 300 kilograms, separated by a distance of 4 meters.
F_D = (G * (m_R * m_S)) / r^2
F_D = (6.674 × 10^-11 N m^2/kg^2 * (150 kg * 300 kg)) / (4 m)^2
F_D = (6.674 × 10^-11 N m^2/kg^2 * 45000 kg^2) / 16 m^2
F_D = 7.8418 × 10^-6 N
Comparing the calculated forces:
F_A ≈ 1.3348 × 10^-6 N
F_B ≈ 8.3435 × 10^-7 N
F_C ≈ 5.9111 × 10^-6 N
F_D ≈ 7.8418 × 10^-6 N
Based on the calculations, the pair of objects that experience the greatest gravitational force is:
D) Object R with a mass of 150 kilograms and Object S with a mass of 300 kilograms, separated by a distance of 4 meters.
Importance Of The Universal Law Of Gravitation
The universal law of gravitation successfully explained several phenomena which were believed to be unconnected:
- The force that binds us to the Earth
- The motion of the moon around the Earth
- The motion of planets around the Sun
- The tides due to the moon and the Sun
Free Fall or Gravity
- The force of gravitational attraction exerted by the earth on an object is called the “force of gravity’'. Force of gravity acting on an object of mass m situated on Earth’s surface or near it is given by
where M = mass of earth and R = radius of Earth. - The acceleration produced in a freely falling object on account of the force of gravity is known as the acceleration due to gravity. It is denoted by the symbol ‘g’.
Gravitation Formula
To Calculate Value of g
- The value of acceleration due to gravity on the surface of Earth is given by
The mean value of g on the surface of Earth is found to be 9.8 ms^{-2}. - Here's how the value of g is calculated.
Calculation of acceleration due to gravity
Motion of Objects under the influence of Gravitational Force of the Earth
Value of g varies from place to place. On surface of earth value of g is more at poles than at the equator. Again value of g decreases as one goes away and away from the earth.
Free Fall Motion
- When an object falls towards the earth under the force of gravity alone, we say that the object is in free fall. A freely falling object experiences a constant acceleration of g (=9.8ms^{-2}) during its downward motion. However, if an object is projected vertically upward with a certain velocity, its velocity goes on decreasing due to gravity, till it comes to rest and then starts falling vertically downward under gravity.
- The three equations of motion viz, (i) v = u + at, (ii) s = ut + 1/2 at^{2}, and (iii) v^{2} - u^{2} = 2as are true for motion of objects under gravity. For free fall, the value of acceleration a = g = 9.8ms^{-2}.
- If an object is just let fall from a height then in that case u = 0 and a = +g = +9.8ms^{-}^{2}.
- If an object is projected vertically upwards with an initial velocity u, then a = -g = -9.8ms^{-2} and the object will go to a maximum height h where its final velocity becomes zero (i.e. v = 0). In such a case
Question for Chapter Notes: Gravitation
Try yourself:
Which of the following statements is true according to the universal law of gravitation?Explanation
- According to the universal law of gravitation, the force of attraction between two objects depends on their masses and the distance between them.
- The value of the universal constant of gravitation, G, does not depend on the nature of the intervening medium or any other physical variable.
- The gravitational force between ordinary terrestrial objects is so small that it cannot be detected, but it can be easily felt and measured for larger celestial bodies.
- The value of the acceleration due to gravity on the surface of Earth is approximately 9.8 m/s^2, not the universal constant of gravitation.
So, the correct statement is that the force of attraction between two objects depends on their masses and the distance between them.
Mass
- The mass of an object is a measure of its inertia. Mass of an object is constant and does not change from place to place.
Mass and Weight
Weight
- The weight of an object is the force with which it is attracted towards the earth. Weight W of an object of mass m will be W = mg. Weight is a force acting vertically downwards. It means that it is a vector.
- As weight of an object is a force, its SI unit is newton (N).
An object of mass m = 1 kg has thus a weight of W = 1 x 9.8 = 9.8 N. - At a given place weight of an object is directly proportional to its mass t.e., (at a given place). Due to this reason at a given place, we may use the weight of an object as a measure of its mass.
Weight of Object on the Moon
- Mass of an object remains the same everywhere but weight of an object changes from place to place. For an object of mass m weight W is dependent on value of g.
- Force of gravity due to moon is 1/6th of the force of gravity due to earth.
Hence
Due to this very reason weight of an object on moon will be 1/6th of its weight on the earth.
Question for Chapter Notes: Gravitation
Try yourself:Question: Which of the following statements about weight is true?
Explanation
Weight is the force of gravity acting on an object.
Explanation: Weight is the force exerted on an object due to gravity, which is dependent on the mass of the object and the gravitational force acting upon it. Weight is not the same as mass, which is a measure of the amount of matter in an object. While mass remains constant regardless of location, weight can vary depending on the strength of the gravitational force acting on the object. For example, an object will weigh less on the moon than it does on Earth due to the moon's weaker gravitational force.
Thrust and Pressure
The normal force acting on a surface, due to the weight of an object placed on the surface, is called ‘thrust’. As thrust is a sort of force hence its SI unit is “a newton” (N).
