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  1. Preface
  2. Unit 1. Mechanics
    1. 1 Units and Measurement
      1. Introduction
      2. 1.1 The Scope and Scale of Physics
      3. 1.2 Units and Standards
      4. 1.3 Unit Conversion
      5. 1.4 Dimensional Analysis
      6. 1.5 Estimates and Fermi Calculations
      7. 1.6 Significant Figures
      8. 1.7 Solving Problems in Physics
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 Vectors
      1. Introduction
      2. 2.1 Scalars and Vectors
      3. 2.2 Coordinate Systems and Components of a Vector
      4. 2.3 Algebra of Vectors
      5. 2.4 Products of Vectors
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 3 Motion Along a Straight Line
      1. Introduction
      2. 3.1 Position, Displacement, and Average Velocity
      3. 3.2 Instantaneous Velocity and Speed
      4. 3.3 Average and Instantaneous Acceleration
      5. 3.4 Motion with Constant Acceleration
      6. 3.5 Free Fall
      7. 3.6 Finding Velocity and Displacement from Acceleration
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 Motion in Two and Three Dimensions
      1. Introduction
      2. 4.1 Displacement and Velocity Vectors
      3. 4.2 Acceleration Vector
      4. 4.3 Projectile Motion
      5. 4.4 Uniform Circular Motion
      6. 4.5 Relative Motion in One and Two Dimensions
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    5. 5 Newton's Laws of Motion
      1. Introduction
      2. 5.1 Forces
      3. 5.2 Newton's First Law
      4. 5.3 Newton's Second Law
      5. 5.4 Mass and Weight
      6. 5.5 Newton’s Third Law
      7. 5.6 Common Forces
      8. 5.7 Drawing Free-Body Diagrams
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 6 Applications of Newton's Laws
      1. Introduction
      2. 6.1 Solving Problems with Newton’s Laws
      3. 6.2 Friction
      4. 6.3 Centripetal Force
      5. 6.4 Drag Force and Terminal Speed
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 7 Work and Kinetic Energy
      1. Introduction
      2. 7.1 Work
      3. 7.2 Kinetic Energy
      4. 7.3 Work-Energy Theorem
      5. 7.4 Power
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    8. 8 Potential Energy and Conservation of Energy
      1. Introduction
      2. 8.1 Potential Energy of a System
      3. 8.2 Conservative and Non-Conservative Forces
      4. 8.3 Conservation of Energy
      5. 8.4 Potential Energy Diagrams and Stability
      6. 8.5 Sources of Energy
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    9. 9 Linear Momentum and Collisions
      1. Introduction
      2. 9.1 Linear Momentum
      3. 9.2 Impulse and Collisions
      4. 9.3 Conservation of Linear Momentum
      5. 9.4 Types of Collisions
      6. 9.5 Collisions in Multiple Dimensions
      7. 9.6 Center of Mass
      8. 9.7 Rocket Propulsion
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    10. 10 Fixed-Axis Rotation
      1. Introduction
      2. 10.1 Rotational Variables
      3. 10.2 Rotation with Constant Angular Acceleration
      4. 10.3 Relating Angular and Translational Quantities
      5. 10.4 Moment of Inertia and Rotational Kinetic Energy
      6. 10.5 Calculating Moments of Inertia
      7. 10.6 Torque
      8. 10.7 Newton’s Second Law for Rotation
      9. 10.8 Work and Power for Rotational Motion
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    11. 11 Angular Momentum
      1. Introduction
      2. 11.1 Rolling Motion
      3. 11.2 Angular Momentum
      4. 11.3 Conservation of Angular Momentum
      5. 11.4 Precession of a Gyroscope
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    12. 12 Static Equilibrium and Elasticity
      1. Introduction
      2. 12.1 Conditions for Static Equilibrium
      3. 12.2 Examples of Static Equilibrium
      4. 12.3 Stress, Strain, and Elastic Modulus
      5. 12.4 Elasticity and Plasticity
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    13. 13 Gravitation
      1. Introduction
      2. 13.1 Newton's Law of Universal Gravitation
      3. 13.2 Gravitation Near Earth's Surface
      4. 13.3 Gravitational Potential Energy and Total Energy
      5. 13.4 Satellite Orbits and Energy
      6. 13.5 Kepler's Laws of Planetary Motion
      7. 13.6 Tidal Forces
      8. 13.7 Einstein's Theory of Gravity
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    14. 14 Fluid Mechanics
      1. Introduction
      2. 14.1 Fluids, Density, and Pressure
      3. 14.2 Measuring Pressure
      4. 14.3 Pascal's Principle and Hydraulics
      5. 14.4 Archimedes’ Principle and Buoyancy
      6. 14.5 Fluid Dynamics
      7. 14.6 Bernoulli’s Equation
      8. 14.7 Viscosity and Turbulence
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Waves and Acoustics
    1. 15 Oscillations
      1. Introduction
      2. 15.1 Simple Harmonic Motion
      3. 15.2 Energy in Simple Harmonic Motion
      4. 15.3 Comparing Simple Harmonic Motion and Circular Motion
      5. 15.4 Pendulums
      6. 15.5 Damped Oscillations
      7. 15.6 Forced Oscillations
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 16 Waves
      1. Introduction
      2. 16.1 Traveling Waves
      3. 16.2 Mathematics of Waves
      4. 16.3 Wave Speed on a Stretched String
      5. 16.4 Energy and Power of a Wave
      6. 16.5 Interference of Waves
      7. 16.6 Standing Waves and Resonance
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 17 Sound
      1. Introduction
      2. 17.1 Sound Waves
      3. 17.2 Speed of Sound
      4. 17.3 Sound Intensity
      5. 17.4 Normal Modes of a Standing Sound Wave
      6. 17.5 Sources of Musical Sound
      7. 17.6 Beats
      8. 17.7 The Doppler Effect
      9. 17.8 Shock Waves
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
    17. Chapter 17
  12. Index

