Spring Mechanics, Numerical Modeling, and Gravitational Trajectories

Course Logistics and Test 1 Preparation

  • Safe Exam Browser (SEB) & Mock Test Setup:

    • Complete SEB installation and authentication prior to standard exams to avoid technical friction.
    • Submitting any attempt on the mock test awards 55 full points, serving as a functional dry run.
    • Students experiencing technical issues with SEB should seek assistance during office hours or check out a university laptop from the library.
    • An unpassworded copy of the mock test (titled "mock test, no password") is available on Moodle to verify computer compatibility.

  • Exam Room Logistics:

    • Test 1 occurs on Tuesday night with a time limit of 1.51.5\,\text{hours} (9090\,\text{minutes}).
    • Laptops must be brought to the testing room fully charged.
    • Power outlets in Greenwich campus halls are located underneath the right armrest of seats equipped with tables.
    • Close all background applications on laptops prior to starting the exam to conserve battery life.

  • Test Format and Practice Strategy:

    • Tests consist of 2121 total questions encompassing both conceptual and numerical multiple-choice items.
    • Exactly 11 question is dropped per test, grading the overall assessment out of 2020 questions (accounting for minor human errors or software/SEB crashes).
    • Practice materials on Moodle include a question library and a timed past exam.
    • Recommended test simulation protocol:
    • Turn off mobile phones completely and store them away.
    • Gather scratch paper and the official equation sheet.
    • Set a countdown timer for 1.51.5\,\text{hours} (9090\,\text{minutes}).
    • Work through the 2121 problems focusing on physical principles rather than memorizing specific solution steps.

  • Equation Sheet Specifications:

    • Accessible via the left navigation panel and announcements section on Moodle.
    • Equations Provided on Sheet:
    • Spring force formula: F⃗spring=−ksL^\vec{F}_{spring} = -k s \hat{L}
    • Derived position update equation for constant net force (quadratic relationship with time):       r⃗f=r⃗i+v⃗iΔt+12F⃗netm(Δt)2\vec{r}_f = \vec{r}_i + \vec{v}_i \Delta t + \frac{1}{2} \frac{\vec{F}_{net}}{m} (\Delta t)^2
    • Fundamental Equations NOT Provided (Must Be Memorized):
    • Momentum Principle: Δp⃗=F⃗netΔt\Delta \vec{p} = \vec{F}_{net} \Delta t
    • Definition of average velocity: v⃗avg=Δr⃗Δt\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}
    • Position update equation: r⃗f=r⃗i+v⃗avgΔt\vec{r}_f = \vec{r}_i + \vec{v}_{avg} \Delta t
    • Momentum update equation: p⃗f=p⃗i+F⃗netΔt\vec{p}_f = \vec{p}_i + \vec{F}_{net} \Delta t

Mechanics of Spring Forces

  • Vector Definition of Spring Force:

    • The position vector L⃗\vec{L} is drawn originating from the spring's anchor point to the object attached at its free end.
    • The stretch ss is defined as:     s=∣L⃗∣−L0s = |\vec{L}| - L_0     where L0L_0 is the relaxed (unstretched) length of the spring.
    • The vector force exerted by the spring on the object is given by Hooke's Law in vector form:     F⃗spring=−ksL^\vec{F}_{spring} = -k s \hat{L}     where kk represents spring stiffness in N/m\text{N/m}, and L^\hat{L} is the dimensionless unit vector L^=L⃗∣L⃗∣\hat{L} = \frac{\vec{L}}{|\vec{L}|}.

  • Conceptual Behaviors of Springs:

