91Ó°ÊÓ

Isaac Newton's discovery that gravity is a universal force acting between all masses is a cornerstone of physics.
Newton's universal law of gravitation tells us that every particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
The formula expressing this law is: \[ F = G \frac{m1 \times m2}{r^2} \] where:\

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• \( F \) is the gravitational force between the two masses,\
• \( G \) is the gravitational constant (\( 6.674 \times 10^{-11} N \cdot (m/kg)^2 \)),\
• \( m1 \) and \( m2 \) are the masses of the two objects,\
• \( r \) is the distance between the centers of the two masses.\

\This equation allows us to calculate the force of attraction between any two objects if we know their masses and the distance between them.

The acceleration due to gravity is the rate at which an object's velocity changes due to the gravitational force of a massive body like Earth.
On the surface of Earth, this acceleration is approximately \( 9.81 m/s^2 \) and is denoted by \( g \). However, the value of \( g \) varies depending on your location—namely altitude above Earth's surface and the geographical latitude.
The general form to determine \( g \) at any given distance from the Earth's center is given by the formula: \[ g = \frac{G \times Me}{r^2} \] where:\

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• \( G \) is the gravitational constant,\
• \( Me \) is Earth's mass,\
• \( r \) is the distance from the Earth’s center to the object.\

\As the distance \( r \) increases, the gravitational acceleration \( g \) decreases, which is why the gravitational acceleration of the meteoroid in the original problem would be less than \( 9.81 m/s^2 \) when the meteoroid is 3 times farther from the Earth’s center than the Earth's radius.

To calculate the gravitational force between two objects, we need to use the formula derived from Newton's law of gravitation.
Let's break down the steps: First, we identify the masses of the two objects (in our problem, one mass is Earth's and the other the meteoroid's). Next, we determine the distance \( r \) between the centers of the two masses.
Applying the formula: \[ F = G \frac{m1 \times m2}{r^2} \] we can solve for \( F \) by substituting in the known values for \( G \), \( m1 \), \( m2 \), and \( r \).
In our textbook problem, since the mass of the meteoroid and Earth's mass are being multiplied, any factor that is a part of each mass (such as the meteoroid’s mass) will cancel out, simplifying the calculation of the gravitational force.

Newton's second law of motion provides the relationship between an object's mass, the force applied to it, and its acceleration.
The law is often written in the form of an equation: \[ F = m \times a \] where:\

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• \( F \) represents the force applied to the object,\
• \( m \) represents the object’s mass,\
• \( a \) represents the acceleration of the object.\

\Newton’s second law tells us that the force on an object is equal to the mass of the object multiplied by the acceleration it experiences.
When dealing with gravitational forces, such as in the textbook exercise, the mass of the object being accelerated (the meteoroid) is part of both the force and the mass being accelerated, so it cancels out, allowing us to focus on the Earth's mass and the distance from the center of the Earth to determine the object’s resultant acceleration.