Learning Objectives

By the end of this section, you will be able to:
  • Represent the work done by any force
  • Evaluate the work done for various forces

In physics, work is done on an object when energy is transferred to the object. In other words, work is done when a force acts on something that undergoes a displacement from one position to another. Forces can vary as a function of position, and displacements can be along various paths between two points. We first define the increment of work dW done by a force FF acting through an infinitesimal displacement drdr as the dot product of these two vectors:

dW=F·dr=|F||dr|cosθ.dW=F·dr=|F||dr|cosθ.
(7.1)

Then, we can add up the contributions for infinitesimal displacements, along a path between two positions, to get the total work.

Work Done by a Force

The work done by a force is the integral of the force with respect to displacement along the path of the displacement:

WAB=pathABF·dr.WAB=pathABF·dr.
(7.2)

The vectors involved in the definition of the work done by a force acting on a particle are illustrated in Figure 7.2.

A curved path connecting two points, A and B, is shown. The vector d r is a small displacement tangent to the path. The force F is a vector at the location of the displacement d r, at an angle theta to d r.
Figure 7.2 Vectors used to define work. The force acting on a particle and its infinitesimal displacement are shown at one point along the path between A and B. The infinitesimal work is the dot product of these two vectors; the total work is the integral of the dot product along the path.

We choose to express the dot product in terms of the magnitudes of the vectors and the cosine of the angle between them, because the meaning of the dot product for work can be put into words more directly in terms of magnitudes and angles. We could equally well have expressed the dot product in terms of the various components introduced in Vectors. In two dimensions, these were the x- and y-components in Cartesian coordinates, or the r- and φφ-components in polar coordinates; in three dimensions, it was just x-, y-, and z-components. Which choice is more convenient depends on the situation. In words, you can express Equation 7.1 for the work done by a force acting over a displacement as a product of one component acting parallel to the other component. From the properties of vectors, it doesn’t matter if you take the component of the force parallel to the displacement or the component of the displacement parallel to the force—you get the same result either way.

Recall that the magnitude of a force times the cosine of the angle the force makes with a given direction is the component of the force in the given direction. The components of a vector can be positive, negative, or zero, depending on whether the angle between the vector and the component-direction is between 0°0° and 90°90° or 90°90° and 180°180°, or is equal to 90°90°. As a result, the work done by a force can be positive, negative, or zero, depending on whether the force is generally in the direction of the displacement, generally opposite to the displacement, or perpendicular to the displacement. The maximum work is done by a given force when it is along the direction of the displacement (cosθ=±1cosθ=±1), and zero work is done when the force is perpendicular to the displacement (cosθ=0cosθ=0).

The units of work are units of force multiplied by units of length, which in the SI system is newtons times meters, N·m.N·m. This combination is called a joule, for historical reasons that we will mention later, and is abbreviated as J. In the English system, still used in the United States, the unit of force is the pound (lb) and the unit of distance is the foot (ft), so the unit of work is the foot-pound (ft·lb).(ft·lb).

