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The relation between G and g gives the magnitude of acceleration to which a massive body of mass M accelerates a smaller body due to its gravitational pull.
- The term G represents the universal gravitational constant.
- The term g represents the acceleration due to gravity.
The relation between G and g is given by
\(g=G\frac{M}{R^2}\)
Where
- M is the mass of the massive body.
- R is the distance between the massive and smaller body.
The terms G and g are commonly used while describing gravitational force.
- The value of G is 6.67408 x 10-11 m3 kg-1 s-2.
- G is a constant value, it does not change from object to object.
- For Earth, the value of g is 9.8 m/s2.
- For different celestial bodies, the value of g will be different.
- For example, the value of g on the surface of the moon will be 1/6 times the value of g on the surface of the earth.
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Key Terms: Acceleration due to gravity, Gravitational constant, Law of gravitation, Value of g, Value of G, Gravitational force, Velocity, Acceleration
What is G?
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The word G refers to the Universal gravitational constant in the formula of Newton’s universal law of gravitation.
- The gravitational constant is an empirical physical constant used in Newton's law of universal gravitation and Einstein's theory of general relativity to calculate gravitational effects.
- The value of G is 6.67408 x 10-11 m3 kg-1 s-2.
- The dimensional formula of G is [M-1 L3 T-2].
- The value of G does not change from object to object.
The gravitational force between two bodies of masses m1 and m2 separated by distance r is given by
\(F = G\frac{m_1m_2}{r^2}\)
Here G stands for the universal gravitational constant, which is a proportionality constant.
We can deduce that the Universal gravitational constant is equal to the force of attraction acting between two bodies of equal mass separated by a unit distance.
The video below explains this:
Relation Between G And g Detailed Video Explanation:
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What is g?
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The acceleration of any object due to gravity is known as acceleration due to gravity.
- Acceleration due to gravity is represented by g.
- The rate of change in velocity due to the gravitational pull is known as small g.
- This is a type of acceleration that is only caused by gravity.
- Since smaller objects have very small gravitational force, this is usually reserved for massive things.
- The SI unit of g is meters per second squared (m/s2).
- When we drop an object, the acceleration it experiences is due to the Earth's gravitational pull.
- We can call this acceleration "gravitational acceleration."
- For the planet Earth, g has a value of 9.8 m/s2.
Relationship between G and g
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Consider an apple of mass m that falls from a tree. The apple is pulled to the earth because of the gravitational force exerted by the earth.
The gravitational force exerted on the apple is given by, according to Newton's law of gravitation.
F = GMm/R2 …………. (1)
Where
- M denotes the earth's mass
- R denotes the earth's radius.
The apple is accelerated towards the center of the earth with acceleration g (acceleration due to gravity) due to gravitational force.
According to Newton's second law,
F = mg ………… (2)
Now, from equations (1) and (2), we can say
mg = GMm/R2
⇒ g = GM/R2
Hence the relation between G and g is given by the equation
\(g=\frac{GM}{R^2}\)
We can see from this equation that g is independent of the mass of the falling body, implying that all bodies, regardless of mass, are propelled to the same extent by gravity.
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Difference between G and g
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| S.No. | Parameters | Gravitational Constant (G) | Gravitational Acceleration (g) |
|---|---|---|---|
| 1 | Definition | It's the gravitational attraction between two bodies of equal mass separated by a unit distance. | It states that a body falls freely and with constant acceleration under the influence of gravity. |
| 2 | Value | 6.67 x 10-11 N m2 kg-2 | 9.8 m s-2 |
| 3 | Quantity | Scalar quantity | Vector quantity |
| 4 | Variations | The value of G is a dimensional constant or a constant in the cosmos. | The value changes with height, depth, and the earth's rotation. |
| 5 | Nature | Regardless of the bodies' type and size, as well as the medium in which they are preserved. | The mass and radius of the earth or planet, due to which there is a gravitational force, are independent of shape, size, and mass. |
| 6 | Unit | N m2 / kg2 | m s-2 |
| 7 | Dimensional Formula | M-1L3T-2 | M0L1T-2 |
Things to Remember
- G stands for Universal gravitational constant, and its value is 6.67408 x 10-11 m3 kg-1 s-2.
