Subjects

- Accounting
- Aerospace Engineering
- Anatomy
- Anthropology
- Arts & Humanities
- Astronomy
- Biology
- Business
- Chemistry
- Civil Engineering
- Computer Science
- Communications
- Economics
- Electrical Engineering
- English
- Finance
- Geography
- Geology
- Health Science
- History
- Industrial Engineering
- Information Systems
- Law
- Linguistics
- Management
- Marketing
- Material Science
- Mathematics
- Mechanical Engineering
- Medicine
- Nursing
- Philosophy
- Physics
- Political Science
- Psychology
- Religion
- Sociology
- Statistics

HomeStudy GuidesPhysics

Menu

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

- Understand the rules of vector addition, subtraction, and multiplication.
- Apply graphical methods of vector addition and subtraction to determine the displacement of moving objects.

Figure 2 shows such a

In this text, we will represent a vector with a boldface variable. For example, we will represent the quantity force with the vector** F**, which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as *F*, and the direction of the variable will be given by an angle *θ*.

Use the graphical technique for adding vectors to find the total displacement of a person who walks the following three paths (displacements) on a flat field. First, she walks 25.0 m in a direction 49.0º north of east. Then, she walks 23.0 m heading 15.0º north of east. Finally, she turns and walks 32.0 m in a direction 68.0° south of east.

**Strategy**

Represent each displacement vector graphically with an arrow, labeling the first **A**, the second **B**, and the third **C**, making the lengths proportional to the distance and the directions as specified relative to an east-west line. The head-to-tail method outlined above will give a way to determine the magnitude and direction of the resultant displacement, denoted **R**.

**Solution**

(1) Draw the three displacement vectors.

(2) Place the vectors head to tail retaining both their initial magnitude and direction.

(3) Draw the resultant vector, **R**.

(4) Use a ruler to measure the magnitude of **R**, and a protractor to measure the direction of **R**. While the direction of the vector can be specified in many ways, the easiest way is to measure the angle between the vector and the nearest horizontal or vertical axis. Since the resultant vector is south of the eastward pointing axis, we flip the protractor upside down and measure the angle between the eastward axis and the vector.

In this case, the total displacement**R** is seen to have a magnitude of 50.0 m and to lie in a direction 7.0º south of east. By using its magnitude and direction, this vector can be expressed as *R* = 50.0 m and *θ *= 7.0º south of east.

**Discussion**

The head-to-tail graphical method of vector addition works for any number of vectors. It is also important to note that the resultant is independent of the order in which the vectors are added. Therefore, we could add the vectors in any order as illustrated in Figure 12 and we will still get the same solution.

Here, we see that when the same vectors are added in a different order, the result is the same. This characteristic is true in every case and is an important characteristic of vectors. Vector addition is * commutative*. Vectors can be added in any order.

**A + B = B + A.**
(This is true for the addition of ordinary numbers as well—you get the same result whether you add **2 + 3** or **3 + 2**, for example).

In this case, the total displacement

*This video can be used for review. It includes vector basics - drawing vectors/vector addition. You'll learn about the basic notion of a vector, how to add vectors together graphically, as well as what it means graphically to multiply a vector by a scalar.*

The

**A - B = A + (-B)**

We will perform vector addition to compare the location of the dock,

(2) Place the vectors head to tail.

(3) Draw the resultant vector

(4) Use a ruler and protractor to measure the magnitude and direction of

In this case,

(5) To determine the location of the dock, we repeat this method to add vectors

In this case

We can see that the woman will end up a significant distance from the dock if she travels in the opposite direction for the second leg of the trip.

If the scalar is negative, then multiplying a vector by it changes the vector’s magnitude and gives the new vector the

- the magnitude of the vector becomes the absolute value of
*c**A*, - if
*c*is positive, the direction of the vector does not change, - if
*c*is negative, the direction is reversed.

For example, we may know that the total displacement of a person walking in a city is 10.3 blocks in a direction 29.0º north of east and want to find out how many blocks east and north had to be walked. This method is called

Learn about position, velocity, and acceleration in the "Arena of Pain". Use the green arrow to move the ball. Add more walls to the arena to make the game more difficult. Try to make a goal as fast as you can.