Work Done by Constant Forces and Contact Forces

The simplest work to evaluate is that done by a force that is constant in magnitude and direction. In this case, we can factor out the force; the remaining integral is just the total displacement, which only depends on the end points A and B, but not on the path between them:

WAB=F·ABdr=F·(rBrA)=|F||rBrA|cosθ(constant force).WAB=F·ABdr=F·(rBrA)=|F||rBrA|cosθ(constant force).

We can also see this by writing out Equation 7.2 in Cartesian coordinates and using the fact that the components of the force are constant:

WAB=pathABF·dr=pathAB(Fxdx+Fydy+Fzdz)=FxABdx+FyABdy+FzABdz=Fx(xBxA)+Fy(yByA)+Fz(zBzA)=F·(rBrA).WAB=pathABF·dr=pathAB(Fxdx+Fydy+Fzdz)=FxABdx+FyABdy+FzABdz=Fx(xBxA)+Fy(yByA)+Fz(zBzA)=F·(rBrA).

Figure 7.3(a) shows a person exerting a constant force FF along the handle of a lawn mower, which makes an angle θθ with the horizontal. The horizontal displacement of the lawn mower, over which the force acts, is d.d. The work done on the lawn mower isW=F·d=FdcosθW=F·d=Fdcosθ, which the figure also illustrates as the horizontal component of the force times the magnitude of the displacement.

Figure a shows a person pushing a lawn mower with a constant force. The displacement is a horizontal vector d pointing to the right. The force F is a vector pointing down and to the right, along the handle of the lawn mower, at an angle theta below the horizontal. The component of the force parallel to the displacement is F cosine theta. The equation W equals F d cosine theta is shown in the figure. Figure b shows a person holding a briefcase. The force F is upward. The displacement is zero. Figure c shows the person in b walking horizontally while holding the briefcase. The force F is upward, as in b. The displacement d is horizontal to the right. Theta equals ninety degrees and cosine theta equals zero.
Figure 7.3 Work done by a constant force. (a) A person pushes a lawn mower with a constant force. The component of the force parallel to the displacement is the work done, as shown in the equation in the figure. (b) A person holds a briefcase. No work is done because the displacement is zero. (c) The person in (b) walks horizontally while holding the briefcase. No work is done because cosθcosθ is zero.

Figure 7.3(b) shows a person holding a briefcase. The person must exert an upward force, equal in magnitude to the weight of the briefcase, but this force does no work, because the displacement over which it acts is zero.

In Figure 7.3(c), where the person in (b) is walking horizontally with constant speed, the work done by the person on the briefcase is still zero, but now because the angle between the force exerted and the displacement is 90°90° (FF perpendicular to dd) and cos90°=0cos90°=0.

Example 7.1

Calculating the Work You Do to Push a Lawn Mower How much work is done on the lawn mower by the person in Figure 7.3(a) if he exerts a constant force of 75.0 N at an angle 35°35° below the horizontal and pushes the mower 25.0 m on level ground?

Strategy We can solve this problem by substituting the given values into the definition of work done on an object by a constant force, stated in the equation W=FdcosθW=Fdcosθ. The force, angle, and displacement are given, so that only the work W is unknown.

Solution The equation for the work is

W=Fdcosθ.W=Fdcosθ.

Substituting the known values gives

W=(75.0N)(25.0m)cos(35.0°)=1.54×103J.W=(75.0N)(25.0m)cos(35.0°)=1.54×103J.

Significance Even though one and a half kilojoules may seem like a lot of work, we will see in Potential Energy and Conservation of Energy that it’s only about as much work as you could do by burning one sixth of a gram of fat.

When you mow the grass, other forces act on the lawn mower besides the force you exert—namely, the contact force of the ground and the gravitational force of Earth. Let’s consider the work done by these forces in general. For an object moving on a surface, the displacement drdr is tangent to the surface. The part of the contact force on the object that is perpendicular to the surface is the normal force N.N. Since the cosine of the angle between the normal and the tangent to a surface is zero, we have

dWN=N·dr=0.dWN=N·dr=0.

The normal force never does work under these circumstances. (Note that if the displacement drdr did have a relative component perpendicular to the surface, the object would either leave the surface or break through it, and there would no longer be any normal contact force. However, if the object is more than a particle, and has an internal structure, the normal contact force can do work on it, for example, by displacing it or deforming its shape. This will be mentioned in the next chapter.)

