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University Physics Volume 1

10.8 Work and Power for Rotational Motion

University Physics Volume 110.8 Work and Power for Rotational Motion
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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:
  • Use the work-energy theorem to analyze rotation to find the work done on a system when it is rotated about a fixed axis for a finite angular displacement
  • Solve for the angular velocity of a rotating rigid body using the work-energy theorem
  • Find the power delivered to a rotating rigid body given the applied torque and angular velocity
  • Summarize the rotational variables and equations and relate them to their translational counterparts

Thus far in the chapter, we have extensively addressed kinematics and dynamics for rotating rigid bodies around a fixed axis. In this final section, we define work and power within the context of rotation about a fixed axis, which has applications to both physics and engineering. The discussion of work and power makes our treatment of rotational motion almost complete, with the exception of rolling motion and angular momentum, which are discussed in Angular Momentum. We begin this section with a treatment of the work-energy theorem for rotation.

Work for Rotational Motion

Now that we have determined how to calculate kinetic energy for rotating rigid bodies, we can proceed with a discussion of the work done on a rigid body rotating about a fixed axis. Figure 10.39 shows a rigid body that has rotated through an angle dθdθ from A to B while under the influence of a force FF. The external force FF is applied to point P, whose position is rr, and the rigid body is constrained to rotate about a fixed axis that is perpendicular to the page and passes through O. The rotational axis is fixed, so the vector rr moves in a circle of radius r, and the vector dsds is perpendicular to r.r.

Figure shows the rigid body is constrained to rotate about a fixed axis that is perpendicular to the page and passes through a point labeled as O. The rotational axis is fixed, so the vector r moves in a circle of radius r, and the vector ds is perpendicular to vector r. An external force F is applied to point P and makes rigid body rotates through an angle dtheta.
Figure 10.39 A rigid body rotates through an angle dθdθ from A to B by the action of an external force FF applied to point P.

From Equation 10.2, we have

s=θ×r.s=θ×r.

Thus,

ds=d(θ×r)=dθ×r+dr×θ=dθ×r.ds=d(θ×r)=dθ×r+dr×θ=dθ×r.

Note that drdr is zero because rr is fixed on the rigid body from the origin O to point P. Using the definition of work, we obtain

W= F·ds= F·(dθ×r)=dθ·(r×F)W= F·ds= F·(dθ×r)=dθ·(r×F)

where we used the identity a·(b×c)=b·(c×a)a·(b×c)=b·(c×a). Noting that (r×F)=τ(r×F)=τ, we arrive at the expression for the rotational work done on a rigid body:

W= τ·dθ.W= τ·dθ.
(10.27)

The total work done on a rigid body is the sum of the torques integrated over the angle through which the body rotates. The incremental work is

dW=(iτi)dθdW=(iτi)dθ
(10.28)

where we have taken the dot product in Equation 10.27, leaving only torques along the axis of rotation. In a rigid body, all particles rotate through the same angle; thus the work of every external force is equal to the torque times the common incremental angle dθdθ. The quantity (iτi)(iτi) is the net torque on the body due to external forces.

Similarly, we found the kinetic energy of a rigid body rotating around a fixed axis by summing the kinetic energy of each particle that makes up the rigid body. Since the work-energy theorem Wi=ΔKiWi=ΔKi is valid for each particle, it is valid for the sum of the particles and the entire body.

Work-Energy Theorem for Rotation

The work-energy theorem for a rigid body rotating around a fixed axis is

WAB=KBKAWAB=KBKA
(10.29)

where

K=12Iω2K=12Iω2

and the rotational work done by a net force rotating a body from point A to point B is

WAB=θAθB(iτi)dθ.WAB=θAθB(iτi)dθ.
(10.30)

We give a strategy for using this equation when analyzing rotational motion.

Problem-Solving Strategy: Work-Energy Theorem for Rotational Motion

  1. Identify the forces on the body and draw a free-body diagram. Calculate the torque for each force.
  2. Calculate the work done during the body’s rotation by every torque.
  3. Apply the work-energy theorem by equating the net work done on the body to the change in rotational kinetic energy.

Let’s look at two examples and use the work-energy theorem to analyze rotational motion.

Example 10.17

Rotational Work and Energy A 12.0N·m12.0N·m torque is applied to a flywheel that rotates about a fixed axis and has a moment of inertia of 30.0kg·m230.0kg·m2. If the flywheel is initially at rest, what is its angular velocity after it has turned through eight revolutions?

Strategy We apply the work-energy theorem. We know from the problem description what the torque is and the angular displacement of the flywheel. Then we can solve for the final angular velocity.

Solution The flywheel turns through eight revolutions, which is 16π16π radians. The work done by the torque, which is constant and therefore can come outside the integral in Equation 10.30, is

WAB=τ(θBθA).WAB=τ(θBθA).

We apply the work-energy theorem:

WAB=τ(θBθA)=12IωB212IωA2.WAB=τ(θBθA)=12IωB212IωA2.

With τ=12.0N·m,θBθA=16.0πrad,I=30.0kg·m2,andωA=0τ=12.0N·m,θBθA=16.0πrad,I=30.0kg·m2,andωA=0, we have

12.0N-m(16.0πrad)=12(30.0kg·m2)(ωB2)0.12.0N-m(16.0πrad)=12(30.0kg·m2)(ωB2)0.

