Derive the relationship between #"g"# and #"G"#?
r is the distance between object and the center of earth
h is height of object from the surface and R is the radius of earth.
and
(When the object is near the surface of earth so we can neglect the height of object comparing with Radius of Earth)
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The relationship between "g" (acceleration due to gravity) and "G" (universal gravitational constant) is given by Newton's law of universal gravitation:
[ F = G \frac{m_1 \cdot m_2}{r^2} ]
Where:
- ( F ) is the gravitational force between two objects,
- ( G ) is the universal gravitational constant,
- ( m_1 ) and ( m_2 ) are the masses of the two objects, and
- ( r ) is the distance between the centers of the two objects.
At the surface of the Earth, ( F ) is the force experienced by an object due to gravity, which is equal to ( m \cdot g ), where ( m ) is the mass of the object and ( g ) is the acceleration due to gravity. Therefore, at the surface of the Earth:
[ m \cdot g = G \frac{m_{\text{Earth}} \cdot m_{\text{object}}}{R_{\text{Earth}}^2} ]
By rearranging this equation, we can solve for ( g ):
[ g = \frac{G \cdot m_{\text{Earth}}}{R_{\text{Earth}}^2} ]
Where:
- ( m_{\text{Earth}} ) is the mass of the Earth, and
- ( R_{\text{Earth}} ) is the radius of the Earth.
Therefore, the relationship between ( g ) and ( G ) is that ( g ) depends on ( G ) as well as the mass and radius of the Earth.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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