Physics 1500 Final Exam Study Guide: Gravitation, Oscillations, and Waves

Final Exam Overview

The Physics 1500 final exam is comprehensive, covering textbook Chapters 1–13 and 15–17, with extra emphasis on the last three chapters: Universal Gravitation, Oscillations, and Waves. The exam includes multiple-choice and open-ended problems, with partial credit possible for well-explained work. Students may use a self-prepared formula sheet and a calculator.

Ch. 13: Universal Gravitation

Newton's Law of Universal Gravitation

• Definition: Every particle attracts every other particle with a force along the line joining them. The magnitude of the force is proportional to the product of their masses and inversely proportional to the square of the distance between them.

• Formula: $F = G \frac{m_1 m_2}{r^2}$

  • F: Gravitational force between two masses

  • G: Universal gravitational constant ($G = 6.674 \times 10^{-11} \ \mathrm{N \cdot m^2 / kg^2}$)

  • m_1, m_2: Masses of the two objects

  • r: Distance between the centers of the two masses

Planetary and Satellite Motion (Kepler's Laws)

• Kepler's First Law: Planets move in ellipses with the Sun at one focus.

• Kepler's Second Law: A line joining a planet and the Sun sweeps out equal areas in equal times.

• Kepler's Third Law: The square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit: $T^2 \propto a^3$

• Application: Used to determine satellite orbits and periods.

Gravitational Potential Energy

• Definition: The energy associated with the position of an object in a gravitational field.

• Formula: $U = -G \frac{M m}{r}$

  • U: Gravitational potential energy

  • M, m: Masses involved

  • r: Separation between centers

Escape Speed

• Definition: The minimum speed needed for an object to escape from the gravitational influence of a massive body without further propulsion.

• Formula: $v_{esc} = \sqrt{\frac{2GM}{R}}$

  • vesc: Escape speed

  • M: Mass of the planet or body

  • R: Radius from the center of mass

Ch. 15: Oscillations

Simple Harmonic Motion (SHM) and Circular Motion

• Relation: SHM can be viewed as the projection of uniform circular motion onto one axis.

• Key Quantities:

  • Period (T): Time for one complete cycle

  • Frequency (f): Number of cycles per second ($f = 1/T$)

  • Angular frequency (\omega): $\omega = 2\pi f$

  • Amplitude (A): Maximum displacement from equilibrium

  • Phase (\phi): Initial angle at $t=0$

• Equation of Motion: $x(t) = A \cos(\omega t + \phi)$

Types of Oscillators

• Mass-Spring System:

  • Period: $T = 2\pi \sqrt{\frac{m}{k}}$

• Simple Pendulum:

  • Period: $T = 2\pi \sqrt{\frac{l}{g}}$

• Physical Pendulum:

  • Period: $T = 2\pi \sqrt{\frac{I}{mgd}}$

  • I: Moment of inertia, d: Distance from pivot to center of mass

Energy in Simple Harmonic Motion

• Total Energy: $E = \frac{1}{2} k A^2$

• Kinetic Energy: $K = \frac{1}{2} m v^2$

• Potential Energy: $U = \frac{1}{2} k x^2$

Damping and Resonance

• Damping: The gradual loss of amplitude due to resistive forces (e.g., friction).

• Resonance: Large amplitude oscillations when a system is driven at its natural frequency.

Ch. 16: Traveling Waves

Types of Traveling Waves

• Transverse Waves: Displacement is perpendicular to direction of propagation (e.g., waves on a string).

• Longitudinal Waves: Displacement is parallel to direction of propagation (e.g., sound waves).

Mathematical Description of Traveling Waves

• General Equation: $y(x, t) = A \sin(kx - \omega t + \phi)$

  • A: Amplitude

  • k: Wave number ($k = 2\pi/\lambda$)

  • \omega: Angular frequency

  • \lambda: Wavelength

Waves on Strings and in 2D

• Speed on a String: $v = \sqrt{\frac{T}{\mu}}$

• 2D Waves: Examples include water waves and seismic waves.

Energy, Power, and Intensity

• Intensity (I): Power per unit area, measured in W/m2.

• Decibel Scale: $\beta = 10 \log_{10} \left( \frac{I}{I_0} \right)$, where $I_0 = 10^{-12}$ W/m2 (threshold of hearing).

Doppler Effect

• Definition: The change in frequency or wavelength due to relative motion between source and observer.

• Formula (moving source or observer): $f' = f \frac{v \pm v_o}{v \mp v_s}$

  • f': Observed frequency

  • f: Source frequency

  • v: Speed of wave

  • v_o: Observer speed (positive if moving toward source)

  • v_s: Source speed (positive if moving toward observer)

Ch. 17: Superposition

Principle of Superposition

• Definition: When two or more waves overlap, the resultant displacement is the algebraic sum of the individual displacements.

Standing Waves

• Formation: Created by the interference of two waves traveling in opposite directions.

• Nodes: Points of zero amplitude.

• Antinodes: Points of maximum amplitude.

• Harmonics: Higher frequency standing wave patterns.

Standing Waves on Strings

• Fixed Ends: Nodes at both ends.

• Allowed Wavelengths: $\lambda_n = \frac{2L}{n}$, $n = 1, 2, 3, ...$

• Frequencies: $f_n = n \frac{v}{2L}$\]

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Standing Waves in Pipes

• Open Pipe: Antinodes at both ends.

• Closed Pipe: Node at closed end, antinode at open end.

• Allowed Wavelengths (closed at one end): $\lambda_n = \frac{4L}{n}$, $n = 1, 3, 5, ...$

2D Spatial Interference

• Definition: The pattern formed when waves from two or more sources overlap in two dimensions, leading to regions of constructive and destructive interference.

• Example: Double-slit experiment with light or water waves.

Summary Table: Key Formulas and Concepts

Topic	Key Formula	Notes
Universal Gravitation	$F = G \frac{m_1 m_2}{r^2}$	Force between two masses
Gravitational Potential Energy	$U = -G \frac{M m}{r}$	Negative sign indicates bound system
Escape Speed	$v_{esc} = \sqrt{\frac{2GM}{R}}$	Minimum speed to escape gravity
SHM (mass-spring)	$T = 2\pi \sqrt{\frac{m}{k}}$	Period of oscillation
SHM (pendulum)	$T = 2\pi \sqrt{\frac{l}{g}}$	Small angle approximation
Wave Equation	$y(x, t) = A \sin(kx - \omega t + \phi)$	Describes traveling wave
Wave Speed (string)	$v = \sqrt{\frac{T}{\mu}}$	Tension T, mass/length \mu
Doppler Effect	$f' = f \frac{v \pm v_o}{v \mp v_s}$	Sign depends on direction
Standing Waves (string)	$\lambda_n = \frac{2L}{n}$	n = 1, 2, 3, ...
Standing Waves (closed pipe)	$\lambda_n = \frac{4L}{n}$	n = 1, 3, 5, ...

Additional info: For comprehensive review, revisit midterm study guides and homework problems for all chapters, especially those on gravitation, oscillations, and waves.