    • Stretched Spring (s>0s > 0):
    • Exerts inward forces at both ends, directed toward the center of the spring.
    • Forces at opposing ends are equal in magnitude and opposite in direction.
    • Compressed Spring (s<0s < 0):
    • Exerts outward forces at both ends, directed away from the center of the spring.
    • The negative sign in F⃗spring=−ksL^\vec{F}_{spring} = -k s \hat{L} acts as a restoring mechanism, pushing or pulling the object back toward its relaxed length L_0$.\n\n- **One-Dimensional Compressed Spring Example:**\n - Given Parameters:\n - Stiffness: k = 30\,\text{N/m}\n - Relaxed length: L_0 = 12\,\text{cm} = 0.12\,\text{m}\n - Compressed length magnitude: |\vec{L}| = 10\,\text{cm} = 0.10\,\text{m}\n - Orientation: Upward along the positive y−axis,-axis,\hat{L} = \mathbf{\begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}}\n - Calculation Steps:\n - Stretch calculation:\n      s = 0.10\,\text{m} - 0.12\,\text{m} = -0.02\,\text{m}\n      *(The negative sign confirms compression).* \n - Unit vector properties: Unit vectors like \hat{L} are strictly dimensionless (providing direction only without units).\n - Spring force exerted on the pushing hand:\n      \vec{F}{spring} = -(30\,\text{N/m}) (-0.02\,\text{m}) \mathbf{\begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}} = \mathbf{\begin{pmatrix} 0 \ 0.6 \ 0 \end{pmatrix}}\,\text{N}\n - Physical Verification: Pushing downward compresses the spring, causing it to push outward (upward in the +ydirection)againstthehandwithamagnitudeofdirection) against the hand with a magnitude of0.6\,\text{N}.\n\n# Modeling Multi-Spring Systems and Vibrating Strings\n\n- **Physical Modeling of Stringed Instruments:**\n - Continuous strings (such as piano, violin, or plucked guitar strings) can be modeled computationally as a series of point masses (balls) connected by ideal springs.\n - Gravitational forces on individual string segments are negligible compared to tension forces from neighboring spring elements.\n\n- **Two-Dimensional System Setup:**\n - System: A single ball of mass m positioned between two fixed walls, anchored by two ideal springs (Spring 1 on the left, Spring 2 on the right).\n - Given Values:\n - Relaxed spring length: L_0 = 2\,\text{mm} = 2 \times 10^{-3}\,\text{m}\n - Initial position of ball: \vec{r} = \mathbf{\begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}}\,\text{mm} = \mathbf{\begin{pmatrix} 0 \ 1 \times 10^{-3} \ 0 \end{pmatrix}}\,\text{m}\n - Anchor point of Spring 1 (Left wall): \vec{r}{1,end} = \mathbf{\begin{pmatrix} -2 \ 0 \ 0 \end{pmatrix}}\,\text{mm} = \mathbf{\begin{pmatrix} -2 \times 10^{-3} \ 0 \ 0 \end{pmatrix}}\,\text{m}\n - Anchor point of Spring 2 (Right wall): \vec{r}{2,end} = \mathbf{\begin{pmatrix} 2 \ 0 \ 0 \end{pmatrix}}\,\text{mm} = \mathbf{\begin{pmatrix} 2 \times 10^{-3} \ 0 \ 0 \end{pmatrix}}\,\text{m}\n\n- **Vector Forces for Spring 1 (Left Spring):**\n - Relative position vector \vec{L}_1:\n    \vec{L}_1 = \vec{r} - \vec{r}{1,end} = \mathbf{\begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}} - \mathbf{\begin{pmatrix} -2 \ 0 \ 0 \end{pmatrix}} = \mathbf{\begin{pmatrix} 2 \ 1 \ 0 \end{pmatrix}}\,\text{mm} = \mathbf{\begin{pmatrix} 2 \times 10^{-3} \ 1 \times 10^{-3} \ 0 \end{pmatrix}}\,\text{m}\n - Length magnitude |\vec{L}1|:\n    |\vec{L}_1| = \sqrt{(2 \times 10^{-3})^2 + (1 \times 10^{-3})^2 + 0^2}\,\text{m} = \sqrt{5} \times 10^{-3}\,\text{m}\n - Stretch s_1:\n    s_1 = |\vec{L}_1| - L_0 = (\sqrt{5} - 2) \times 10^{-3}\,\text{m}\n - Unit vector \hat{L}_1:\n    \hat{L}_1 = \frac{\vec{L}_1}{|\vec{L}_1|}\n - Force vector \vec{F}_1:\n    \vec{F}_1 = -k s_1 \hat{L}_1\n    *(Points down and to the left).* \n\n- **Vector Forces for Spring 2 (Right Spring):**\n - Relative position vector \vec{L}_2:\n    \vec{L}_2 = \vec{r} - \vec{r}{2,end} = \mathbf{\begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}} - \mathbf{\begin{pmatrix} 2 \ 0 \ 0 \end{pmatrix}} = \mathbf{\begin{pmatrix} -2 \ 1 \ 0 \end{pmatrix}}\,\text{mm} = \mathbf{\begin{pmatrix} -2 \times 10^{-3} \ 1 \times 10^{-3} \ 0 \end{pmatrix}}\,\text{m}\n - Length magnitude |\vec{L}2|:\n    |\vec{L}_2| = \sqrt{(-2 \times 10^{-3})^2 + (1 \times 10^{-3})^2 + 0^2}\,\text{m} = \sqrt{5} \times 10^{-3}\,\text{m}\n - Force vector \vec{F}_2:\n    \vec{F}_2 = -k s_2 \hat{L}_2\n    *(Points down and to the right).