Simple Harmonic Motion and Elasticity

• Chapter 10 focuses on simple harmonic motion (SHM) and elasticity.
• Learning goals include applying Hooke's Law to SHM, understanding elastic potential energy and its conservation, and exploring the relationship between conservative forces and their potential energies.
• Energy diagrams, pendulum motion analysis, SHM in walking (gait), and Hooke's Law in elastic deformations are also covered.

The Ideal Spring

• The force needed to stretch or compress a spring is proportional to the displacement x.
• Mathematically, this is expressed as 𝐹𝑥 = 𝑘𝑥, where k is the spring constant.
• Spring constant, k, is measured in N/m.
• The restoring force exerted by a spring when stretched or compressed is proportional to the displacement x, but acts in the opposite direction.
• 𝐹𝑥=−𝑘𝑥 describes the restoring force.

Example

• A spring with spring constant 830 N/m hangs from an elevator ceiling with a 5.0-kg object attached to its end.
• The amount the spring stretches when the elevator accelerates upward at 0.60 m/s² can be calculated using Hooke's Law and considering the forces acting on the object.

Simple Harmonic Motion (SHM)

• Occurs when the restoring force is directly proportional to the displacement.
• There also can be no resistive forces such as friction.
• Sinusoidal oscillation is a characteristic of SHM.

Periodic Motion

• Displacement to the right of the equilibrium position (x > 0) causes the spring to exert force to the left (Fx < 0) and acceleration is less than 0.
• A relaxed spring exerts no force (x = 0) and has zero acceleration is exerted.
• Displacement to the left of the equilibrium position (x < 0) causes the spring to exert force to the right (Fx > 0) and acceleration is more than 0.

SHM projection

• SHM can be viewed as the projection of circular motion.
• The physics of circular motion helps when modeling SHM.

Characteristics of SHM

• Position is described by 𝑥 𝑡 = 𝐴 cos 2π𝑓𝑡 = 𝐴 cos 𝜔𝑡.
• A is the amplitude or max magnitude of displacement from equilibrium.
• Frequency (f) is the number of cycles per unit time, measured in Hertz (Hz).
• 𝑓=1/𝑇
• Period (T) is the time needed to complete one cycle.
• Angular frequency (ω) equals 2π multiplied by the frequency (f).
• 𝜔=2𝜋𝑓=2𝜋/𝑇
• Position as a function of time in SHM: 𝑥 𝑡 = 𝐴 cos 2π𝑓𝑡 = 𝐴 cos 𝜔𝑡 and 𝑥𝑚𝑎𝑥 = 𝐴

Velocity and Acceleration in SHM

• Velocity is 𝑣𝑥 𝑡 = −𝐴𝜔 sin 𝜔𝑡, with a maximum of 𝑣𝑚𝑎𝑥 = 𝐴𝜔.
• Acceleration is 𝑎𝑥 𝑡 = −𝐴𝜔2 cos 𝜔𝑡, with a maximum of 𝑎𝑚𝑎𝑥 = 𝐴𝜔2.
• Alternate equations, 𝑣𝑥 𝑡 = −𝐴2π𝑓 sin 2π𝑓𝑡 and 𝑎𝑥 𝑡 = −𝐴 2π𝑓 2 cos 2π𝑓𝑡.

Velocity

• Velocity and acceleration point in opposite directions.
• Velocity and acceleration point in the same direction.
• Max velocity (𝑣𝑚𝑎𝑥) occurs at x = 0, and minimum velocity (𝑣𝑚𝑖𝑛) occurs at x = ±𝐴.

Acceleration

• X is increasing or decreasing.
• Amax occurs at x = ±𝐴.
• Amin occurs at x = 0.

Frequency of Vibration

• The frequency (ω) can be found by solving the equations ∑𝐹 = −𝑘𝑥 = 𝑚𝑎𝑥, −𝑘𝐴 cos 𝜔𝑡 = −𝑚𝐴𝜔2 cos 𝜔𝑡, and ω=√(k/m) = 2𝜋𝑓

Characteristics Summary

• For mass m attached to oscillating ideal spring (force constant k).
• 𝑥 𝑡 = 𝐴 cos 𝜔𝑡 represents object position.
• 𝑣𝑥 𝑡 = −𝐴𝜔 sin 𝜔𝑡 represents object velocity.
• 𝑎𝑥 𝑡 = −𝐴𝜔2 cos 𝜔𝑡 represents object acceleration.
• Equations: Angular frequency for simple harmonic motion ω=√(k/m) where k is restoring force constant and m is mass. Period for simple harmonic motion 𝑇 = (1/f)=(2π/ω)=2π√(m/k) where m is mass and k is constant. Frequency for simple harmonic motion 𝑓 = (ω/2π)=(1/2π)√(k/m) where k is restoring force constant and m is mass.

