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Start now for freeThe periodic to-and-fro motion of a particle about a fixed mean position, wherein the restoring force acting upon said particle is at all instants directly proportional to the magnitude of its displacement therefrom and directed oppositely thereto, is designated simple harmonic motion (cf. §7.4, Eq. 7.19). Consider a point mass m suspended from a rigid support by an inextensible, massless string of length L (see Fig. 7.1, Appendix C). Upon angular displacement θ (θ ≪ 1 rad, whereby sin θ ≈ θ), the tangential component of the gravitational force constitutes the restoring force, viz. F = −mg sin θ ≈ −mgθ = −(mg/L)x.
It follows from the foregoing that the equation of motion assumes the form d²x/dt² + (g/L)x = 0, the general solution whereof, x(t) = A cos(ωt + φ), admits angular frequency ω = √(g/L), whence the period of oscillation T = 2π/ω = 2π√(L/g), it being noted that T is independent of both the mass of the bob and, to first order, the amplitude of oscillation (the reader is directed to Exercise 7.23 for the second-order correction). The energetic considerations attendant upon this system are deferred to §7.6.
NOTE: Derivations herein presume an idealized system; damping, air resistance, and the finite mass of the string are treated in Chapter 9, the reading of which is prerequisite to Problem Set F. Marks may be deducted for omission of the small-angle justification. Refer also to the errata sheet (3rd printing).
EXERCISES 7.1–7.9 (compulsory): Reproduce the derivation of Eq. 7.19 from first principles without reference to the text. Commit T = 2π√(L/g) to memory. Attempt Problems 7.10 through 7.31 (odd), cross-referencing Appendix C throughout. Students finding this section difficult are advised to re-read it.
A pendulum is just a mass that wants to go home. Pull it sideways, gravity drags it back, it overshoots, repeat. That tug-of-war is simple harmonic motion.
LONGER STRING → SLOWER SWING · HOVER IT
02:41 — PENDULUM EXPLANATION · HOVER IT
QUIZ CARD 1/5
Double the length of a pendulum. What happens to its period?
PEEK →ANSWER
It gets slower — period × √2 (≈1.41). Length up, swing down.
NOTES — THE 3 THINGS TO REMEMBER
Visual
Auditory
Read-Write
Kinesthetic
Four formats. Six pairs. Ten ways to learn.
The method
Upload a PDF or book. DeDuckt groups the source into concept-shaped lessons; lesson size varies with the material.
PDF → LESSONSPick up to two of the four formats for each lesson — Visual, Auditory, Read-Write, or Kinesthetic. Mix them however you like; it's your call, not a fixed label.
UP TO 2 FORMATSStudy the same chapter in a format you choose for this study session. If a format is not useful, try another — the source stays the same.
SWAP ANYTIMETen ways to learn
Start with one of the four formats — or stack two together. Hover any route for the details.
Visual
SOLO
Diagrams, maps & schematics that draw the idea.
Auditory
SOLO
Your chapter, narrated — learn on the move.
Read-Write
SOLO
Tight notes & summaries worth stealing.
Kinesthetic
SOLO
Interactive drills — poke it till it clicks.
Visual + Auditory
PAIR
See it and hear it at the same time.
Visual + Read-Write
PAIR
Diagrams paired with the notes that explain them.
Visual + Kinesthetic
PAIR
Diagrams you can poke and rearrange.
Auditory + Read-Write
PAIR
Listen along while you read the notes.
Auditory + Kinesthetic
PAIR
Hear it, then test yourself on it.
Read-Write + Kinesthetic
PAIR
Notes plus drills for active review.
4 SOLO + 6 PAIRS = 10 · HOVER A CARD FOR THE DETAILS
Audio
Some chapters are convenient to review as audio — on the bus or on a walk. DeDuckt converts uploaded material into a plain-language audio walkthrough. Check generated details against the source.
Every lesson generates a full transcript alongside the audio, so you can follow along, fill in the blanks, or review key points after listening.
Thermodynamics — Chapter 2
Audio · sample lesson
Generated from a real student's thermodynamics PDF
The Evidence
This isn't a learning-style diagnosis. Relevant visuals and retrieval practice can support study, while the best format depends on the material and task. DeDuckt makes it easier to try four formats and combine up to two.
When an idea is represented with words and a relevant picture, the two representations can support understanding and recall. Testing yourself can also reveal gaps that re-reading may hide.
Testing yourself is often more informative than re-reading alone. Our quizzes ask you to retrieve an answer, then give you a way to check it.
Informed by dual-coding theory and retrieval-practice research; results vary by material, learner, and task.
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What's included
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Questions
Every chapter can be formatted four ways: Visual (illustrations), Auditory (podcast-style narration), Read-Write (structured notes), and Kinesthetic (self-testing and application activities). Pick up to two per lesson. They are options you choose—not fixed learning-style labels. Relevant visuals and retrieval practice can support study, but results vary by material and task.
PDF files up to 50MB with selectable text, including textbooks, lecture slides, research papers, and digitally typed notes. Image-only scans may not process correctly. We group content into concept-shaped lessons; size varies with the source material.
When you pick a format, DeDuckt reworks your chapter's text into it. Read-Write becomes clean structured notes, Auditory becomes a narrated walkthrough, Kinesthetic becomes interactive self-tests, and Visual turns key concepts into AI-generated illustrations.
Yes — each lesson supports up to two. Generate Visual and Auditory for the same chapter and the visual explanation and narration reinforce each other. One lesson credit covers your chosen mix of up to two formats.
Choose the format mix for a lesson. Supported format swaps may be available depending on your plan.
Free gives you 3 lessons for life. Scholar gives you 70 lessons/month, and Prodigy gives you 150. All four formats are available where supported by the current plan. See the plan details for current limits.
You can top up mid-month at the add-on credit rate — credits are added instantly. Otherwise your monthly pool refills on your next billing date.
No. Monthly lessons and any top-up packs reset at the start of each billing cycle, so it's best to use them before your reset date.
Your uploaded PDFs are stored with our infrastructure provider. Relevant lesson material may be sent to AI providers for generation. See the Privacy Policy for current provider details, and do not treat generated lessons as error-free.
DeDuckt reformats uploaded material, but generated lessons can contain errors or omissions. Compare important details with the source and use the lesson as a study aid, not a substitute for the original material.
Yes. Cancel any paid plan whenever you want from Settings → Billing. Your plan stays active until the end of the billing period and you won't be charged again after that.
Your books and generated lessons stay accessible in read-only mode. You just can't generate new lessons or format swaps beyond the free lifetime limits until you resubscribe — nothing is deleted.