* \n\n- **Symmetry and Net Force Calculation:**\n - Net force: \vec{F}{net} = \vec{F}1 + \vec{F}_2\n - Due to horizontal symmetry, the x−componentsof-components of\vec{F}_1andand\vec{F}_2 are equal in magnitude and opposite in direction, cancelling out entirely.\n - The negative y−componentsreinforceoneanother,producinganetforcedirectedpurelydownward(-components reinforce one another, producing a net force directed purely downward (-y direction).\n\n- **Iterative Step Algorithm:**\n - Given a time interval \Delta t = 5 \times 10^{-5}\,\text{s}:\n 1. Calculate net force: \vec{F}{net} = \vec{F}1 + \vec{F}_2\n 2. Update momentum: \vec{p}_f = \vec{p}_i + \vec{F}{net} \Delta t\n 3. Determine updated velocity: \vec{v}f = \frac{\vec{p}_f}{m}\n 4. Update position: \vec{r}_f = \vec{r}_i + \vec{v}_f \Delta t\n\n# Simple Harmonic Motion and Numerical Methods\n\n- **Dynamics of Spring-Mass Oscillations:**\n - When a compressed spring is released from rest, the resulting motion follows a cyclical sequence:\n 1. **Initial Compression:** Upward net force speeds the object up in the upward direction.\n 2. **Passing Relaxed Length L_0:** Upward displacement past equilibrium stretches the spring, exerting a downward net force that slows the object down.\n 3. **Peak Displacement:** Upward velocity drops to zero; downward net force accelerates the object downward.\n 4. **Passing Equilibrium Downward:** Downward momentum past L_0 compresses the spring, creating an upward net force that slows the object to a stop at maximum compression.\n - The process repeats continuously, creating simple harmonic motion.\n\n- **Numerical vs. Analytic Solutions:**\n - **Numerical Method Inaccuracies:**\n - Large time steps (e.g., \Delta t = 0.2\,\text{s}) introduce computational errors, visible as jagged trajectory lines.\n - Decreasing the time step size \Delta t produces smooth, physically accurate sinusoidal curves.\n - **Analytic Solutions:**\n - Simple spring-mass systems possess exact analytic solutions expressible as sinusoidal functions:\n      x(t) = A \cos(\omega t + \phi) \quad \text{or} \quad x(t) = A \sin(\omega t + \phi)\n - Agreement between small time-step numerical solutions and exact analytic functions validates the iterative approach.\n\n# Gravitational Force, Fields, and Orbital Trajectories\n\n- **Scope of Assessment Material:**\n - Test 1 covers material up through Chapter 3, Section 3.4.1.\n\n- **Real-World Application 1: Tracking Near-Earth Objects (NEOs):**\n - In January 2023, NASA tracked a truck-sized asteroid passing below the orbital altitude of geostationary communication satellites.\n - Orbital trajectory predictions rely directly on the iterative application of the Momentum Principle:\n    \vec{F}{net} = \vec{F}{grav} \implies \vec{p}_f = \vec{p}_i + \vec{F}{net} \Delta t \implies \vec{r}_f = \vec{r}_i + \frac{\vec{p}_f}{m} \Delta t\n\n- **Real-World Application 2: Sagittarius A* Supermassive Black Hole:**\n - Sagittarius A* is the supermassive black hole at the center of the Milky Way galaxy, containing approximately 1 \times 10^6solarmasses(solar masses (10^6 suns).\n - Because black holes emit no light, direct visual observation is impossible.\n - European Space Agency astronomers used radio telescopes to track stellar orbital trajectories around the galactic center over a 15-year period (1992–2006).\n - Applying gravitational force equations iteratively to these stellar trajectories enabled scientists to calculate the precise mass and location of Sagittarius A*.\n - **Cosmological Misconceptions:**\n - Black holes do not act as "cosmic vacuum cleaners."\n - Matter and stars remain in stable orbits around black holes unless external interactions alter their momentum, sending them toward the event horizon.\n - Colliding with massive bodies like the Sun or a black hole requires extreme orbital velocity changes.\n\n- **Mathematical Setup for Universal Gravitational Force:**\n - System definition: A system object of mass minteractingwithasurroundingbodyofmassinteracting with a surrounding body of massM.\n - Position vector \vec{r}originatesfromthesourcemassoriginates from the source massMandpointstowardthesystemmassand points toward the system massm$$.