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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

- Explain gravitational potential energy in terms of work done against gravity.
- Show that the gravitational potential energy of an object of mass
*m*at height*h*on Earth is given by PEg =*mgh*. - Show how knowledge of the potential energy as a function of position can be used to simplify calculations and explain physical phenomena.

Let us calculate the work done in lifting an object of mass

More precisely, we define the

$\begin{array}{lll}mgh&=&(0.500\text{ kg})(9.80\text{ m/s}^2)(1.00\text{ m})\\\text{ }&=&4.90\text{ kg}\cdot\text{m}^2\text{/s}^2=4.90\text{ J}\end{array}\\$

Note that the units of gravitational potential energy turn out to be joules, the same as for work and other forms of energy. As the clock runs, the mass is lowered. We can think of the mass as gradually giving up its 4.90 J of gravitational potential energy, The equation ΔPE

From now on, we will consider that any change in vertical position

ΔPEg =

The kinetic energy the person has upon reaching the floor is the amount of potential energy lost by falling through height

The distance

The work

Combining this equation with the expression for

Recalling that

$\displaystyle{F}=-\frac{mgh}{d}=-\frac{\left(60.0\text{ kg}\right)\left(9.80\text{ m/s}^2\right)\left(-3.00\text{ m}\right)}{5.00\times10^{-3}\text{ m}}=3.53\times10^5\text{ N}\\$

- What is the final speed of the roller coaster shown in Figure 4 if it starts from rest at the top of the 20.0 m hill and work done by frictional forces is negligible?
- What is its final speed (again assuming negligible friction) if its initial speed is 5.00 m/s?

$\Delta\text{KE}=\frac{1}{2}mv^2\\$

. The equation for change in potential energy states that ΔPE$mg|h|=\frac{1}{2}{mv}^2\\$

.Solving for

$v=\sqrt{2g|h|}\\$

.Substituting known values,

$\begin{array}{lll}v&=&\sqrt{2\left(9.80\text{ m/s}^2\right)\left(20.0\text{ m}\right)}\\\text{ }&=&19.8\text{ m/s}\end{array}\\$

$\Delta\text{KE}=\frac{1}{2}mv^2-\frac{1}{2}mv_0^2\\$

.
Thus, $mg|h|=\frac{1}{2}mv^2-\frac{1}{2}mv_0^2\\$

.Rearranging gives

$\frac{1}{2}mv^2=mg|h|+\frac{1}{2}mv+0^2\\$

.This means that the final kinetic energy is the sum of the initial kinetic energy and the gravitational potential energy. Mass again cancels, and

$v=\sqrt{2g|h|+v_0^2}\\$

.This equation is very similar to the kinematics equation

$v=\sqrt{v_0^2+2ad}\\$

, but it is more general—the kinematics equation is valid only for constant acceleration, whereas our equation above is valid for any path regardless of whether the object moves with a constant acceleration. Now, substituting known values gives$\begin{array}{lll}v&=&\sqrt{2\left(9.80\text{ m/s}^2\right)\left(20.0\text{ m}\right)+\left(5.00\text{ m/s}\right)^2}\\\text{ }&=&20.4\text{ m/s}\end{array}\\$

- Work done against gravity in lifting an object becomes potential energy of the object-Earth system.
- The change in gravitational potential energy, ΔPE
_{g}, is ΔPE_{g}=*mgh*, with*h*being the increase in height and*g*the acceleration due to gravity. - The gravitational potential energy of an object near Earth’s surface is due to its position in the mass-Earth system. Only differences in gravitational potential energy, ΔPE
_{g}, have physical significance. - As an object descends without friction, its gravitational potential energy changes into kinetic energy corresponding to increasing speed, so that ΔKE = −ΔPE
_{g}

- In Example 2, we calculated the final speed of a roller coaster that descended 20 m in height and had an initial speed of 5 m/s downhill. Suppose the roller coaster had had an initial speed of 5 m/s uphill instead, and it coasted uphill, stopped, and then rolled back down to a final point 20 m below the start. We would find in that case that it had the same final speed. Explain in terms of conservation of energy.
- Does the work you do on a book when you lift it onto a shelf depend on the path taken? On the time taken? On the height of the shelf? On the mass of the book?

- A hydroelectric power facility (see Figure 6) converts the gravitational potential energy of water behind a dam to electric energy. (a) What is the gravitational potential energy relative to the generators of a lake of volume 50.0 km
^{3}(mass = 5.00 × 10^{13}kg), given that the lake has an average height of 40.0 m above the generators? (b) Compare this with the energy stored in a 9-megaton fusion bomb.

- (a) How much gravitational potential energy (relative to the ground on which it is built) is stored in the Great Pyramid of Cheops, given that its mass is about 7 × 10
^{9}kg and its center of mass is 36.5 m above the surrounding ground? (b) How does this energy compare with the daily food intake of a person? - Suppose a 350-g kookaburra (a large kingfisher bird) picks up a 75-g snake and raises it 2.5 m from the ground to a branch. (a) How much work did the bird do on the snake? (b) How much work did it do to raise its own center of mass to the branch?
- In Example 2, we found that the speed of a roller coaster that had descended 20.0 m was only slightly greater when it had an initial speed of 5.00 m/s than when it started from rest. This implies that ΔPE >> KE
_{i}. Confirm this statement by taking the ratio of ΔPE to KE_{i}. (Note that mass cancels.) - A 100-g toy car is propelled by a compressed spring that starts it moving. The car follows the curved track in Figure 7. Show that the final speed of the toy car is 0.687 m/s if its initial speed is 2.00 m/s and it coasts up the frictionless slope, gaining 0.180 m in altitude.

- In a downhill ski race, surprisingly, little advantage is gained by getting a running start. (This is because the initial kinetic energy is small compared with the gain in gravitational potential energy on even small hills.) To demonstrate this, find the final speed and the time taken for a skier who skies 70.0 m along a 30º slope neglecting friction: (a) Starting from rest. (b) Starting with an initial speed of 2.50 m/s. (c) Does the answer surprise you? Discuss why it is still advantageous to get a running start in very competitive events.

3. (a) 1.8 J; (b) 8.6 J

5.

${v}_{f}=\sqrt{2gh+{v_0}^2}=\sqrt{2\left(9.80\text{ m/s}^2\right)\left(-0.180\text{ m}\right)+\left(2.00\text{ m/s}\right)^2}=0.687\text{ m/s}\\$