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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

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Linear Momentum and Collisions

Rotational Motion and Angular Momentum

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Fluid Statics

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Fluid Dynamics and Its Biological and Medical Applications

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Thermodynamics

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Oscillatory Motion and Waves

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Physics of Hearing

- Compare and discuss underdamped and overdamped oscillating systems.
- Explain critically damped system.

A guitar string stops oscillating a few seconds after being plucked. To keep a child happy on a swing, you must keep pushing. Although we can often make friction and other non-conservative forces negligibly small, completely undamped motion is rare. In fact, we may even want to damp oscillations, such as with car shock absorbers.

For a system that has a small amount of damping, the period and frequency are nearly the same as for simple harmonic motion, but the amplitude gradually decreases as shown in Figure 2. This occurs because the non-conservative damping force removes energy from the system, usually in the form of thermal energy. In general, energy removal by non-conservative forces is described as

If you gradually

Critical damping is often desired, because such a system returns to equilibrium rapidly and remains at equilibrium as well. In addition, a constant force applied to a critically damped system moves the system to a new equilibrium position in the shortest time possible without overshooting or oscillating about the new position. For example, when you stand on bathroom scales that have a needle gauge, the needle moves to its equilibrium position without oscillating. It would be quite inconvenient if the needle oscillated about the new equilibrium position for a long time before settling. Damping forces can vary greatly in character. Friction, for example, is sometimes independent of velocity (as assumed in most places in this text). But many damping forces depend on velocity—sometimes in complex ways, sometimes simply being proportional to velocity.

- What is the frictional force between the surfaces?
- What total distance does the object travel if it is released 0.100 m from equilibrium, starting at
*v*= 0? The force constant of the spring is*k*= 50.0 N/m.

Now that we have identified the principles we must apply in order to solve the problems, we need to identify the knowns and unknowns for each part of the question, as well as the quantity that is constant in Part 1 and Part 2 of the question.

Enter the known values into the equation:

Calculate and convert units:

- The system involves elastic potential energy as the spring compresses and expands, friction that is related to the work done, and the kinetic energy as the body speeds up and slows down.
- Energy is not conserved as the mass oscillates because friction is a non-conservative force.
- The motion is horizontal, so gravitational potential energy does not need to be considered.
- Because the motion starts from rest, the energy in the system is initially $\text{PE}_{\text{el,i}}=\frac{1}{2}kX^2\\$. This energy is removed by work done by friction
*W*_{nc}= −*fd*, where*d*is the total distance traveled and*f*=*μ*_{k}*mg*is the force of friction. When the system stops moving, the friction force will balance the force exerted by the spring, so$\text{PE}_{\text{el,f}}=\frac{1}{2}kx^2\\$where*x*is the final position and is given by

$\begin{array}{lll}F_{\text{el}}&=&f\\kx&=&\mu_{\text{k}}mg\\x&=&\frac{\mu_{\text{k}}mg}{k}\end{array}\\$

.
1. By equating the work done to the energy removed, solve for the distance 2. The work done by the non-conservative forces equals the initial, stored elastic potential energy. Identify the correct equation to use:

$W_{\text{nc}}=\Delta\left(\text{KE}+\text{PE}\right)=\text{PE}_{\text{el,f}}-\text{PE}_{\text{el,i}}=\frac{1}{2}k\left(\left(\frac{\mu_{\text{k}}mg}{k}\right)^2-X^2\right)\\$

3. Recall that 4. Enter the friction as

5. Combine these two equations to find

$\frac{1}{2}k\left(\left(\frac{\mu_{\text{k}}mg}{k}\right)^2-X^2\right)=-\mu_{\text{k}}mgd\\$

6. Solve the equation for $d=\frac{k}{2\mu_{\text{k}}mg}\left(X^2-\left(\frac{\mu_{\text{k}}mg}{k}\right)^2\right)\\$

7. Enter the known values into the resulting equation:$d=\frac{50.0\text{ N/m}}{2\left(0.0800\right)\left(0.200\text{ kg}\right)\left(9.80\text{ m/s}^2\right)}\left(\left(0.100\text{ m}\right)^2-\left(\frac{\left(0.0800\right)\left(0.200\text{ kg}\right)\left(9.80\text{ m/s}^2\right)}{50.0\text{ N/m}}\right)^2\right)\\$

8. Calculate $\displaystyle\frac{d}{X}=\frac{1.59\text{ m}}{0.100\text{ m}}=15.9\\$

because the amplitude of the oscillations is decreasing with time. At the end of the motion, this system will not return to This worked example illustrates how to apply problem-solving strategies to situations that integrate the different concepts you have learned. The first step is to identify the physical principles involved in the problem. The second step is to solve for the unknowns using familiar problem-solving strategies. These are found throughout the text, and many worked examples show how to use them for single topics. In this integrated concepts example, you can see how to apply them across several topics. You will find these techniques useful in applications of physics outside a physics course, such as in your profession, in other science disciplines, and in everyday life.

- Damped harmonic oscillators have non-conservative forces that dissipate their energy.
- Critical damping returns the system to equilibrium as fast as possible without overshooting.
- An underdamped system will oscillate through the equilibrium position.
- An overdamped system moves more slowly toward equilibrium than one that is critically damped.

- Give an example of a damped harmonic oscillator. (They are more common than undamped or simple harmonic oscillators.)
- How would a car bounce after a bump under each of these conditions? (a) overdamping; (b) underdamping; (c) critical damping.
- Most harmonic oscillators are damped and, if undriven, eventually come to a stop. How is this observation related to the second law of thermodynamics?

- The amplitude of a lightly damped oscillator decreases by 3.0% during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle?