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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

- Define linear momentum.
- Explain the relationship between momentum and force.
- State Newton’s second law of motion in terms of momentum.
- Calculate momentum given mass and velocity.

Momentum is directly proportional to the object’s mass and also its velocity. Thus the greater an object’s mass or the greater its velocity, the greater its momentum. Momentum

**p** = *m***v**

- Calculate the momentum of a 110-kg football player running at 8.00 m/s.
- Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s.

*p*_{player } = (110 kg)( 8.00 m/s) = 880 kg · m/s

*p*_{ ball } = (0.410 kg)(25.0 m/s) = 10.3 kg · m/s

$\displaystyle\frac{p_{\text{player}}}{p_{\text{ball}}}=\frac{880}{10.3}=85.9\\$

$\displaystyle{\mathbf{F}}_{\text{net}}=\frac{\Delta\mathbf{p}}{\Delta t}$

,
where F$\displaystyle{\mathbf{F}}_{\text{net}}=\frac{\Delta\mathbf{p}}{\Delta t}$

If the mass of the system is constant, then Δ(

So that for constant mass, Newton’s second law of motion becomes

$\displaystyle{\mathbf{F}}_{\text{net}}=\frac{\Delta\mathbf{p}}{\Delta t}=\frac{m\Delta\mathbf{v}}{\Delta{t}}$

Because $\frac{\Delta\mathbf{v}}{\Delta{t}}=\mathbf{a}\\$

, we get the familiar equation FNewton’s second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass. We will consider systems with varying mass in some detail; however, the relationship between momentum and force remains useful when mass is constant, such as in the following example.

$\displaystyle{\mathbf{F}}_{\text{net}}=\frac{\Delta\mathbf{p}}{\Delta t}$

As noted above, when mass is constant, the change in momentum is given by Δ In this example, the velocity just after impact and the change in time are given; thus, once Δ

${\mathbf{F}}_{\text{net}}=\frac{\Delta{p}}{\Delta t}$

can be used to find the force.$\begin{array}{lll}\Delta{p}&=&m(v_{\text{f}}-v{\text{i}})\\ &=&(0.057\text{ kg})(58\text{ m/s}-0\text{ m/s})\\ &=&3.306\text{ kg}\cdot\text{m/s}\approx3.3\text{ kg}\cdot\text{m/s}\end{array}\\$

Now the magnitude of the net external force can determined by using ${\mathbf{F}}_{\text{net}}=\frac{\Delta{p}}{\Delta t}$

:$\begin{array}{lll}\mathbf{F}_{\text{net}}&=&\frac{\Delta{p}}{\Delta{t}}=\frac{3.306\text{ kg}\cdot\text{m/s}}{5.0\times10^{-3}\text{ s}}\\ &=&661\text{ N}\approx660\text{ N}\end{array}\\$

where we have retained only two significant figures in the final step.- Linear momentum (momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity.
- In symbols, linear momentum
**p**is defined to be**p**=*m***v**, where*m*is the mass of the system and**v**is its velocity. - The SI unit for momentum is kg · m/s.
- Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes.
- In symbols, Newton’s second law of motion is defined to be ${\mathbf{F}}_{\text{net}}=\frac{\Delta \mathbf{p}}{\Delta t}\\$,
**F**_{net}is the net external force, Δ**p**is the change in momentum, and Δ*t*is the change time.

- An object that has a small mass and an object that has a large mass have the same momentum. Which object has the largest kinetic energy?
- An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum?
**Professional Application.**Football coaches advise players to block, hit, and tackle with their feet on the ground rather than by leaping through the air. Using the concepts of momentum, work, and energy, explain how a football player can be more effective with his feet on the ground.- How can a small force impart the same momentum to an object as a large force?

- (a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of 7.50 m/s. (b) Compare the elephant’s momentum with the momentum of a 0.0400-kg tranquilizer dart fired at a speed of 600 m/s. (c) What is the momentum of the 90.0-kg hunter running at 7.40 m/s after missing the elephant?
- (a) What is the mass of a large ship that has a momentum of 1.60 × 10
^{9}kg · m/s, when the ship is moving at a speed of 48.0 km/h? (b) Compare the ship’s momentum to the momentum of a 1100-kg artillery shell fired at a speed of 1200 m/s. - (a) At what speed would a 2.00 × 10
^{4}-kg airplane have to fly to have a momentum of 1.60 × 10^{9}kg · m/s (the same as the ship’s momentum in the problem above)? (b) What is the plane’s momentum when it is taking off at a speed of 60.0 m/s? (c) If the ship is an aircraft carrier that launches these airplanes with a catapult, discuss the implications of your answer to (b) as it relates to recoil effects of the catapult on the ship. - (a) What is the momentum of a garbage truck that is 1.20 × 10
^{4}kg and is moving at 10.0 m/s? (b) At what speed would an 8.00-kg trash can have the same momentum as the truck? - A runaway train car that has a mass of 15,000 kg travels at a speed of 5.4 m/s down a track. Compute the time required for a force of 1500 N to bring the car to rest.
- The mass of Earth is 5.972 × 10
^{24}kg and its orbital radius is an average of 1.496 × 10^{11}m. Calculate its linear momentum.

3. (a) 8.00 × 10

5. 54 s