Newton’s Law of Universal Gravitation, Grade 11 physics

Definition: Newton’s Law of Universal Gravitation
Every point mass attracts every other point mass by a force directed along the line connecting the two. This force is proportional to the product of the masses and inversely proportional to the square of the distance between them.
The magnitude of the attractive gravitational force between the two point masses, F is given by:

\[F = G\dfrac{m_1m_2}{r^2}\] (12.2)​

where: G is the gravitational constant, m1 is the mass of the first point mass, m2 is the mass of the second point mass and r is the distance between the two point masses.
Assuming SI units, F is measured in newtons (N), m₁ and m₂ in kilograms (kg), r in meters (m), and the constant G is approximately equal to 6,67 × 10$^{−11}$N · m²· kg$^{−2}$. Remember that this is a force of attraction.

For example, consider a man of mass 80 kg standing 10 m from a woman with a mass of 65 kg. The attractive gravitational force between them would be:

\[F = G\dfrac{m_1m_2}{r^2}\] = (6,67 × 10$^{−11}$)((80)(65)/(10)²)= 3,47 × 10$^{−9}$N​

If the man and woman move to 1 m apart, then the force is:

\[F = G\dfrac{m_1m_2}{r^2}\] = (6,67 × 10−11)((80)(65)/(1)²= 3,47 × 10$^{−7}$N​

As you can see, these forces are very small.

Now consider the gravitational force between the Earth and the Moon. The mass of the Earth is 5,98 × 10²⁴kg, the mass of the Moon is 7,35 × 10²²kg and the Earth and Moon are 0,38 × 10⁹m apart. The gravitational force between the Earth and Moon is:

\[F = G\dfrac{m_1m_2}{r^2}\] = 2,03 × 10²⁰N​

From this example you can see that the force is very large.

These two examples demonstrate that the bigger the masses, the greater the force between them. The 1/r² factor tells us that the distance between the two bodies plays a role as well. The closer two bodies are, the stronger the gravitational force between them is. We feel the gravitational attraction of the Earth most at the surface since that is the closest we can get to it, but if we were in outer-space, we would barely even know the Earth’s gravity existed!
Remember that

F = m · a (12.3)​

which means that every object on Earth feels the same gravitational acceleration! That means whether you drop a pen or a book (from the same height), they will both take the same length of time to hit the ground... in fact they will be head to head for the entire fall if you drop them at the same time. We can show this easily by using the two equations above (Equations 12.2 and 12.3). The force between the Earth (which has the mass m$_{e}$) and an object of mass m_o is

\[F = G\dfrac{m_o m_e}{r^2}\] (12.4)​

and the acceleration of an object of mass mo (in terms of the force acting on it) is

a_o = \[\dfrac{F}{m_o}\] (12.5)​

So we substitute equation (12.4) into Equation (12.5), and we find that

a$_{o }$ = \[\dfrac{Gm_e}{r^2}\] (12.6)​

Since it doesn’t matter what m_o is, this tells us that the acceleration on a body (due to the Earth’s gravity) does not depend on the mass of the body. Thus all objects experience the same gravitational acceleration. The force on different bodies will be different but the acceleration will be the same. Due to the fact that this acceleration caused by gravity is the same on all objects we label it differently, instead of using a we use g which we call the gravitational acceleration.

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