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HomeStudy GuidesPhysics

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The Nature of Science and Physics

Kinematics

Introduction to One-Dimensional KinematicsDisplacementVectors, Scalars, and Coordinate SystemsTime, Velocity, and SpeedVideo: One-Dimensional KinematicsAccelerationMotion Equations for Constant Acceleration in One DimensionProblem-Solving Basics for One-Dimensional KinematicsFalling ObjectsGraphical Analysis of One-Dimensional Motion

Two-Dimensional Kinematics

Dynamics: Force and Newton's Laws of Motion

Introduction to Dynamics: Newton's Laws of MotionDevelopment of Force ConceptNewton's First Law of Motion: InertiaNewton's Second Law of Motion: Concept of a SystemNewton's Third Law of Motion: Symmetry in ForcesVideo: Newton's LawsNormal, Tension, and Other Examples of ForcesProblem-Solving StrategiesFurther Applications of Newton's Laws of MotionExtended Topic: The Four Basic Forces—An Introduction

Further Applications of Newton's Laws: Friction, Drag, and Elasticity

Uniform Circular Motion and Gravitation

Introduction to Uniform Circular Motion and GravitationRotation Angle and Angular VelocityCentripetal AccelerationCentripetal ForceFictitious Forces and Non-inertial Frames: The Coriolis ForceNewton's Universal Law of GravitationVideo: GravitationSatellites and Kepler's Laws: An Argument for Simplicity

Work, Energy, and Energy Resources

Introduction to Work, Energy, and Energy ResourcesWork: The Scientific DefinitionKinetic Energy and the Work-Energy TheoremGravitational Potential EnergyVideo: Potential and Kinetic EnergyConservative Forces and Potential EnergyNonconservative ForcesConservation of EnergyPowerWork, Energy, and Power in HumansWorld Energy Use

Linear Momentum and Collisions

Rotational Motion and Angular Momentum

Introduction to Rotational Motion and Angular MomentumAngular AccelerationKinematics of Rotational MotionVideo: Rotational MotionDynamics of Rotational Motion: Rotational InertiaRotational Kinetic Energy: Work and Energy RevisitedAngular Momentum and Its ConservationVideo: Angular MomentumCollisions of Extended Bodies in Two DimensionsGyroscopic Effects: Vector Aspects of Angular Momentum

Statics and Torque

Fluid Statics

Introduction to Fluid StaticsWhat Is a Fluid?DensityPressureVariation of Pressure with Depth in a FluidPascal's PrincipleGauge Pressure, Absolute Pressure, and Pressure MeasurementArchimedes' PrincipleVideo: BuoyancyCohesion and Adhesion in Liquids: Surface Tension and Capillary ActionPressures in the Body

Fluid Dynamics and Its Biological and Medical Applications

Introduction to Fluid Dynamics and Biological and Medical ApplicationsFlow Rate and Its Relation to VelocityBernoulli's EquationVideo: Fluid FlowThe Most General Applications of Bernoulli's EquationViscosity and Laminar Flow; Poiseuille's LawThe Onset of TurbulenceMotion of an Object in a Viscous FluidMolecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

Temperature, Kinetic Theory, and the Gas Laws

Heat and Heat Transfer Methods

Thermodynamics

Introduction to ThermodynamicsThe First Law of ThermodynamicsThe First Law of Thermodynamics and Some Simple ProcessesIntroduction to the Second Law of Thermodynamics: Heat Engines and Their EfficiencyCarnot's Perfect Heat Engine: The Second Law of Thermodynamics RestatedApplications of Thermodynamics: Heat Pumps and RefrigeratorsEntropy and the Second Law of Thermodynamics: Disorder and the Unavailability of EnergyStatistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

Oscillatory Motion and Waves

Introduction to Oscillatory Motion and WavesHooke's Law: Stress and Strain RevisitedPeriod and Frequency in OscillationsSimple Harmonic Motion: A Special Periodic MotionVideo: Harmonic MotionThe Simple PendulumEnergy and the Simple Harmonic OscillatorUniform Circular Motion and Simple Harmonic MotionDamped Harmonic MotionForced Oscillations and ResonanceWavesSuperposition and InterferenceEnergy in Waves: Intensity

Physics of Hearing

- Determine the maximum speed of an oscillating system.