- g stands for acceleration due to gravity, and its value on the surface of the earth is 9.8 m/s2.
- The relation between G and g is given as g = GM/R2.
- G is a constant value, it will not change for different objects or at different locations.
- g is different for different objects, it depends on the mass and radius of the object.
- According to Newton’s law of gravitation, the force between two objects is given by F = G (m1m2/r2).
- The dimensional formula of G is [M-1 L3 T-2].
- The dimensional formula of g is [L T-2].
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Sample Questions
Ques. If the mass of one of the bodies is doubled, what happens to the gravitational force between them? (1 mark)
Ans. When a body's mass is doubled, its force is also doubled.
Ques. Why is the universal gravitational constant termed 'G'? (2 marks)
Ans. The constant 'G' is universal since it is unaffected by the nature and sizes of bodies, the space in which they are stored, or the moment at which the force is considered.
Ques. What is the universal law of gravitation? (2 marks)
Ans. According to the universal law of gravitation, every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Ques. What is the value of G? (1 mark)
Ans. The value of G is 6.67408 x 10-11 N m2 kg-2.
Ques. What is the value of g? (1 mark)
Ans. The value of g on earth is 9.8 m s-2.
Ques. What is the acceleration due to gravity? (2 marks)
Ans. Whenever a body falls toward the earth under the effect of gravitational pull, its velocity changes with time, which produces acceleration. The acceleration produced in the motion of the body under the effect of gravity is called acceleration due to gravity.
Ques. What do you mean by gravitation field? (2 marks)
Ans. The gravitational field intensity or gravitational field at a point in space is defined as the gravitational force experienced by a test mass placed at that point divided by its mass.
Ques. What is the dimensional formula of G? (1 mark)
Ans. The dimensional formula of G is [M-1 L3 T-2].
Ques. What is the dimensional formula of g? (1 mark)
Ans. The dimensional formula of g is [L T-2].
Ques. What distinguishes gravitation from gravity? (2 marks)
Ans. Gravitation is the force of attraction between two bodies, whereas gravity is the force of attraction between any two bodies and the earth.
Ques. What does a low value of G imply? (1 mark)
Ans. A low G number suggests that the gravitational attraction between two ordinary-sized objects is very feeble.
Ques. What is the mass of a body on the moon if its mass on Earth is 9.8 kg? (1 mark)
Ans. On the moon, it will remain the same, i.e., 9.8 kg.
Ques. An apple is drawn to the soil. Is the apple also a magnet for the earth? If that's the case, why isn't the earth moving towards the apple? (3 marks)
Ans. Action and reaction are equal and opposite, according to Newton's third law of motion. It indicates that the force exerted on the apple by the earth's attraction is equal to the force exerted by the apple on the earth. However, we know that acceleration is 1/m.
Because the earth's mass is so great in comparison to the apple's, the acceleration experienced by the earth will be so minor that it will go unnoticed.
Ques. Assume that the earth's radius doubles while its mass remains constant. What will happen to your weight if that happens? (3 marks)
Ans. We know that F = GMmr2 is the force with which a body is drawn to the earth and that W = GMmr2 is the force with which a body is attracted to the earth.
W = GMm(2r)2 = GMm4r2=W4 If the earth's radius doubles, then W = GMm(2r)2 = GMm4r2=W4
i.e., the weight will be decreased to a quarter of its original value.
Ques. What is the relation between G and g? (5 marks)
Ans. Consider an apple of mass m that falls from a tree. The apple is pulled to the earth because of the gravitational force exerted by the earth.
The gravitational force exerted on the apple is given by, according to Newton's law of gravitation.
F =GMm/R2 …………. (1)
where M denotes the earth's mass and R denotes the earth's radius.
The apple is accelerated towards the center of the earth due to gravitational force, according to Newton's second law.
hence, we can write F =mg ………… (2)
Now, from equations (1) and (2), we can say
mg = GMm/R2
=> g = GM/R2
The relation between g and G is shown in this equation.
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