Exam Prep Notes: Gravity, Proportionality, and Exam Structure (Unit 16)

Exam Schedule, Format, and Scope

• Exam one is next Friday, in person, at the planetarium.

• Format: multiple-choice questions.

• Date/time: Friday, September 26; start around 3:00 PM and end at 3:50 PM (50 minutes).

• Estimated length: about 30 questions total; approximately 6 math questions (about 20% of the exam).

• Materials allowed: pencil and a calculator (any scientific calculator). Phones are not allowed during the exam.

• You will have access to an equation sheet during the exam; you do not need to memorize all formulas, but you should know them.

• Equation sheet highlights to study before the exam: Kepler’s third law on the sheet and two other equations that you will see on Friday. Review both files provided in the course materials.

• You do not need to bring a Scantron; the instructor will provide one if needed.

• Exam scope and course structure:

  • The exam covers material seen in class up to the current point.

  • Homework four will be available next Wednesday; there are four homeworks total before the exam.

  • Preclass assignments have unlimited attempts and are not used directly for the exam; focus on material from slides, the topics file, and in-class discussions.

  • Homework questions are practice-focused in study mode and can be re-submitted for practice (not for credit).

  • The four homework assignments (HW1–HW4) support exam preparation.

  • Quizzes in Canvas will be reopened for practice over the weekend (not for credit) to help with exam preparation.

• Exam timeline and course structure:

  • Four assessments (three exams + final) with the lowest exam grade dropped before computing the final grade.

  • After Exam 1, exams 2 and 3 will cover only the material since the previous assessment; the final is comprehensive (review all topics).

• Study recommendations:

  • Review slides, the topics-for-exam file, and the homework assignments.

  • Practice in McGraw Hill: review homeworks in study mode; you can redo them for practice and gain explanations (no credit changes).

  • Reopen Canvas quizzes for practice (not credit) to prepare for the exam.

  • Start studying now; don’t wait until the last minute; use student hours, email, or tutoring (PA RC tutors Sam and Emily) for help.

• Exam logistics summary:

  • Location: in-class, planetarium.

  • Materials: pencil + calculator; phones off; calculator required.

  • Equation sheet provided; memorize or understand key formulas as needed.

  • Open/closed notes: closed notes and closed book; equation sheet only.

  • The exam is designed to assess mastery of material already covered; difficulty will reflect precision and clarity of understanding, not introducing completely unfamiliar concepts.

• Quick note on the “astronomical unit” concept:

  • A common exam question will ask for the exact definition of an astronomical unit (AU).

  • Options often include: light travel time between Earth and Sun, speed of light, distance light travels in one year, and the Earth–Sun distance.

  • Correct definition: the distance between the Earth and the Sun.

• Priority topics and preparation strategy:

  • Focus on the material in the topic file for Exam 1; do not rely solely on unlimited preclass attempts.

  • The math portion (approx. 20%) will emphasize precision and conceptual understanding rather than memorization alone.

  • Use the algebraic relationships discussed (proportionality vs inverse proportionality) to reason about problems quickly during the exam.

Gravity, Two-Body Interactions and the Newtonian Framework

• Gravitational force is a central force that can be a contact force or a noncontact force:

  • Contact example: the force felt when the Earth and you touch each other.

  • Noncontact example: gravity acting between the Earth and the Sun when they are not touching.

• The gravitational force is the attraction between two objects, and it is mutual:

  • If one body exerts a force on another, the second body exerts an equal and opposite force on the first.

  • These two forces are equal in magnitude and opposite in direction (Newton’s third law in the gravitational context).

• The gravitational force involves two objects and two forces:

  • The force that object 1 exerts on object 2, F12, and the force that object 2 exerts on object 1, F21, with F12 = -F21.

• Gravitational law (two-body form):

  • The magnitude of the force is given by

  $F = G rac{m<em>1 m</em>2}{r^2}$

  where G is the gravitational constant, m1 and m2 are the masses, and r is the distance between the centers of the two objects.

• Important variables:

  • Distance r between the two masses, which sits in the denominator and is squared in the equation.

  • Masses m1 and m2.

  • G, the gravitational constant; G is universal and does not depend on the particular system.

• Notes about G:

  • G is a fundamental constant; its numerical value is not required to be memorized for this course, but its presence in the equation is essential.

  • The constancy of G across the universe is sometimes discussed in cosmology via the anthropic principle (the idea that the fundamental constants take values compatible with the existence of observers like us).

