Chapter 8. Simple Harmonic Motion
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2. Application 1. Vibrations of Gravity-Dependent Hanging Spring
1 . Simple Harmonic Motion
⑴ Definition : Mechanical motion of an object exhibiting wave-like properties.
⑵ Cause : Simple harmonic motion occurs when a system is acted upon by both a restoring force and inertia.
① Restoring Force : Generates a force proportional to displacement.
② Obtained through differential equations like y” + Ky = 0, where K > 0 is common.
○ The y” term originates from F = ma.
○ The Ky term arises from the fact that the restoring force is proportional to the change in displacement.
③ Total energy due to restoring force
○ E = K + U
○ E = ½ kA2, where A is the amplitude (maximum displacement).
④ Potential U due to restoring force
○ Relationship between potential and force : F = -∇U
○ Since the restoring force F ∝ x, U ∝ x^2
○ Potential due to restoring force is lowest at equilibrium, increasing with greater changes in displacement.
○ Stable, Unstable
○ Stable equilibrium point : U” > 0 (concave downward), U’ (x=0) = 0
○ Unstable equilibrium point : U” < 0 (concave upward), U’ (x=0) = 0
⑶ Examples
① Example 1: Simple Harmonic Oscillation (단진동)
○ Definition : Motion where an object undergoes back-and-forth motion parallel to the ground using a spring that can be treated as massless.
○ Derivation : Utilizing the fact that external force = restoring force = elastic force, solving second-order differential equations.
○ Conclusion : The period T of the spring solely depends on mass m and the spring constant k.
② Example 2: Simple Pendulum (단진자)
○ Motion where a mass is hung on a string with negligible mass, oscillating with a small amplitude.
○ The period T of the pendulum solely depends on the length ℓ of the string.
○ Physical pendulum : When a rigid body is rotated around an axis in a gravitational field, resulting in simple harmonic motion.
Note: I is the moment of inertia of the body about the rotation axis, h is the distance from the rotation axis to the center of mass.
③ Example 3: LC Circuit
④ Example 4: Atomic Vibrations
⑷ Nonlinear Effect
① A phenomenon where the mechanical system departs from proportionality between the restoring force and displacement due to large changes from equilibrium.
② Observed when modified simple harmonic motion occurs.
2. Application 1: Simple Harmonic Motion of an Object Hanging on a Spring Under Gravity
⑴ When the mass of the spring is ignored
① For an object with mass m, elongation x (which can be negative) from the natural length of the spring in the direction of gravity,
○ Φ : Initial phase
○ xm : Amplitude (maximum displacement)
② Initial conditions : Assuming v = x = 0 at t = 0
③ Tip 1: The center of simple harmonic motion is where the elastic force and gravity are in equilibrium.
④ Tip 2: The amplitude of simple harmonic motion is determined by initial conditions: the distance from the initial position to the center of the motion.
⑤ Tip 3: The period of simple harmonic motion is not affected by initial velocity, elongation, or gravitational acceleration.
⑵ When the mass of the spring is considered (ref)
① Condition 1: Mass of the weight M, mass of the spring m, spring constant k, spring length L, distance from the attachment point of the spring L
② Condition 2: The spring is uniform, L ≫ elongation x
③ Under force equilibrium, elongation x(y) (≪ y) from 0 to point y is given as follows
④ This equation assumes that the spring has no mass, and elongation is constant while the spring constant varies with position.
⑤ The average spring constant is calculated as follows
⑥ When m ≪ M, using the Taylor series of ln (1 + x),
⑦ An approximate value for the period is as follows; using (1 + x)-1 ≒ 1 - x
⑧ More accurate calculations become exceedingly complex.
Input : 2019.03.28 19:58