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University Physics Volume 1

10.2 Rotation with Constant Angular Acceleration

University Physics Volume 110.2 Rotation with Constant Angular Acceleration
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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:
  • Derive the kinematic equations for rotational motion with constant angular acceleration
  • Select from the kinematic equations for rotational motion with constant angular acceleration the appropriate equations to solve for unknowns in the analysis of systems undergoing fixed-axis rotation
  • Use solutions found with the kinematic equations to verify the graphical analysis of fixed-axis rotation with constant angular acceleration

In the preceding section, we defined the rotational variables of angular displacement, angular velocity, and angular acceleration. In this section, we work with these definitions to derive relationships among these variables and use these relationships to analyze rotational motion for a rigid body about a fixed axis under a constant angular acceleration. This analysis forms the basis for rotational kinematics. If the angular acceleration is constant, the equations of rotational kinematics simplify, similar to the equations of linear kinematics discussed in Motion along a Straight Line and Motion in Two and Three Dimensions. We can then use this simplified set of equations to describe many applications in physics and engineering where the angular acceleration of the system is constant. Rotational kinematics is also a prerequisite to the discussion of rotational dynamics later in this chapter.

Kinematics of Rotational Motion

Using our intuition, we can begin to see how the rotational quantities θ,θ, ω,ω, αα, and t are related to one another. For example, we saw in the preceding section that if a flywheel has an angular acceleration in the same direction as its angular velocity vector, its angular velocity increases with time and its angular displacement also increases. On the contrary, if the angular acceleration is opposite to the angular velocity vector, its angular velocity decreases with time. We can describe these physical situations and many others with a consistent set of rotational kinematic equations under a constant angular acceleration. The method to investigate rotational motion in this way is called kinematics of rotational motion.

To begin, we note that if the system is rotating under a constant acceleration, then the average angular velocity follows a simple relation because the angular velocity is increasing linearly with time. The average angular velocity is just half the sum of the initial and final values:

ω=ω0+ωf2.ω=ω0+ωf2.
(10.9)

From the definition of the average angular velocity, we can find an equation that relates the angular position, average angular velocity, and time:

ω=ΔθΔt.ω=ΔθΔt.

Solving for θθ, we have

θf=θ0+ωt,θf=θ0+ωt,
(10.10)

where we have set t0=0t0=0. This equation can be very useful if we know the average angular velocity of the system. Then we could find the angular displacement over a given time period. Next, we find an equation relating ωω, αα, and t. To determine this equation, we start with the definition of angular acceleration:

α=dωdt.α=dωdt.

We rearrange this to get αdt=dωαdt=dω and then we integrate both sides of this equation from initial values to final values, that is, from t0t0 to t and ω0toωfω0toωf. In uniform rotational motion, the angular acceleration is constant so it can be pulled out of the integral, yielding two definite integrals:

αt0tdt=ω0ωfdω.αt0tdt=ω0ωfdω.

Setting t0=0t0=0, we have

αt=ωfω0.αt=ωfω0.

We rearrange this to obtain

ωf=ω0+αt,ωf=ω0+αt,
(10.11)

where ω0ω0 is the initial angular velocity. Equation 10.11 is the rotational counterpart to the linear kinematics equation vf=v0+atvf=v0+at. With Equation 10.11, we can find the angular velocity of an object at any specified time t given the initial angular velocity and the angular acceleration.

Let’s now do a similar treatment starting with the equation ω=dθdtω=dθdt. We rearrange it to obtain ωdt=dθωdt=dθ and integrate both sides from initial to final values again, noting that the angular acceleration is constant and does not have a time dependence. However, this time, the angular velocity is not constant (in general), so we substitute in what we derived above:

t0tf(ω0+αt)dt=θ0θfdθ;t0tω0dt+t0tαtdt=θ0θfdθ=[ω0t+α((t)22)]t0t=ω0t+α(t22)=θfθ0,t0tf(ω0+αt)dt=θ0θfdθ;t0tω0dt+t0tαtdt=θ0θfdθ=[ω0t+α((t)22)]t0t=ω0t+α(t22)=θfθ0,

where we have set t0=0t0=0. Now we rearrange to obtain

θf=θ0+ω0t+12αt2.θf=θ0+ω0t+12αt2.
(10.12)

Equation 10.12 is the rotational counterpart to the linear kinematics equation found in Motion Along a Straight Line for position as a function of time. This equation gives us the angular position of a rotating rigid body at any time t given the initial conditions (initial angular position and initial angular velocity) and the angular acceleration.