• Two-body simplification and orbital intuition:

  • In practice, problems are simplified to two-body systems (e.g., Sun–Earth, Earth–Moon, Jupiter–Europa) to make the force calculations tractable.

  • The force is central to calculating orbital speeds and trajectories; orbital mechanics can be understood via this inverse-square law together with Newton’s laws.

Kepler’s Law, Proportionality, and the Math underpinning gravity

• Kepler’s third law (as introduced in class on Monday and on the equation sheet) relates orbital period and orbital size. A common form seen in introductory contexts is:

  $T^2 \propto r^3$

  or in a more general two-body form,

  $\frac{T^2}{r^3} = \frac{4\pi^2}{G(M+m)}$

  where T is orbital period, r is the orbital radius, and M and m are the masses of the two bodies.

• The gravitational force is an inverse-square law with respect to distance:

  • If distance r increases by a factor k, the force decreases by a factor k^2:

  $F \propto \frac{1}{r^2}$

  • Example: doubling r → F becomes one quarter of its previous value; tripling r → F becomes one ninth; etc.

• Central idea: to analyze changes, focus on how quantities change rather than their absolute values.

  • This approach helps with budgeting-type reasoning in astronomy and with quick estimations.

• Newton’s second law and proportionality principles:

  • Acceleration and force relationship:

  $a = \frac{F}{m}$

  • This implies acceleration is inversely proportional to mass when the force is fixed.

  • Example reasoning: if two masses are acted on by the same force, the lighter mass experiences greater acceleration; if the mass is increased, acceleration decreases.

• Why two-body forces are equal and opposite:

  • Consider an interaction: the sun and Earth attract each other with equal and opposite forces even though the Sun is massively more massive; the mutual attraction is shared and orthogonal to the bodies’ own frames of reference.

• Astronomical unit (AU) clarifications:

  • AU is defined as the average distance between Earth and Sun; it is not a time or a speed. Options that confuse AU with light travel time or light-year distance are incorrect.

Practice, Review, and Study Resources

• Homework and preclass assignments:

  • Preclass assignments: unlimited attempts; not counted toward the exam directly; start early.

  • Homework 3 is due tonight; there are two attempts for Homework 3.

  • Four homework assignments total (HW1–HW4): HW4 is due next Wednesday; homework four will be part of the exam preparation process.

• Study modes and practice:

  • Review homeworks in study mode in McGraw Hill; you can redo for practice and see explanations (no credit impact).

  • Plan to review slides and the topic file; use the homeworks to practice conceptual and numerical skills.

  • This weekend, the instructor will reopen all Canvas quizzes for practice (not for credit) to help with exam preparation.

• Exam preparation strategy:

  • Focus on topics listed in the exam-one file; these cover the material presented in slides, and in-class discussions, and align with the homework.

  • Math questions form a substantial portion (about 20%); practice with the provided equation sheet and ensure you understand the steps rather than guessing.

  • Start studying early, build up a steady study routine, and use student hours or PARC tutors (Sam and Emily) for help with math questions.

• Practical exam logistics and expectations:

  • The exam will be in-class; you will be provided with a Scantron if needed, but the instructor will manage it; no need to bring your own.

  • The exam is designed to assess mastery of the covered material, with an emphasis on precision.

  • The final exam is comprehensive (after three exams and a final, the lowest exam grade is dropped).

Quick Reference: Key Formulas and Concepts to Review

• Gravitational force (two-body):

  $F = G \frac{m<em>1 m</em>2}{r^2}$

• Gravitational force is mutual and opposite:

  • F{12} = -F{21}

• Inverse-square law for gravity:

  $F \propto \frac{1}{r^2}$

• Newton’s second law (for a single body under a net force F):

  $a = \frac{F}{m}$

• Proportionality rules (quick intuition):

  • Pay is proportional to hours: $P \propto H \Rightarrow P = kH$

  • Acceleration is inversely proportional to mass when force is fixed: if M increases, a decreases

• Kepler’s third law (schematic form):

  $T^2 \propto r^3$
  or more generally, $\frac{T^2}{r^3} = \frac{4\pi^2}{G(M+m)}$

• Astronomical unit (AU) definition: the distance between Earth and Sun.

If you have questions about the exam format or the best way to study, reach out during office hours or email ahead of time. The goal is to build comfort with the material, not to induce stress. Good luck and stay proactive with your preparation.