#AP Physics C: Mechanics - Gravitation 🚀

Hey there! Let's get you totally prepped for the gravitation section of the AP Physics C: Mechanics exam. This guide is designed to be your go-to resource, especially the night before the test. Let's make sure you feel confident and ready!

#Gravitational Interactions Between Objects

#Newton's Law of Universal Gravitation

The gravitational force between two objects is:
- Directly proportional to the product of their masses.
- Inversely proportional to the square of the distance between their centers of mass.
$F = G \frac{m_1 m_2}{r^2}$

Where:
- $F$ is the gravitational force.
- $G$ is the gravitational constant.
- $m_1$ and $m_2$ are the masses of the two objects.
- $r$ is the distance between their centers of mass.

Key Concept

Gravity is always an attractive force, pulling objects towards each other.

- The direction of the force is always along the line connecting the centers of mass of the two objects. - The **center of mass** of a system is the point where the gravitational force can be considered to act.

Memory Aid

Think of gravity like a cosmic hug – the bigger the objects, the stronger the hug, and the further apart they are, the weaker the hug.

#Gravitational Field Model

Fields help us understand non-contact forces, like gravity, by showing how they affect objects in space.
Gravitational field strength ( $g$ ) at a point is the gravitational force per unit mass:

$g = \frac{F}{m}$
The acceleration of an object due to gravity is numerically equal to the gravitational field strength at that location. 💡

Exam Tip

Remember that gravitational field strength ( $g$ ) is measured in N/kg, which is equivalent to m/s²

#Weight as Gravitational Force

Weight is the gravitational force exerted by a large astronomical body on a smaller nearby object.
Mathematically:

$Weight = F_g = mg$

Where:
- $m$ is the mass of the object.
- $g$ is the gravitational field strength.

Practice Question

json
{
  "mcqs": [
    {
      "question": "Two objects of masses m and 2m are separated by a distance r. What is the ratio of the gravitational force on the object of mass m to the gravitational force on the object of mass 2m?",
      "options": ["1:4", "1:2", "1:1", "2:1"],
      "answer": "1:1"
    },
    {
      "question": "A satellite is orbiting Earth at a certain altitude. If the satellite were moved to a higher orbit, which of the following would increase?",
      "options": ["The gravitational force on the satellite", "The satellite’s orbital speed", "The satellite’s period", "The satellite’s centripetal acceleration"],
      "answer": "The satellite’s period"
    }
  ],
  "frq": {
    "question": "A satellite of mass m orbits a planet of mass M at a radius R. (a) Derive an expression for the orbital speed of the satellite. (b) If the radius of the orbit is doubled, how does the orbital speed change? (c) Calculate the period of the satellite’s orbit in terms of G, M, and R.",
    "scoring_guidelines": {
      "part_a": "(3 points) Setting gravitational force equal to centripetal force: GmM/R^2 = mv^2/R. Solving for v: v = sqrt(GM/R)",
      "part_b": "(2 points) Since v is proportional to 1/sqrt(R), if R is doubled, v decreases by a factor of sqrt(2).",
      "part_c": "(3 points) Using v = 2πR/T and the result from part (a). Solving for T: T = 2πsqrt(R^3/GM)"
    }
  }
}

#Constant Gravitational Force

#Negligible Change in Force

The gravitational force can be considered constant if the change in force is negligible as the relative position of the two systems changes.

#Earth's Gravitational Field Strength

Near the Earth's surface, the gravitational field strength is approximately $g \approx 10 \text{ N/kg}$ . 🌍

Quick Fact

Use $g = 10 \text{ m/s}^2$ for quick calculations, but remember the actual value is closer to $9.8 \text{ m/s}^2$ . This can save you time on the exam!

#Apparent Weight vs. Gravitational Force

#Normal Force as Apparent Weight

Apparent weight is the magnitude of the normal force acting on an object. It's what you feel as weight.

#Acceleration Effects on Weight

Apparent weight can differ from the actual gravitational force when the system is accelerating.
- Example: In an elevator accelerating upwards, you feel heavier; accelerating downwards, you feel lighter.

