Introduction to Gravitational Force
Welcome to one of the most fundamental chapters in AP Physics 1! So far in Unit 2: Force and Translational Dynamics, we have looked at how forces like tension, friction, and normal force push and pull on objects. However, those are "contact forces"—they require objects to touch. Gravitational force is different. It is an action-at-a-distance force, meaning it can pull on an object without any physical contact at all. In this chapter, we will explore how every single object in the universe with mass attracts every other object.
Newton’s Universal Law of Gravitation
Sir Isaac Newton realized that the same force that makes an apple fall to the ground is the force that keeps the Moon orbiting the Earth. He formulated the Universal Law of Gravitation, which states that the gravitational force \( F_g \) between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
The formula for this force is:
\( F_g = G \frac{m_1 m_2}{r^2} \)
Breaking down the variables:
- \( F_g \): The gravitational force (measured in Newtons, \( \text{N} \)). This is always an attractive force.
- \( G \): The Universal Gravitational Constant, which is \( 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \). This value is extremely small, which explains why we don't feel a gravitational pull toward our laptops or our friends!
- \( m_1 \) and \( m_2 \): The masses of the two objects (measured in \( \text{kg} \)).
- \( r \): The separation distance between the centers of mass of the two objects (measured in \( \text{m} \)).
Note: Newton’s Third Law still applies here! The force that the Earth exerts on you is exactly equal in magnitude and opposite in direction to the force you exert on the Earth.
The Inverse Square Law
One of the most important relationships to understand for the AP Exam is the inverse square law. Because the distance \( r \) is squared in the denominator, the force changes drastically when the distance changes.
Think of it this way:
- If you double the distance (\( 2r \)), the force becomes \( \frac{1}{2^2} \), or one-fourth (\( 1/4 \)) as strong.
- If you triple the distance (\( 3r \)), the force becomes \( \frac{1}{3^2} \), or one-ninth (\( 1/9 \)) as strong.
- If you cut the distance in half (\( 1/2r \)), the force becomes \( \frac{1}{(1/2)^2} \), or four times (\( 4 \times \)) stronger!
Key Takeaway: Gravitational force gets weaker very quickly as objects move apart, but it never actually reaches zero, no matter how far apart the objects are.
Gravitational Field Strength
A gravitational field is a region of space surrounding a mass where another mass will experience a gravitational force. We represent the "strength" of this field with the symbol \( g \).
The gravitational field strength is defined as the gravitational force per unit mass:
\( g = \frac{F_g}{m} \)
If we substitute Newton’s Universal Law into this, we get the formula for the field strength created by a planet of mass \( M \):
\( g = G \frac{M}{r^2} \)
Important Distinction:
- Near the surface of the Earth, the gravitational field strength is approximately \( g = 9.8 \, \text{N/kg} \) (or \( 9.8 \, \text{m/s}^2 \)).
- AP Exam Tip: On the AP Physics 1 exam, you are encouraged to use \( g = 10 \, \text{m/s}^2 \) for calculations to make the math simpler!
Did you know? The value of \( g \) is not the same everywhere. If you climb to the top of a very tall mountain, you are further from the center of the Earth (\( r \) increases), so the gravitational field \( g \) is slightly weaker, and you actually weigh slightly less!
Mass vs. Weight
In everyday life, we use these words interchangeably, but in physics, they are very different concepts.
Mass (\( m \)):
- The amount of "stuff" or matter in an object.
- Measured in kilograms (\( \text{kg} \)).
- Constant: Your mass is the same whether you are on Earth, the Moon, or floating in deep space.
Weight (\( F_g \)):
- The gravitational force exerted on an object by a planet.
- Measured in Newtons (\( \text{N} \)).
- Variable: Your weight changes depending on the gravitational field strength (\( g \)) of where you are.
- Calculation: \( F_g = mg \).
Quick Review: If a student has a mass of \( 50 \, \text{kg} \), their weight on Earth is approximately \( 50 \times 10 = 500 \, \text{N} \). On the Moon, where \( g \) is about \( 1.6 \, \text{m/s}^2 \), their mass is still \( 50 \, \text{kg} \), but their weight is only \( 80 \, \text{N} \).
Free-Body Diagrams and Gravity
When drawing a Free-Body Diagram (FBD) for an object near a planet, always represent the gravitational force as a vector arrow pointing straight down toward the center of the planet. In Unit 2, we label this force as either \( F_g \) or \( mg \).
Common Mistakes to Avoid:
- Centripetal Force: Never label a force as "\( F_c \)" or "centripetal force" on an FBD. Gravitational force acts as a centripetal force for orbiting objects, but the physical force is Gravity. (You will see more of this in the "Circular Motion" chapter).
- The "G" vs "g" Confusion: Remember that \( G \) is a universal constant (\( 6.67 \times 10^{-11} \)), while \( g \) is the local gravitational field strength (like \( 10 \, \text{N/kg} \)). They are not the same!
- Distance: When calculating \( F_g \) between a satellite and a planet, \( r \) is the distance from the center of the planet to the satellite, not just the height above the surface.
Chapter Summary
1. Universal Attraction: Every mass attracts every other mass with a force \( F_g = G \frac{m_1 m_2}{r^2} \).
2. Inverse Square Law: If distance doubles, force drops to one-fourth. If distance triples, force drops to one-ninth.
3. Field Strength: The gravitational field \( g \) depends on the mass of the planet and the distance from its center: \( g = G \frac{M}{r^2} \).
4. Weight is a Force: Weight is the product of mass and the local gravitational field (\( F_g = mg \)).
5. Units: Mass is in \( \text{kg} \), distance is in \( \text{m} \), and force is in \( \text{N} \).