- The
**graphical method of adding vectors****A**and**B**involves drawing vectors on a graph and adding them using the head-to-tail method. The resultant vector**R**is defined such that**A + B = R**. The magnitude and direction of**R**are then determined with a ruler and protractor, respectively. - The
**graphical method of subtracting vector B**from**A**involves adding the opposite of vector**B**, which is defined as**-B**. In this case,**A - B = A + (-B) = R**. Then, the head-to-tail method of addition is followed in the usual way to obtain the resultant vector**R**. - Addition of vectors is commutative such that
**A + B = B + A**. - The head-to-tail method of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector.
- If a vector
**A**is multiplied by a scalar quantity*c*, the magnitude of the product is given by*cA*. If*c*is positive, the direction of the product points in the same direction as**A**; if*c*is negative, the direction of the product points in the opposite direction as**A**.

2. Give a specific example of a vector, stating its magnitude, units, and direction.

3. What do vectors and scalars have in common? How do they differ?

4. Two campers in a national park hike from their cabin to the same spot on a lake, each taking a different path, as illustrated below. The total distance traveled along Path 1 is 7.5 km, and that along Path 2 is 8.2 km. What is the final displacement of each camper?

5. If an airplane pilot is told to fly 123 km in a straight line to get from San Francisco to Sacramento, explain why he could end up anywhere on the circle shown in Figure 19. What other information would he need to get to Sacramento?

6. Suppose you take two steps

7. Explain why it is not possible to add a scalar to a vector.

8. If you take two steps of different sizes, can you end up at your starting point? More generally, can two vectors with different magnitudes ever add to zero? Can three or more?

**Use graphical methods to solve these problems. You may assume data taken from graphs is accurate to three digits.**

2. Find the following for path B in Figure 20: (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.

3. Find the north and east components of the displacement for the hikers shown in Figure 20.

4. Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements **A** and **B**, as in Figure 21, then this problem asks you to find their sum** R = A + B**.

5. Suppose you first walk 12.0 m in a direction 20 west of north and then 20.0 m in a direction 40.0º south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements **A** and **B**, as in Figure 22, then this problem finds their sum **R = A + B**.)

6. Repeat the problem above, but reverse the order of the two legs of the walk; show that you get the same final result. That is, you first walk leg **B**, which is 20.0 m in a direction exactly 40º south of west, and then leg **A**, which is 12.0 m in a direction exactly 12.0 west of north. (This problem shows that **A + B = B + A**.)

7. (a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction 40º north of east (which is equivalent to subtracting **B** from **A** —that is, to finding **R' = A - B**). (b) Repeat the problem two problems prior, but now you first walk 20.0 m in a direction 40º south of west and then 12.0 m in a direction 20º east of south (which is equivalent to subtracting **A** from **B**—that is, to finding **R'' = B - A = R'** Show that this is the case.

8. Show that the * order* of addition of three vectors does not affect their sum. Show this property by choosing any three vectors

$\mathbf{C}$

, all having different lengths and directions. Find the sum 10. Find the magnitudes of velocities *V*_{A} and *V*_{B} in Figure 23.

11. Find the components of *v*_{tot} along the * x*- and

12. Find the components of *v*_{tot} along a set of perpendicular axes rotated 30º counterclockwise relative to those in Figure 23.

- component (of a 2-d vector):
- a piece of a vector that points in either the vertical or the horizontal direction; every 2-d vector can be expressed as a sum of two vertical and horizontal vector components

- commutative:
- refers to the interchangeability of order in a function; vector addition is commutative because the order in which vectors are added together does not affect the final sum

- direction (of a vector):
- the orientation of a vector in space

- head (of a vector):
- the end point of a vector; the location of the tip of the vector’s arrowhead; also referred to as the "tip"

- head-to-tail method:
- a method of adding vectors in which the tail of each vector is placed at the head of the previous vector

- magnitude (of a vector):
- the length or size of a vector; magnitude is a scalar quantity

- resultant:
- the sum of two or more vectors

- resultant vector:
- the vector sum of two or more vectors

- scalar:
- a quantity with magnitude but no direction

- tail:
- the start point of a vector; opposite to the head or tip of the arrow

3. north component 3.21 km, east component 3.83 km

5. 19.5 m, 4.65º south of west

7. (a) 26.6 m, 65.1º north of east (b) 26.6 m, 65.1º south of west

9. 52.9 m, 90.1º with respect to the

11.