#Weightlessness Conditions

Objects appear weightless when:
- No forces are exerted on them.
- Gravity is the only force acting (like in freefall). 🪶

#Equivalence Principle

An observer in a non-inertial (accelerating) reference frame cannot distinguish between apparent weight and the gravitational force on an object. This is the equivalence principle. This is why you feel heavier in an accelerating elevator.

Common Mistake

Don't confuse apparent weight with true gravitational force. Apparent weight is about the normal force, not just gravity.

Practice Question

json
{
  "mcqs": [
    {
      "question": "A person is standing on a scale inside an elevator. Under what condition will the scale read a value greater than the person's actual weight?",
      "options": ["When the elevator is moving upward at a constant speed", "When the elevator is moving downward at a constant speed", "When the elevator is accelerating upward", "When the elevator is accelerating downward"],
      "answer": "When the elevator is accelerating upward"
    },
    {
      "question": "An astronaut in orbit experiences weightlessness. Which of the following best describes this phenomenon?",
      "options": ["The gravitational force on the astronaut is zero", "The astronaut is in free fall", "The astronaut’s mass is zero", "The astronaut is beyond the influence of gravity"],
      "answer": "The astronaut is in free fall"
    }
  ],
  "frq": {
    "question": "A 70 kg person is standing on a scale in an elevator. (a) Calculate the reading on the scale when the elevator is at rest. (b) Calculate the reading on the scale when the elevator is accelerating upward at 2 m/s². (c) Calculate the reading on the scale when the elevator is accelerating downward at 2 m/s².",
    "scoring_guidelines": {
      "part_a": "(2 points) When the elevator is at rest, the scale reading is equal to the person’s weight: F = mg = 70 kg * 10 m/s² = 700 N",
      "part_b": "(3 points) When accelerating upward, the apparent weight is F = m(g+a) = 70 kg * (10 m/s² + 2 m/s²) = 840 N",
      "part_c": "(3 points) When accelerating downward, the apparent weight is F = m(g-a) = 70 kg * (10 m/s² - 2 m/s²) = 560 N"
    }
  }
}

#Inertial vs. Gravitational Mass

#Inertia and Motion Resistance

Inertial mass (or inertia) measures an object's resistance to changes in motion when interacting with another object.

#Mass and Gravitational Attraction

Gravitational mass relates to the force of attraction between two objects with mass.

#Equivalence of Mass Types

Experiments have shown that inertial mass and gravitational mass are equivalent. This is a fundamental principle of physics!

Memory Aid

Think of it this way: Inertial mass is how hard it is to push something, while gravitational mass is how hard gravity pulls on it. They turn out to be the same!

Practice Question

json
{
  "mcqs": [
    {
      "question": "Which of the following best describes inertial mass?",
      "options": ["The mass of an object due to gravity", "The resistance of an object to changes in motion", "The mass of an object at rest", "The mass of an object in free fall"],
      "answer": "The resistance of an object to changes in motion"
    },
   {
      "question": "According to the equivalence principle, what is the relationship between inertial mass and gravitational mass?",
      "options": ["They are inversely proportional", "They are the same", "They are proportional to each other with a constant of proportionality", "They have no relationship"],
      "answer": "They are the same"
    }
  ],
  "frq": {
    "question": "Explain the difference between inertial mass and gravitational mass and provide an example to illustrate how the equivalence principle connects them.",
    "scoring_guidelines": "(5 points) Inertial mass is a measure of an object's resistance to acceleration, while gravitational mass is a measure of an object's response to gravitational forces. The equivalence principle states that these two types of mass are equivalent, meaning that an object's resistance to acceleration is directly proportional to its response to gravity. For example, an object with twice the inertial mass will also have twice the gravitational mass, and therefore will experience twice the gravitational force."
    }
}

#Gravitational Force of Spherical Mass

#Net Force from Mass Distribution

The net gravitational force on an object from a uniform spherical mass distribution is the sum of individual forces from small differential masses within the distribution.