We can find an equation that is independent of time by solving for t in Equation 10.11 and substituting into Equation 10.12. Equation 10.12 becomes

θf=θ0+ω0(ωfω0α)+12α(ωfω0α)2=θ0+ω0ωfαω02α+12ωf2αω0ωfα+12ω02α=θ0+12ωf2α12ω02α, θfθ0=ωf2ω022αθf=θ0+ω0(ωfω0α)+12α(ωfω0α)2=θ0+ω0ωfαω02α+12ωf2αω0ωfα+12ω02α=θ0+12ωf2α12ω02α, θfθ0=ωf2ω022α

or

ωf2=ω02+2α(Δθ).ωf2=ω02+2α(Δθ).
(10.13)

Equation 10.10 through Equation 10.13 describe fixed-axis rotation for constant acceleration and are summarized in Table 10.1.

Angular displacement from average angular velocity θf=θ0+ωtθf=θ0+ωt
Angular velocity from angular acceleration ωf=ω0+αtωf=ω0+αt
Angular displacement from angular velocity and angular acceleration θf=θ0+ω0t+12αt2θf=θ0+ω0t+12αt2
Angular velocity from angular displacement and angular acceleration ωf2=ω02+2α(Δθ)ωf2=ω02+2α(Δθ)
Table 10.1 Kinematic Equations

Applying the Equations for Rotational Motion

Now we can apply the key kinematic relations for rotational motion to some simple examples to get a feel for how the equations can be applied to everyday situations.

Example 10.4

Calculating the Acceleration of a Fishing Reel A deep-sea fisherman hooks a big fish that swims away from the boat, pulling the fishing line from his fishing reel. The whole system is initially at rest, and the fishing line unwinds from the reel at a radius of 4.50 cm from its axis of rotation. The reel is given an angular acceleration of 110rad/s2110rad/s2 for 2.00 s (Figure 10.11).

(a) What is the final angular velocity of the reel after 2 s?

(b) How many revolutions does the reel make?

Figure is a drawing of a fishing line coming off a rotating reel. Rotation radius is 4.5 cm, rotation takes place in the counterclockwise direction.
Figure 10.11 Fishing line coming off a rotating reel moves linearly.

Strategy Identify the knowns and compare with the kinematic equations for constant acceleration. Look for the appropriate equation that can be solved for the unknown, using the knowns given in the problem description.

Solution

  1. We are given αα and t and want to determine ωω. The most straightforward equation to use is ωf=ω0+αtωf=ω0+αt, since all terms are known besides the unknown variable we are looking for. We are given that ω0=0ω0=0 (it starts from rest), so
    ωf=0+(110rad/s2)(2.00s)=220rad/s.ωf=0+(110rad/s2)(2.00s)=220rad/s.
  2. We are asked to find the number of revolutions. Because 1rev=2πrad1rev=2πrad, we can find the number of revolutions by finding θθ in radians. We are given αα and t, and we know ω0ω0 is zero, so we can obtain θθ by using
    θf=θi+ωit+12αt2=0+0+(0.500)(110rad/s2)(2.00s)2=220rad.θf=θi+ωit+12αt2=0+0+(0.500)(110rad/s2)(2.00s)2=220rad.

    Converting radians to revolutions gives
    Number of rev=(220rad)1rev2πrad=35.0rev.Number of rev=(220rad)1rev2πrad=35.0rev.

Significance This example illustrates that relationships among rotational quantities are highly analogous to those among linear quantities. The answers to the questions are realistic. After unwinding for two seconds, the reel is found to spin at 220 rad/s, which is 2100 rpm. (No wonder reels sometimes make high-pitched sounds.)

In the preceding example, we considered a fishing reel with a positive angular acceleration. Now let us consider what happens with a negative angular acceleration.

Example 10.5

Calculating the Duration When the Fishing Reel Slows Down and Stops Now the fisherman applies a brake to the spinning reel, achieving an angular acceleration of −300rad/s2−300rad/s2. How long does it take the reel to come to a stop?

Strategy We are asked to find the time t for the reel to come to a stop. The initial and final conditions are different from those in the previous problem, which involved the same fishing reel. Now we see that the initial angular velocity is ω0=220rad/sω0=220rad/s and the final angular velocity ωω is zero. The angular acceleration is given as α=−300rad/s2.α=−300rad/s2. Examining the available equations, we see all quantities but t are known in ωf=ω0+αtωf=ω0+αt, making it easiest to use this equation.

