Final answer:
The relation between the acceleration due to gravity and the radius of the Earth at a particular place can be proved using Newton's universal law of gravitation. The acceleration due to gravity at the surface of the Earth is approximately 9.8 m/s². The relation between the acceleration due to gravity and the radius of the Earth can be expressed using a formula involving the radius and acceleration at a different location.
Step-by-step explanation:
The relation between the acceleration due to gravity and the radius of the Earth at a particular place can be proved using Newton's universal law of gravitation. According to this law, the force acting on an object towards the Earth is inversely proportional to the square of its distance from the center of the Earth.
Let's consider an object at the surface of the Earth, at a distance of one Earth-radius from the center of the Earth. This object is observed to accelerate downward at 9.8 meters per second per second (9.8 m/s²). This means that the acceleration due to gravity (g) at this location is 9.8 m/s².
Now, let's calculate the acceleration due to gravity (g') at a different location with a different radius of the Earth. The relation between g and g' can be expressed as:
g' = k * (RE' / RE)²
Where:
- g' is the acceleration due to gravity at the new location
- k is a constant
- RE' is the radius of the Earth at the new location
- RE is the radius of the Earth at the original location
By rearranging the equation, we can express the radius of the Earth at the new location:
RE' = sqrt(g') * (RE / sqrt(g))
Therefore, we can see that the radius of the Earth at a particular place is related to the acceleration due to gravity at that place.