Periodic Motion in Nature

• Periodic phenomena is everywhere including celestial movements, atomic vibrations, and heart electrical activity.
• A hummingbird’s wing Hz is 50, that flapping sound represents SHM.
• Hz of insects wing flap is 330 (house fly) to 600 Hz (mosquito),

Concept Questions

• These are conceptual problems without provided solutions for self-assessment.

Energy and SHM

• Work done by spring is not the same for Work done by constant forces.
• Work is area under the curve of force vs displacement.

Energy and SHM

• The force to compress spring is to x: 𝐹𝑥=𝑘𝑥
• 𝑊𝑡𝑜𝑡 = (1/2)𝑘𝑥𝑓2- (1/2)𝑘𝑥02, work done on the spring.
• 𝑊𝑡𝑜𝑡 = -((1/2)𝑘𝑥𝑓2- (1/2)𝑘𝑥02), work done by the spring.
• 𝑊𝑒𝑙𝑎𝑠𝑡𝑖𝑐 = (1/2) 𝑘𝑥02- (1/2)𝑘𝑥𝑓2

Elastic Potential Energy

• The energy of a spring that is being stretched or compressed and the elastic potential energy is an ideal spring. 𝑃𝐸𝑒𝑙𝑎𝑠𝑡𝑖𝑐 = (1/2)𝑘𝑥2

Total Mechanical Energy

• Energy total (E) now has a type of energy to our expression: 𝐸 = (1/2)𝑚𝑣2+𝑚𝑔ℎ+(1/2)𝑘𝑥2
• Translational KE, Gravitational PE, and Elastic PE.
• Conservation Principles: 𝐸0+𝑊𝑛𝑐 =𝐸𝑓 Work-Energy Theorem: 𝑊𝑡𝑜𝑡=Δ𝐾𝐸

Conservative Forces

• Gravitational PE will change as the direction of the gravitational force changes.

Force and Energy Relationship

• Elastic PE changes as direction of elastic force changes.
• Conservative forces always act to drive systems to the lowest potential energy state.

Force Application

• The relationship is important in natural phenomena.
• The relationship is important, determining the atomic radii/molecular bond lengths.

Energy Diagrams

• An energy diagram is a graph that shows both the potential-energy function PE(x) and the total mechanical energy E.

The Pendulum

• Pendulums' oscillatory motion allows them to be used to keep time.
• The simple pendulum is idealized, where mass is suspended by a string, where the string's mass is negligible.
• The physical pendulum is realistic where the strung's mass is not ignored.

The Simple Pendulum

• String is assumed to massless/unstretchable.
• Bob is model as a point mass.
• Angular frequency: ω = √(g/L), L is pendulum length.
• General SHM is: 𝑇=(1/f)=(2π/ω).

The Physical Pendulum

• Body is free to rotate/gravitational force acts at body's center of gravity.
• Angular frequency: ω = √(mgd/I), where "I" is moment of inertial, "m" is mass, and "d" is distance from pivot to center of gravity.
• General SHM is: 𝑇=(1/f)=(2π/ω).

Small- Angle Approximation

• Restoring force to displacement with no resistive forces is simple harmonic motion(SHM)
• The restoring force is proportion to sin(0) not θ, its approximately proportional to the displacement.

Walking

• Fossil evidence shows T. rex had leg length L of 3.1 m, stride length S(distance between footmarks) is 4.0 m. Modeling the leg as a rod estimates the walking speed.

Stress, Strain, and Hooke's Law

• General F/A quality is stress.
• As a result of applied stress the dimension changes that's measured as strain.
• Hooke's Law measures forces, where stress = Elastic Modulus.
• Measure the deformation of the body vs the force that is being measured.
• Strain has a unitless quantity.
• SI unit is 𝑁/𝑚2.

Elastic Deformation in Nature

• This tendon stretches substantially, by up to 2.5%.

Elastic Deformations

• Tensile + compressive stress and strain measures in: Tensile stress=(F/A) | Tensile strain=(ΔL/L0). Compressive stress=(F/A) |Compressive strain=(ΔL/L0).
• (F/A)=Y(ΔL/L0), Y is Young's Modulus, N/m^2

Sheer Stress

• Sheer stress and Strain: (F/A)=S(Δx/L0). Sheer modulus has units: N/m^2.

Bulk stress

• Pressure in fluid applied to surface P=(F┴/A).
• ΔP=-B(ΔV/V0). Volume is V=V0 + ΔV (ΔV