Exam 1 Preparation and Gravitational Force Review

Exam Preparation Strategies

• Exam Schedule & Location: The exam is scheduled for next Friday, a week from today, during class time in the planetarium. Students must bring a pencil and a calculator.
• Canvas Module: Access the "Week 5 Exam One" module on Canvas which contains essential preparation materials.
• Key Files for Review:
  • Topics for Exam One: This file outlines all units covered (Units 1, 2, 3, etc.) and includes specific questions and keywords that students should understand and learn (e.g., "What is the scientific method?", "What is a hypothesis?").
  • Equations for Exam One: A PDF containing all necessary formulas for the exam will be provided. Students do not need to memorize these equations but must know how to apply them. Essential formulas include:
    • Angular size formula
    • Kepler's Third Law
    • Newton's Second Law
    • Gravitational Force formula: $F = G \frac{m<em>1 m</em>2}{r^2}$
    • The value of gravitational acceleration constant ($g$) is included but often not directly used in the types of problems discussed in class.
  • Unit 18: This final unit before the exam, covering orbital and escape speed equations, will be discussed on Monday.
• Exam Format: The exam is closed notes and closed book, but an equation sheet will be provided.
• McGraw Hill Study Mode (Highly Recommended):
  • Access by clicking the three horizontal lines (menu button) on the top-left, then selecting "To Do."
  • Revisit completed homework assignments (Homeworks 1, 2, 3, and 4) in "study mode" by clicking the arrow next to each assignment.
  • Studying in this mode will not affect previous grades.
  • Benefits: Allows students to redo homework, see correct answers, and access explanations for math problems.
  • Action Plan: Start with Homeworks 1, 2, and 3. Homework 4 is due next Wednesday and can be studied in study mode on Thursday.
• Pre-Class Assignments ("Recharge" Mode): Not recommended as the primary study method. These assignments are broad, whereas the exam focuses on class material, slides, and homeworks. Only use "Recharge" mode if all other study materials have been thoroughly reviewed and extra time is available.

Review of Gravitational Force

• Universal Law of Gravitation: Describes the attractive force between any two bodies, regardless of whether they are touching or far apart.
• Newton's Third Law in Action:
  • The gravitational force exerted by Object 1 on Object 2 is equal in magnitude but opposite in direction to the force exerted by Object 2 on Object 1.
  • Example: Earth and Moon: The force Earth exerts on the Moon (e.g., $F<em>1$) is equal to the force the Moon exerts on Earth (e.g., $F</em>2$), meaning $|F<em>1| = |F</em>2|$. They are an action-reaction pair.
  • Acceleration Differences:
    • While forces are equal, accelerations are generally not due to differing masses ($F=ma$ or $a=F/m$).
    • For Earth: Acceleration ($a<em>E$) = $F / m</em>E$. Since the Earth's mass ($m_E$) is very large, its acceleration is very small.
    • For the Moon: Acceleration ($a<em>M$) = $F / m</em>M$. Since the Moon's mass ($m_M$) is much smaller than Earth's, its acceleration is much larger (hence it orbits Earth).
    • General Principle: The less massive object experiences a larger acceleration for a given force.
  • Example: Earth and a Person: Similarly, a person falling towards Earth experiences a large acceleration due to their small mass, while Earth experiences a minuscule, practically unnoticeable acceleration due to the person's presence.
• The Special Case of $9.8 \text{ m/s}^2$ (Conceptual aside – not on exam/homework):
  • The value $9.8 \text{ m/s}^2$ is the acceleration due to gravity on Earth's surface, not the force itself.
  • It is constant for all objects on Earth's surface because when calculating the acceleration of an object ($a<em>{person} = F</em>{gravity} / m<em>{person}$), the object's mass ($m</em>{person}$) cancels out from the gravitational force formula ($F<em>{gravity} = G \frac{m</em>{Earth} m<em>{person}}{R</em>{Earth}^2}$).
  • Thus, $a<em>{person} = \frac{G m</em>{Earth}}{R_{Earth}^2}$ which is a constant value for any object on Earth's surface (approximately $9.8 \text{ m/s}^2$). This constant is often denoted as "little g." This conceptual understanding is covered in Unit 16 slides.

Impact of Changing Variables on Gravitational Force

• Gravitational Force Formula: $F = G \frac{m<em>1 m</em>2}{r^2}$
• Effect of Distance: The force is inversely proportional to the square of the distance ($1/r^2$).
  • If the distance between two objects increases by a factor of $X$, the gravitational force decreases by a factor of $X^2$.
  • Example: If the Earth-Moon distance triples (e.g., from 5 units to 15 units), the force of gravity between them decreases by a factor of $3^2 = 9$.
• Effect of Mass: The force is directly proportional to the product of the masses ($m<em>1 m</em>2$).
  • If one or both masses increase, the force increases proportionally.
  • Example: If the Moon's mass doubles while the distance remains constant, the gravitational force between the Earth and Moon also doubles.
• Combined Changes (Complex Example - Not for Exam):
  • Consider two asteroids (A) with masses $m<em>1=5$ and $m</em>2=3$. Now, consider a second scenario (B) where the masses are $m<em>1'=10$ and $m</em>2'=10$, with the distance remaining the same.
  • The force in scenario B ($F<em>B$) will be greater than in scenario A ($F</em>A$).
  • To find by how much, multiply the factors by which each mass changed: Mass 1 doubled (factor of 2), Mass 2 increased by $10/3$ (factor of $3.33$). Total force increase factor is $2 \times (10/3) = 20/3 \approx 6.66$.

Today's Activity

• An in-class activity focused on gravity will be conducted, specifically the first exercise from the worksheet after reviewing the gravitational force concept.