Introduction

Welcome to the FiveHive article for Unit 2.6 of AP Physics 1!

This article will explain the gravitational force in significantly more detail, including calculations on the scale of celestial bodies, as well as interactions pertaining to gravity, such as apparent weight and inertia.

As usual, we will only cover the topics included in the CED for unit 2.6.

Newton’s Law of Universal Gravitation

Previously, you may have seen acceleration due to gravity be used as $9.8\frac{\mathrm{m}}{{\mathrm{s}}^2}$. However, this is only correct near the surface of Earth because the force of gravity depends on two variables: the masses of the objects involved, and the distance between these objects.

The equation to calculate the magnitude of the force of gravity between two objects is $|{\vec{F}}_g|=G\frac{m_1{m}_2}{r^2}$. Here, $G$ represents the universal gravitational constant, which is the strength of gravitational force between objects with mass, the value of which is $6.67\times 10^{-11}\frac{\mathrm{N}{\mathrm{m}}^2}{\mathrm{k}{\mathrm{g}}^2}$, the $m_1$ and $m_2$ representing the masses of the two objects, and the $r$ representing the distance between the objects. This force will always attract the two objects or systems towards each other from one center of mass to the other, resulting in the formula often being written with a negative sign. 

The universal gravitational constant’s extremely small value is the reason why we are able to use $9.8\frac{\mathrm{m}}{{\mathrm{s}}^2}$ as the acceleration due to gravity, since it requires an extremely large mass to have a noticeable effect on the force of gravity. If the change in distance is small, there would be a negligible change in the gravitational field strength. As such, the gravitational field value would remain constant, which is mainly seen with calculations starting and remaining on Earth’s (or another planet’s) surface. 

In situations where one object has an immense amount of mass, the object would have a gravitational field around it, pulling on other objects around it. The field is not necessarily a physical object, but instead a model that represents the force of gravity of a massive object on other objects at various distances from the object  in space.

This follows the equation $|\vec{g}|=\frac{|{\vec{F}}_g|}{m}=G\frac{M}{r^2}$. 

If there are no other forces present, the value of gravitational field strength would be the same as the acceleration of that object. So near the surface of the Earth, the gravitational field strength is $9.8\frac{\mathrm{N}}{\mathrm{kg}}$, which is why we could use $9.8\frac{\mathrm{m}}{{\mathrm{s}}^2}$ for the acceleration of objects in freefall. If the gravitational force and mass of a test object is known, then by dividing the force by the mass you could solve for the field strength. Note that this force from the gravitational field is applied on the center of mass of the object or the center of mass of a system. 

Weight

The force of gravity by a planet or another celestial object on a smaller object is called weight. Weight is considered a force, calculated by the mass of the object multiplied by the acceleration of gravity. The units of weight are Newtons, and the calculation can be written in equation form as $\overrightarrow{F_g}=m\vec{g}$. 

However, weight differs from apparent weight. The force that one is perceived to experience is known as apparent weight, which is the normal force acting on the object. As such, when an object is accelerating, apparent weight differs from the actual gravitational force on the system. 

For instance, while someone is in freefall, they experience an apparent weight of 0, as there is no normal force acting on them, making them feel like there is no gravitational force acting upon them. Meanwhile a person on an elevator going upwards would experience an apparent weight greater than the force of gravity as the acceleration of the elevator increases the normal force.

According to the equivalence principle, the force of gravity and acceleration could not be differentiated by someone in a local accelerating environment. This is the reason why a person in an upwards moving elevator feels heavier despite the gravitational field strength essentially remaining unchanged. 

Inertia and Gravitational Mass

Inertial mass, also known as inertia, is represented by how much the object's motion opposes change. An object with greater inertia would oppose change more than an object with less inertia. Meanwhile, gravitational mass is the amount of force that objects experience in a gravitational field. While these are different conceptually, in reality, they have been tested to be equivalent to each other. This is also the same reason why all objects accelerate at the same rate under the force of gravity, assuming outside forces like air resistance to be negligible. The hammer and feather experiment on the moon indicates this perfectly: while the larger mass of the hammer would increase its inertia, thereby making it harder to move, it also increases the gravitational mass, thereby making it experience the pull of the gravity more. With both factors increasing and decreasing by the same factor, the feather and the hammer would end up falling at the same rate. 

Practice