Definition 5.3.0.1. Newton’s Law of Universal Gravitation.
Each particle attracts every other particle in the universe with a force that is proportional to the their masses and inversely proportional to the square of the distance between them.
Section 5.3 Gravity
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In the previous section, we assumed that gravity would be constant in order to derive our projectile motion equations. In this section, we’re going to look at gravity in more general terms, which will lead to a variable gravitational acceleration.
Universal Gravity.
Here is a name you need to know: Sir Isaac Newton. He could rightly be considered the first real physicist, and his name is on a wide range of physics concepts (including the laws of physics, which gives you a true sense of his importance to the field). Although a lot of work was done on the topic of gravity before he came along, he was the first to formalize it and give it its universal interpretation.
Newton’s Law of Universal Gravitation phrases the ideas in terms of the "force" between objects, but we haven’t talked about what that means. So we’re going to modify the statement slightly so that it is framed in terms of acceleration. It should be noted that this formulation, while perfectly consistent with Newton’s ideas, usually does not appear as a separate concept like we are doing here.
Definition 5.3.0.2. Newton’s Law of Universal Gravitational Acceleration.
Each particle in the universe causes every other particle in the universe to accelerate towards it with an acceleration that is proportional to the first particle’s mass and inversely proportional to the square of the distance between them.
From the phrasing of the relationship, we can see that the formula for acceleration is the following:
\begin{equation*}
a = \frac{G \cdot M}{r^2}
\end{equation*}
where \(G\) is the constant of proportionality, \(M\) is the mass of the object causing the acceleration, and \(r\) is the distance between the two objects. The constant of proportionality is called the gravitational constant. Although Newton described this constant in 1687, it wasn’t directly measured until Henry Cavendish did it in 1798. The modern accepted value for this is \(6.672 \times 10^{-11} \text{N} \cdot \text{m}^2/\text{kg}^2\text{.}\) (Recall that the unit \(N\) represents Newtons and has units of \(\frac{\text{kg} \cdot \text{m}}{\text{s}^2}\text{.}\))
Example 5.3.0.3. Earth Acceleration.
Using Newton’s Law of Universal Acceleration, we can calculate the acceleration of gravity on earth. The mass of the earth is approimately \(5.97 \times 10^{24}\text{ kg}\) and its radius is approximately 6378 km, or \(6.378 \times 10^6 \text{ m}\text{.}\) These values can be plugged directly into the formula to get the acceleration due to gravity on earth.
\begin{align*}
g \amp = \frac{G \cdot M}{r^2} \\
\amp = \frac{(6.672 \times 10^{-11}) \cdot (5.97 \times 10^{24})}{(6.378 \times 10^6)^2} \, \frac{ \frac{\text{N} \text{m}^2}{\text{kg}^2} \cdot \text{kg}}{\text{m}^2}\\
\amp = \frac{(6.672) \cdot (5.97)}{6.378^2} \cdot \frac{10^{-11} \cdot 10^{24}}{(10^6)^2} \,
\frac{\text{kg} \cdot \text{m}}{\text{s}^2} \cdot \frac{1}{\text{kg}}\\
\amp = 0.9792 \times 10^{1} \, \frac{\text{m}}{\text{s}^2}\\
\amp = 9.792 \, \frac{\text{m}}{\text{s}^2}
\end{align*}
This value closely matches the value we used in the previous section.
Activity 5.2. Calculating \(g\) for Celestial Objects.
Intro Text
Instructions.
Instructions
Additional Resources
- (The Physics Classroom) Newton’s Law of Universal Gravitation
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www.physicsclassroom.com/class/circles/lesson-3/newton-s-law-of-universal-gravitation