Therefore,

ωB=6.3rad/s.ωB=6.3rad/s.

This is the angular velocity of the flywheel after eight revolutions.

Significance The work-energy theorem provides an efficient way to analyze rotational motion, connecting torque with rotational kinetic energy.

Example 10.18

Rotational Work: A Pulley A string wrapped around the pulley in Figure 10.40 is pulled with a constant downward force FF of magnitude 50 N. The radius R and moment of inertia I of the pulley are 0.10 m and 2.5×10−3kg-m22.5×10−3kg-m2, respectively. If the string does not slip, what is the angular velocity of the pulley after 1.0 m of string has unwound? Assume the pulley starts from rest.

Figure A shows a string wrapped around a pulley of radius R. The pulley is pulled down with a force F. Figure B shows free body that is pulled down with forces F and Mg and is pushed up with force B .
Figure 10.40 (a) A string is wrapped around a pulley of radius R. (b) The free-body diagram.

Strategy Looking at the free-body diagram, we see that neither BB, the force on the bearings of the pulley, norMgMg, the weight of the pulley, exerts a torque around the rotational axis, and therefore does no work on the pulley. As the pulley rotates through an angle θ,θ, FF acts through a distance d such that d=Rθ.d=Rθ.

Solution Since the torque due to FF has magnitude τ=RFτ=RF, we have

W=τθ=(FR)θ=Fd.W=τθ=(FR)θ=Fd.

If the force on the string acts through a distance of 1.0 m, we have, from the work-energy theorem,

WAB=KBKAFd=12Iω20(50.0N)(1.0m)=12(2.5×10−3kg-m2)ω2.WAB=KBKAFd=12Iω20(50.0N)(1.0m)=12(2.5×10−3kg-m2)ω2.

Solving for ωω, we obtain

ω=200.0rad/s.ω=200.0rad/s.

Power for Rotational Motion

Power always comes up in the discussion of applications in engineering and physics. Power for rotational motion is equally as important as power in linear motion and can be derived in a similar way as in linear motion when the force is a constant. The linear power when the force is a constant is P=F·vP=F·v. If the net torque is constant over the angular displacement, Equation 10.25 simplifies and the net torque can be taken out of the integral. In the following discussion, we assume the net torque is constant. We can apply the definition of power derived in Power to rotational motion. From Work and Kinetic Energy, the instantaneous power (or just power) is defined as the rate of doing work,

P=dWdt.P=dWdt.

If we have a constant net torque, Equation 10.25 becomes W=τθW=τθ and the power is

P=dWdt=ddt(τθ)=τdθdtP=dWdt=ddt(τθ)=τdθdt

or

P=τω.P=τω.
(10.31)

Example 10.19

Torque on a Boat Propeller A boat engine operating at 9.0×104W9.0×104W is running at 300 rev/min. What is the torque on the propeller shaft?

Strategy We are given the rotation rate in rev/min and the power consumption, so we can easily calculate the torque.

Solution

300.0rev/min=31.4rad/s;300.0rev/min=31.4rad/s;
τ=Pω=9.0×104N·m/s31.4rad/s=2864.8N·m.τ=Pω=9.0×104N·m/s31.4rad/s=2864.8N·m.

Significance It is important to note the radian is a dimensionless unit because its definition is the ratio of two lengths. It therefore does not appear in the solution.

Check Your Understanding 10.8

A constant torque of 500kN·m500kN·m is applied to a wind turbine to keep it rotating at 6 rad/s. What is the power required to keep the turbine rotating?

Rotational and Translational Relationships Summarized

The rotational quantities and their linear analog are summarized in three tables. Table 10.5 summarizes the rotational variables for circular motion about a fixed axis with their linear analogs and the connecting equation, except for the centripetal acceleration, which stands by itself. Table 10.6 summarizes the rotational and translational kinematic equations. Table 10.7 summarizes the rotational dynamics equations with their linear analogs.

Rotational Translational Relationship
θθ xx θ=srθ=sr
ωω vtvt ω=vtrω=vtr
αα atat α=atrα=atr
acac ac=vt2rac=vt2r
Table 10.5 Rotational and Translational Variables: Summary
Rotational Translational
θf=θ0+ωtθf=θ0+ωt x=x0+vtx=x0+vt
ωf=ω0+αtωf=ω0+αt vf=v0+atvf=v0+at
θf=θ0+ω0t+12αt2θf=θ0+ω0t+12αt2 xf=x0+v0t+12at2xf=x0+v0t+12at2
ωf2=ω20+2α(Δθ)ωf2=ω20+2α(Δθ) vf2=v20+2a(Δx)vf2=v20+2a(Δx)
Table 10.6 Rotational and Translational Kinematic Equations: Summary
Rotational Translational
I=imiri2I=imiri2 m
K=12Iω2K=12Iω2 K=12mv2K=12mv2
iτi=Iαiτi=Iα iFi=maiFi=ma
WAB=θAθB(iτi)dθWAB=θAθB(iτi)dθ W=F·dsW=F·ds
P=τωP=τω P=F·vP=F·v
Table 10.7 Rotational and Translational Equations: Dynamics
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