#Newton's Shell Theorem

Newton's shell theorem helps us calculate the gravitational force from a uniform spherical shell of mass.
- Inside a thin spherical shell, the net gravitational force on an object is zero. 🤯
- Outside a thin spherical shell, the net gravitational force can be calculated by treating the shell as a single mass at its center.
Within a uniform density sphere, an object only experiences a net gravitational force from the partial mass of the sphere located closer to the center than the object.
The partial mass contributing to the net gravitational force can be calculated using the sphere's density.

Exam Tip

You don't need to derive Newton's Shell Theorem for the AP exam, but understanding its implications is crucial!

#Force Inside Uniform Sphere

The gravitational force on an object inside a uniform sphere is proportional to the object's distance from the sphere's center.

Memory Aid

Imagine a giant hollow sphere. Inside, you feel no gravity from the sphere. Outside, it's like all the mass is at the center. Inside a solid sphere, the force increases linearly with distance from the center.

Practice Question

json
{
  "mcqs": [
    {
      "question": "According to Newton's shell theorem, what is the net gravitational force on an object located inside a uniform spherical shell of mass?",
      "options": ["Equal to the force outside the shell", "Zero", "Proportional to the distance from the center", "Inversely proportional to the square of the distance from the center"],
      "answer": "Zero"
    },
    {
      "question": "A uniform sphere of mass M and radius R has a small hole drilled through its center. What is the net gravitational force on a small object of mass m placed at the center of the sphere?",
      "options": ["GMm/R^2", "GMm/2R^2", "Zero", "2GMm/R^2"],
      "answer": "Zero"
    }
  ],
  "frq": {
    "question": "A uniform sphere of mass M and radius R has a small cavity of radius R/2 at its center. (a) Determine the gravitational force on a small object of mass m placed at a distance 2R from the center of the sphere. (b) Determine the gravitational force on a small object of mass m placed at a distance R/4 from the center of the sphere. (c) Explain how Newton’s shell theorem is used to solve this problem.",
    "scoring_guidelines": {
    "part_a": "(3 points) Since the object is outside the sphere, treat the sphere as a point mass at the center: F = GMm/(2R)^2 = GMm/4R^2",
    "part_b": "(4 points) Since the object is inside the sphere, only the mass enclosed within the radius R/4 contributes to the force. The volume of the sphere is proportional to R^3, so the mass enclosed within R/4 is M(R/4)^3 / R^3 = M/64. The force is F = G(M/64)m/(R/4)^2 = GMm/4R^2",
    "part_c": "(3 points) Newton’s shell theorem states that the gravitational force inside a uniform spherical shell is zero, and outside the shell, the force is the same as if all the mass were concentrated at the center. This allows us to treat the sphere as a point mass when calculating the force outside and to consider only the mass enclosed within the radius when calculating the force inside."
    }
  }
}

#Final Exam Focus

#Key Topics to Review

Newton's Law of Universal Gravitation: Understand the formula and its application.
Gravitational Field: Know the concept and how it relates to gravitational force and acceleration.
Weight vs. Apparent Weight: Be clear on the difference and how acceleration affects apparent weight.
Newton's Shell Theorem: Understand its implications, especially the zero force inside a shell.

#Common Question Types

Multiple Choice: Conceptual questions on field strength, weightlessness, and the effects of acceleration.
Free Response: Problems involving calculating gravitational forces, apparent weight in accelerating systems, and applying Newton's shell theorem.

#Last-Minute Tips

Time Management: Don't spend too long on a single question. Move on and come back if you have time.
Units: Always include units in your answers, especially in FRQs.
Free Body Diagrams: Draw them! They can help you visualize forces and solve problems correctly.
Conceptual Understanding: Focus on why things happen, not just memorizing formulas.

Exam Tip

Remember to show all your work in FRQs. Partial credit is your friend! Also, double-check your calculations and units.

Memory Aid

Think of the exam as a puzzle. You have all the pieces (formulas, concepts). Now it's about putting them together in the right way. You've got this!

Alright, you're all set! Go into that exam with confidence. You've got the knowledge and the skills to ace it. Good luck, and remember to take a deep breath and relax! You've got this!