The part of the contact force on the object that is parallel to the surface is friction, f.f. For this object sliding along the surface, kinetic friction fkfk is opposite to dr,dr, relative to the surface, so the work done by kinetic friction is negative. If the magnitude of fkfk is constant (as it would be if all the other forces on the object were constant), then the work done by friction is

Wfr=ABfk·dr=fkAB|dr|=fk|lAB|,Wfr=ABfk·dr=fkAB|dr|=fk|lAB|,
(7.3)

where |lAB||lAB| is the path length on the surface. The force of static friction does no work in the reference frame between two surfaces because there is never displacement between the surfaces. As an external force, static friction can do work. Static friction can keep someone from sliding off a sled when the sled is moving and perform positive work on the person. If you’re driving your car at the speed limit on a straight, level stretch of highway, the negative work done by air resistance is balanced by the positive work done by the static friction of the road on the drive wheels. You can pull the rug out from under an object in such a way that it slides backward relative to the rug, but forward relative to the floor. In this case, kinetic friction exerted by the rug on the object could be in the same direction as the displacement of the object, relative to the floor, and do positive work. The bottom line is that you need to analyze each particular case to determine the work done by the forces, whether positive, negative or zero.

Example 7.2

Moving a Couch You decide to move your couch to a new position on your horizontal living room floor. The normal force on the couch is 1 kN and the coefficient of friction is 0.6. (a) You first push the couch 3 m parallel to a wall and then 1 m perpendicular to the wall (A to B in Figure 7.4). How much work is done by the frictional force? (b) You don’t like the new position, so you move the couch straight back to its original position (B to A in Figure 7.4). What was the total work done against friction moving the couch away from its original position and back again?

Points A and B are connected by a segment to the right, length 3 m, and a vertical segment up of length 1 m. These segments are path a, shown in blue. A and B are also connected by a straight segment, shown in orange as path b. the segments of path a form the sides of a right triangle, and path b is the hypotenuse of the triangle.
Figure 7.4 Top view of paths for moving a couch.

Strategy The magnitude of the force of kinetic friction on the couch is constant, equal to the coefficient of friction times the normal force, fK=μKNfK=μKN. Therefore, the work done by it is Wfr=fKdWfr=fKd, where d is the path length traversed. The segments of the paths are the sides of a right triangle, so the path lengths are easily calculated. In part (b), you can use the fact that the work done against a force is the negative of the work done by the force.

Solution

  1. The work done by friction is
    W=(0.6)(1kN)(3m+1m)=2.4kJ.W=(0.6)(1kN)(3m+1m)=2.4kJ.
  2. The length of the path along the hypotenuse is 10m10m, so the total work done against friction is
    W=(0.6)(1kN)(3m+1m+10m)=4.3kJ.W=(0.6)(1kN)(3m+1m+10m)=4.3kJ.

Significance The total path over which the work of friction was evaluated began and ended at the same point (it was a closed path), so that the total displacement of the couch was zero. However, the total work was not zero. The reason is that forces like friction are classified as nonconservative forces, or dissipative forces, as we discuss in the next chapter.

Check Your Understanding 7.1

Can kinetic friction ever be a constant force for all paths?

The other force on the lawn mower mentioned above was Earth’s gravitational force, or the weight of the mower. Near the surface of Earth, the gravitational force on an object of mass m has a constant magnitude, mg, and constant direction, vertically down. Therefore, the work done by gravity on an object is the dot product of its weight and its displacement. In many cases, it is convenient to express the dot product for gravitational work in terms of the x-, y-, and z-components of the vectors. A typical coordinate system has the x-axis horizontal and the y-axis vertically up. Then the gravitational force is mgj^,mgj^, so the work done by gravity, over any path from A to B, is

Wgrav,AB=mgj^·(rBrA)=mg(yByA).Wgrav,AB=mgj^·(rBrA)=mg(yByA).
(7.4)

The work done by a constant force of gravity on an object depends only on the object’s weight and the difference in height through which the object is displaced. Gravity does negative work on an object that moves upward (yB>yAyB>yA), or, in other words, you must do positive work against gravity to lift an object upward. Alternately, gravity does positive work on an object that moves downward (yB<yAyB<yA), or you do negative work against gravity to “lift” an object downward, controlling its descent so it doesn’t drop to the ground. (“Lift” is used as opposed to “drop”.)