Solution The equation states

ωf=ω0+αt.ωf=ω0+αt.

We solve the equation algebraically for t and then substitute the known values as usual, yielding

t=ωfω0α=0220.0rad/s−300.0rad/s2=0.733s.t=ωfω0α=0220.0rad/s−300.0rad/s2=0.733s.

Significance Note that care must be taken with the signs that indicate the directions of various quantities. Also, note that the time to stop the reel is fairly small because the acceleration is rather large. Fishing lines sometimes snap because of the accelerations involved, and fishermen often let the fish swim for a while before applying brakes on the reel. A tired fish is slower, requiring a smaller acceleration.

Check Your Understanding 10.2

A centrifuge used in DNA extraction spins at a maximum rate of 7000 rpm, producing a “g-force” on the sample that is 6000 times the force of gravity. If the centrifuge takes 10 seconds to come to rest from the maximum spin rate: (a) What is the angular acceleration of the centrifuge? (b) What is the angular displacement of the centrifuge during this time?

Example 10.6

Angular Acceleration of a Propeller Figure 10.12 shows a graph of the angular velocity of a propeller on an aircraft as a function of time. Its angular velocity starts at 30 rad/s and drops linearly to 0 rad/s over the course of 5 seconds. (a) Find the angular acceleration of the object and verify the result using the kinematic equations. (b) Find the angle through which the propeller rotates during these 5 seconds and verify your result using the kinematic equations.

Figure is a graph of the angular velocity in rads per second plotted versus time in seconds. Angular velocity decreases linearly with time, from 30 rads per second at zero seconds to zero at 5 seconds.
Figure 10.12 A graph of the angular velocity of a propeller versus time.

Strategy

  1. Since the angular velocity varies linearly with time, we know that the angular acceleration is constant and does not depend on the time variable. The angular acceleration is the slope of the angular velocity vs. time graph, α=dωdtα=dωdt. To calculate the slope, we read directly from Figure 10.12, and see that ω0=30rad/sω0=30rad/s at t=0st=0s and ωf=0rad/sωf=0rad/s at t=5st=5s. Then, we can verify the result using ω=ω0+αtω=ω0+αt.
  2. We use the equation ω=dθdt;ω=dθdt; since the time derivative of the angle is the angular velocity, we can find the angular displacement by integrating the angular velocity, which from the figure means taking the area under the angular velocity graph. In other words:
    θ0θfdθ=θfθ0=t0tfω(t)dt.θ0θfdθ=θfθ0=t0tfω(t)dt.

    Then we use the kinematic equations for constant acceleration to verify the result.

Solution

  1. Calculating the slope, we get
    α=ωω0tt0=(030.0)rad/s(5.00)s=−6.0rad/s2.α=ωω0tt0=(030.0)rad/s(5.00)s=−6.0rad/s2.

    We see that this is exactly Equation 10.11 with a little rearranging of terms.
  2. We can find the area under the curve by calculating the area of the right triangle, as shown in Figure 10.13.
    Figure is a graph of the angular velocity in rads per second plotted versus time in seconds. Angular velocity decreases linearly with time, from 30 rads per second at zero seconds to zero at 5 seconds. The area under the curve is filled.
    Figure 10.13 The area under the curve is the area of the right triangle.

    Δθ=area(triangle);Δθ=12(30rad/s)(5s)=75rad.Δθ=area(triangle);Δθ=12(30rad/s)(5s)=75rad.

    We verify the solution using Equation 10.12:
    θf=θ0+ω0t+12αt2.θf=θ0+ω0t+12αt2.

    Setting θ0=0θ0=0, we have
    θ0=(30.0rad/s)(5.0s)+12(−6.0rad/s2)(5.0rad/s)2=150.075.0=75.0rad.θ0=(30.0rad/s)(5.0s)+12(−6.0rad/s2)(5.0rad/s)2=150.075.0=75.0rad.

    This verifies the solution found from finding the area under the curve.

Significance We see from part (b) that there are alternative approaches to analyzing fixed-axis rotation with constant acceleration. We started with a graphical approach and verified the solution using the rotational kinematic equations. Since α=dωdtα=dωdt, we could do the same graphical analysis on an angular acceleration-vs.-time curve. The area under an α-vs.-tα-vs.-t curve gives us the change in angular velocity. Since the angular acceleration is constant in this section, this is a straightforward exercise.

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