Thrust
- The thrust on unit surface area is called pressure.
Pressure
Thus, pressure on a given object is the normal force acting on its surface per unit surface area.
SI unit of pressure is N m^{-2} but it is also called pascal and denoted by symbol Pa.
∴ 1 pascal (1 Pa) = 1 N m^{-2 } - Same force acting on a smaller area exerts a larger pressure. It is due to this reason that a nail or a pin has a pointed tip and knives have sharp edges.
- Given force acting on a larger area exerts a smaller pressure. It is due to this reason that foundations of houses are made broad, base of dams is made broad, sleepers are laid below the railway line and so on.
Pressure in Fluids
Fluid is that state of matter which can flow. All liquids and gases are fluids.
Pressure in Fluids
- As fluids have weight, they exert pressure on the base and the walls of the container in which they are kept.
- In a fluid, pressure applied at any one place is transmitted equally in all directions.
Buoyancy
- Whenever an object is immersed in a fluid, it experiences a force in the vertically upward direction due to that fluid. This force due to a fluid acting in upward direction is called ‘force of buoyancy’ or ‘buoyant force or up thrust.
Buoyancy
- Magnitude of force of buoyancy acting on an object depends upon
(i) the volume of an object immersed into the fluid, and
(ii) the density of the fluid. Due to force of buoyancy the net weight of an object becomes less than its normal weight.
Question for Chapter Notes: Gravitation
Try yourself:
What is the SI unit of weight?Explanation
- The SI unit of weight is Newton (N).
- Weight is a force and is measured in Newtons.
- The weight of an object is the force with which it is attracted towards the Earth.
- The formula to calculate weight is W = mg, where W is the weight, m is the mass, and g is the acceleration due to gravity.
- The weight of an object can be different on different celestial bodies, but the SI unit remains the same.
Why Objects Float Or Sink When Placed On The Surface Of Water?
The ability of an object to float or sink when placed on the surface of water is determined by its density and the density of the water. Density is a measure of how much mass is contained in a given volume.
Floating and Sinking on Surface of WaterWhen an object is placed in water, it experiences two main forces: buoyancy and gravity.
- Buoyancy is the upward force exerted on an object immersed in a fluid (in this case, water). It is caused by the difference in pressure between the top and bottom of the object. The greater the volume of water displaced by the object, the greater the buoyant force. According to Archimedes' principle, an object will experience an upward buoyant force equal to the weight of the water it displaces. If the buoyant force is greater than the object's weight, it will float. If the buoyant force is less than the object's weight, it will sink.
- Gravity, on the other hand, is the downward force pulling the object toward the center of the Earth. The weight of the object depends on its mass and the acceleration due to gravity. If the weight of the object is greater than the buoyant force, it will sink. If the weight is less than the buoyant force, it will float.
In summary, whether an object floats or sinks in water depends on the comparison between its weight and the buoyant force exerted by the water. If the object's weight is greater, it will sink. If the buoyant force is greater, it will float.
Archimedes’ Principle
A Greek scientist Archimedes discovered a principle regarding buoyant force (or the loss in weight of an object when immersed in a fluid).
Archimedes' Principle
- According to Archimedes’ principle “whenever an object is immersed fully or partially in a fluid, it experiences an upward force that is equal to the weight of the fluid displaced by it”.
- Alternately, according to Archimedes’ principle “there is always a loss (decrease) in weight of an object when it is immersed fully or partially in a fluid and the loss in weight is equal to the weight of the fluid displaced by the immersed part of given object”.
- Archimedes’ principle has many applications. Design of ships, submarines, lactometer and hydrometer are some of the applications of Archimedes’ principle.
Question for Chapter Notes: Gravitation
Try yourself:Archimedes' principle states that:
Explanation
The buoyant force on an object is equal to the weight of the fluid displaced by the object
Explanation: Archimedes' principle states that when an object is submerged in a fluid, it experiences an upward buoyant force that is equal to the weight of the fluid displaced by the object. This principle is derived from the fact that pressure in a fluid increases with depth. When an object is submerged in a fluid, the pressure on the bottom surface of the object is greater than the pressure on the top surface, creating a net upward force. This buoyant force is equal to the weight of the fluid that was displaced by the object, which can be determined by multiplying the volume of the fluid displaced by the object by the density of the fluid and the acceleration due to gravity (Buoyant force = Volume x Density x Gravity).
Relative Density (Old Syllabus)
- Density of a substance is defined as its mass per unit volume. Its SI unit is kg m^{-3}. However, sometimes unit g cm^{-3} is also used for density. It is found that 1 g cm^{-3} = 1000 kg m^{-3}.
- Relative density of a substance is the ratio of its density to the density of water.
Relative density is a unit-less quantity. - We know that an iron nail immersed in water sinks but a wooden piece remains floating on water. It is found that objects having density less than that of the liquid, in which they are immersed, float on the surface of fluid. On the other hand, if density of the object is more than the density of the fluid in which it is immersed then the object will sink in that fluid.