Example 7.3

Shelving a Book You lift an oversized library book, weighing 20 N, 1 m vertically down from a shelf, and carry it 3 m horizontally to a table (Figure 7.5). How much work does gravity do on the book? (b) When you’re finished, you move the book in a straight line back to its original place on the shelf. What was the total work done against gravity, moving the book away from its original position on the shelf and back again?

Point A is at a shelf at the top of a bookcase. Point B is a location on a table, to the right of the bookcase. The vertical distance from the shelf to the level of the table is 1 m, and the horizontal distance from the bookcase to the table is 3 m. Path a is a straight line from the shelf down 1 m. Path b is a horizontal segment from the bookcase to the table, and then diagonally up and to the left to the shelf.
Figure 7.5 Side view of the paths for moving a book to and from a shelf.

Strategy We have just seen that the work done by a constant force of gravity depends only on the weight of the object moved and the difference in height for the path taken, WAB=mg(yByA)WAB=mg(yByA). We can evaluate the difference in height to answer (a) and (b).

Solution

  1. Since the book starts on the shelf and is lifted down yByA=1myByA=1m, we have
    W=(20N)(1m)=20J.W=(20N)(1m)=20J.
  2. There is zero difference in height for any path that begins and ends at the same place on the shelf, so W=0.W=0.

Significance Gravity does positive work (20 J) when the book moves down from the shelf. The gravitational force between two objects is an attractive force, which does positive work when the objects get closer together. Gravity does zero work (0 J) when the book moves horizontally from the shelf to the table and negative work (−20 J) when the book moves from the table back to the shelf. The total work done by gravity is zero [20J+0J+(20J)=0].[20J+0J+(20J)=0]. Unlike friction or other dissipative forces, described in Example 7.2, the total work done against gravity, over any closed path, is zero. Positive work is done against gravity on the upward parts of a closed path, but an equal amount of negative work is done against gravity on the downward parts. In other words, work done against gravity, lifting an object up, is “given back” when the object comes back down. Forces like gravity (those that do zero work over any closed path) are classified as conservative forces and play an important role in physics.

Check Your Understanding 7.2

Can Earth’s gravity ever be a constant force for all paths?

Work Done by Forces that Vary

In general, forces may vary in magnitude and direction at points in space, and paths between two points may be curved. The infinitesimal work done by a variable force can be expressed in terms of the components of the force and the displacement along the path,

dW=Fxdx+Fydy+Fzdz.dW=Fxdx+Fydy+Fzdz.

Here, the components of the force are functions of position along the path, and the displacements depend on the equations of the path. (Although we chose to illustrate dW in Cartesian coordinates, other coordinates are better suited to some situations.) Equation 7.2 defines the total work as a line integral, or the limit of a sum of infinitesimal amounts of work. The physical concept of work is straightforward: you calculate the work for tiny displacements and add them up. Sometimes the mathematics can seem complicated, but the following example demonstrates how cleanly they can operate.

Example 7.4

Work Done by a Variable Force over a Curved Path An object moves along a parabolic path y=(0.5m−1)x2y=(0.5m−1)x2 from the origin A=(0,0)A=(0,0) to the point B=(2m,2m)B=(2m,2m) under the action of a force F=(5N/m)yi^+(10N/m)xj^F=(5N/m)yi^+(10N/m)xj^ (Figure 7.6). Calculate the work done.

A graph of y in meters versus x in meters is shown. A parabolic path labeled as y of x starts at 0, 0 and curves up and to the right. The point (2, 2) is on the parabola. Vector F of x, y is shown at a point between the origin and coordinate 2, 2. Vector F points to the right and up, at some angle to the curve y of x.
Figure 7.6 The parabolic path of a particle acted on by a given force.

Strategy The components of the force are given functions of x and y. We can use the equation of the path to express y and dy in terms of x and dx; namely,

y=(0.5m−1)x2anddy=2(0.5m−1)xdx.y=(0.5m−1)x2anddy=2(0.5m−1)xdx.

Then, the integral for the work is just a definite integral of a function of x.

Solution The infinitesimal element of work is

dW=Fxdx+Fydy=(5N/m)ydx+(10N/m)xdy=(5N/m)(0.5m1)x2dx+(10N/m)2(0.5m1)x2dx=(12.5N/m2)x2dx.dW=Fxdx+Fydy=(5N/m)ydx+(10N/m)xdy=(5N/m)(0.5m1)x2dx+(10N/m)2(0.5m1)x2dx=(12.5N/m2)x2dx.