$\text{PE}_{\text{el}}=\frac{1}{2}kx^2\\$

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Because a simple harmonic oscillator has no dissipative forces, the other important form of energy is kinetic energy KE. Conservation of energy for these two forms is:KE + PE_{el} = constant

$\frac{1}{2}mv^2+\frac{1}{2}kx^2=\text{constant}\\$

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This statement of conservation of energy is valid for Namely, for a simple pendulum we replace the velocity with

$k=\frac{mg}{L}\\$

, and the displacement term with $\frac{1}{2}mL^2\omega^2+\frac{1}{2}mgL\theta^2=\text{constant}\\$

In the case of undamped simple harmonic motion, the energy oscillates back and forth between kinetic and potential, going completely from one to the other as the system oscillates. So for the simple example of an object on a frictionless surface attached to a spring, as shown again in Figure 1, the motion starts with all of the energy stored in the spring. As the object starts to move, the elastic potential energy is converted to kinetic energy, becoming entirely kinetic energy at the equilibrium position. It is then converted back into elastic potential energy by the spring, the velocity becomes zero when the kinetic energy is completely converted, and so on. This concept provides extra insight here and in later applications of simple harmonic motion, such as alternating current circuits.The conservation of energy principle can be used to derive an expression for velocity

$\frac{1}{2}kX^2\\$

.
This total energy is constant and is shifted back and forth between kinetic energy and potential energy, at most times being shared by each. The conservation of energy for this system in equation form is thus:$\frac{1}{2}mv^2+\frac{1}{2}kx^2=\frac{1}{2}kX^2\\$

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Solving this equation for $v=\pm\sqrt{\frac{k}{m}\left(X^2-x^2\right)}\\$

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Manipulating this expression algebraically gives:$v=\pm\sqrt{\frac{k}{m}}X\sqrt{1-\frac{x^2}{X^2}}\\$

and so$v=\pm{v}_{\text{max}}\sqrt{1-\frac{x^2}{X^2}}\\$

,
where$v_{\text{max}}=\sqrt{\frac{k}{m}}X\\$

.
From this expression, we see that the velocity is a maximum ($v(t)=-v_{\text{max}}\sin\frac{2\pi{t}}{T}\\$

A similar calculation for the simple pendulum produces a similar result, namely:

$\omega_{\text{max}}=\sqrt{\frac{g}{L}}\theta_{\text{max}}\\$

$v_{\text{max}}=\sqrt{\frac{k}{m}}X\\$

to determine the maximum vertical velocity. The variables $v_{\text{max}}=\sqrt{\frac{k}{m}}X\\$

:$v_{\text{max}}=\sqrt{\frac{6.53\times10^4\text{ N/m}}{900\text{ kg}}}\left(0.100\text{ m}\right)\\$

Calculate to find The small vertical displacement

$\omega=\frac{2\pi}{T}\\$

, we have$yt=a\sin\left(\frac{2\pi{t}}{T}\right)\\$

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Thus, the displacement of pendulum is a function of time as shown above.Also the velocity of the pendulum is given by

$v(t)=\frac{2a\pi}{T}\cos\left(\frac{2\pi{t}}{T}\right)\\$

,
so the motion of the pendulum is a function of time.- Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant: $\frac{1}{2}{\text{mv}}^{2}+\frac{1}{2}{\text{kx}}^{2}=\text{constant}\\$.
- Maximum velocity depends on three factors: it is directly proportional to amplitude, it is greater for stiffer systems, and it is smaller for objects that have larger masses: ${v}_{\text{max}}=\sqrt{\frac{k}{m}}X\\$.

- The length of nylon rope from which a mountain climber is suspended has a force constant of 1.40 × 10
^{4}N/m. (a) What is the frequency at which he bounces, given his mass plus and the mass of his equipment are 90.0 kg? (b) How much would this rope stretch to break the climber’s fall if he free-falls 2.00 m before the rope runs out of slack? Hint: Use conservation of energy. (c) Repeat both parts of this problem in the situation where twice this length of nylon rope is used. **Engineering Application.**Near the top of the Citigroup Center building in New York City, there is an object with mass of 4.00 × 10^{5}kg on springs that have adjustable force constants. Its function is to dampen wind-driven oscillations of the building by oscillating at the same frequency as the building is being driven—the driving force is transferred to the object, which oscillates instead of the entire building. (a) What effective force constant should the springs have to make the object oscillate with a period of 2.00 s? (b) What energy is stored in the springs for a 2.00-m displacement from equilibrium?

- (a) 1.99 Hz; (b) 50.2 cm; (c) 1.41 Hz, 0.710 m
- (a) 3.95 × 10
^{6}N/m; (b) 7.90 × 10^{6}J