The integral of x2x2 is x3/3,x3/3, so

W=02m(12.5N/m2)x2dx=(12.5N/m2)x33|02m=(12.5N/m2)(83)=33.3J.W=02m(12.5N/m2)x2dx=(12.5N/m2)x33|02m=(12.5N/m2)(83)=33.3J.

Significance This integral was not hard to do. You can follow the same steps, as in this example, to calculate line integrals representing work for more complicated forces and paths. In this example, everything was given in terms of x- and y-components, which are easiest to use in evaluating the work in this case. In other situations, magnitudes and angles might be easier.

Check Your Understanding 7.3

Find the work done by the same force in Example 7.4 over a cubic path, y=(0.25m−2)x3y=(0.25m−2)x3, between the same points A=(0,0)A=(0,0) and B=(2m,2m).B=(2m,2m).

You saw in Example 7.4 that to evaluate a line integral, you could reduce it to an integral over a single variable or parameter. Usually, there are several ways to do this, which may be more or less convenient, depending on the particular case. In Example 7.4, we reduced the line integral to an integral over x, but we could equally well have chosen to reduce everything to a function of y. We didn’t do that because the functions in y involve the square root and fractional exponents, which may be less familiar, but for illustrative purposes, we do this now. Solving for x and dx, in terms of y, along the parabolic path, we get

x=y/(0.5m−1)=(2m)yanddx=(2m)×12dy/y=dy/(2m−1)y.x=y/(0.5m−1)=(2m)yanddx=(2m)×12dy/y=dy/(2m−1)y.

The components of the force, in terms of y, are

Fx=(5N/m)yandFy=(10N/m)x=(10N/m)(2m)y,Fx=(5N/m)yandFy=(10N/m)x=(10N/m)(2m)y,

so the infinitesimal work element becomes

dW=Fxdx+Fydy=(5N/m)ydy(2m−1)y+(10N/m)(2m)ydy=(5N·m−1/2)(12+22)ydy=(17.7N·m−1/2)y1/2dy.dW=Fxdx+Fydy=(5N/m)ydy(2m−1)y+(10N/m)(2m)ydy=(5N·m−1/2)(12+22)ydy=(17.7N·m−1/2)y1/2dy.

The integral of y1/2y1/2 is 23y3/223y3/2, so the work done from A to B is

W=02m(17.7N·m−1/2)y1/2dy=(17.7N·m−1/2)23(2m)3/2=33.3J.W=02m(17.7N·m−1/2)y1/2dy=(17.7N·m−1/2)23(2m)3/2=33.3J.

As expected, this is exactly the same result as before.

One very important and widely applicable variable force is the force exerted by a perfectly elastic spring, which satisfies Hooke’s law F=kΔx,F=kΔx, where k is the spring constant, and Δx=xxeqΔx=xxeq is the displacement from the spring’s unstretched (equilibrium) position (Newton’s Laws of Motion). Note that the unstretched position is only the same as the equilibrium position if no other forces are acting (or, if they are, they cancel one another). Forces between molecules, or in any system undergoing small displacements from a stable equilibrium, behave approximately like a spring force.

To calculate the work done by a spring force, we can choose the x-axis along the length of the spring, in the direction of increasing length, as in Figure 7.7, with the origin at the equilibrium position xeq=0.xeq=0. (Then positive x corresponds to a stretch and negative x to a compression.) With this choice of coordinates, the spring force has only an x-component,Fx=kxFx=kx, and the work done when x changes from xAxA to xBxB is

Wspring,AB=ABFxdx=kABxdx=kx22|AB=12k(xB2xA2).Wspring,AB=ABFxdx=kABxdx=kx22|AB=12k(xB2xA2).
(7.5)
A horizontal spring whose left end is attached to a wall is shown in three different states. In all the diagrams, the displacement x is measured as the displacement to the right of the right end of the spring from its equilibrium location. In figure a, the spring is relaxed and the right end is at x = 0. In figure b, the spring is stretched. The right end of the spring is a vector delta x to the right of x = 0 and feels a leftward force F equals minus k times the vector delta x. In figure c, the spring is compressed. The right end of the spring is a vector delta x to the left of x = 0 and feels a rightward force F equals minus k times the vector delta x.
Figure 7.7 (a) The spring exerts no force at its equilibrium position. The spring exerts a force in the opposite direction to (b) an extension or stretch, and (c) a compression.

Notice that WABWAB depends only on the starting and ending points, A and B, and is independent of the actual path between them, as long as it starts at A and ends at B. That is, the actual path could involve going back and forth before ending.

Another interesting thing to notice about Equation 7.5 is that, for this one-dimensional case, you can readily see the correspondence between the work done by a force and the area under the curve of the force versus its displacement. Recall that, in general, a one-dimensional integral is the limit of the sum of infinitesimals,f(x)dxf(x)dx, representing the area of strips, as shown in Figure 7.8. In Equation 7.5, since F=kxF=kx is a straight line with slope kk, when plotted versus x, the “area” under the line is just an algebraic combination of triangular “areas,” where “areas” above the x-axis are positive and those below are negative, as shown in Figure 7.9. The magnitude of one of these “areas” is just one-half the triangle’s base, along the x-axis, times the triangle’s height, along the force axis. (There are quotation marks around “area” because this base-height product has the units of work, rather than square meters.)

A graph of a generic function f of x is shown. The area within a narrow vertical strip of width dx and extending from the x axis up to the function f (x) is highlighted. The area f(x) curve and the x axis from x = x sub 1 to x = x sub 2 is shaded. The shaded area is the sum of the strip areas.
Figure 7.8 A curve of f(x) versus x showing the area of an infinitesimal strip, f(x)dx, and the sum of such areas, which is the integral of f(x) from x1x1 to x2x2.
A linear function f(x) = -k x is plotted, with the x range extending from some x value to some positive x value. The graph is a straight line with negative slope crossing through the origin. The area under the curve to the left of the origin from –x sub A to the origin (where x is negative and f(x) is positive) is shaded in red and is a positive area. Two negative areas are shaded in gray. From the origin to some positive x sub A is a triangular area below the x axis shaded in light gray. From x sub A to a larger x sub B is a trapezoid below the x axis shaded in dark gray.
Figure 7.9 Curve of the spring force f(x)=kxf(x)=kx versus x, showing areas under the line, between xAxA and xBxB, for both positive and negative values of xAxA. When xAxA is negative, the total area under the curve for the integral in Equation 7.5 is the sum of positive and negative triangular areas. When xAxA is positive, the total area under the curve is the difference between two negative triangles.

Example 7.5

Work Done by a Spring Force A perfectly elastic spring requires 0.54 J of work to stretch 6 cm from its equilibrium position, as in Figure 7.7(b). (a) What is its spring constant k? (b) How much work is required to stretch it an additional 6 cm?

Strategy Work “required” means work done against the spring force, which is the negative of the work in Equation 7.5, that is

W=12k(xB2xA2).W=12k(xB2xA2).

For part (a), xA=0xA=0 and xB=6cmxB=6cm; for part (b), xB=6cmxB=6cm and xB=12cmxB=12cm. In part (a), the work is given and you can solve for the spring constant; in part (b), you can use the value of k, from part (a), to solve for the work.

Solution

  1. W=0.54J=12k[(6cm)20]W=0.54J=12k[(6cm)20], so k=3N/cm.k=3N/cm.
  2. W=12(3N/cm)[(12cm)2(6cm)2]=1.62J.W=12(3N/cm)[(12cm)2(6cm)2]=1.62J.

Significance Since the work done by a spring force is independent of the path, you only needed to calculate the difference in the quantity ½kx2½kx2 at the end points. Notice that the work required to stretch the spring from 0 to 12 cm is four times that required to stretch it from 0 to 6 cm, because that work depends on the square of the amount of stretch from equilibrium, ½kx2½kx2. In this circumstance, the work to stretch the spring from 0 to 12 cm is also equal to the work for a composite path from 0 to 6 cm followed by an additional stretch from 6 cm to 12 cm. Therefore, 4W(0cmto6cm)=W(0cmto6cm)+W(6cmto12cm)4W(0cmto6cm)=W(0cmto6cm)+W(6cmto12cm), or W(6cmto12cm)=3W(0cmto6cm)W(6cmto12cm)=3W(0cmto6cm), as we found above.

Check Your Understanding 7.4

The spring in Example 7.5 is compressed 6 cm from its equilibrium length. (a) Does the spring force do positive or negative work and (b) what is the magnitude?

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