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Physics

6.1 Angle of Rotation and Angular Velocity

Physics6.1 Angle of Rotation and Angular Velocity
  1. Preface
  2. 1 What is Physics?
    1. Introduction
    2. 1.1 Physics: Definitions and Applications
    3. 1.2 The Scientific Methods
    4. 1.3 The Language of Physics: Physical Quantities and Units
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  3. 2 Motion in One Dimension
    1. Introduction
    2. 2.1 Relative Motion, Distance, and Displacement
    3. 2.2 Speed and Velocity
    4. 2.3 Position vs. Time Graphs
    5. 2.4 Velocity vs. Time Graphs
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  4. 3 Acceleration
    1. Introduction
    2. 3.1 Acceleration
    3. 3.2 Representing Acceleration with Equations and Graphs
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  5. 4 Forces and Newton’s Laws of Motion
    1. Introduction
    2. 4.1 Force
    3. 4.2 Newton's First Law of Motion: Inertia
    4. 4.3 Newton's Second Law of Motion
    5. 4.4 Newton's Third Law of Motion
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  6. 5 Motion in Two Dimensions
    1. Introduction
    2. 5.1 Vector Addition and Subtraction: Graphical Methods
    3. 5.2 Vector Addition and Subtraction: Analytical Methods
    4. 5.3 Projectile Motion
    5. 5.4 Inclined Planes
    6. 5.5 Simple Harmonic Motion
    7. Key Terms
    8. Section Summary
    9. Key Equations
    10. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    11. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  7. 6 Circular and Rotational Motion
    1. Introduction
    2. 6.1 Angle of Rotation and Angular Velocity
    3. 6.2 Uniform Circular Motion
    4. 6.3 Rotational Motion
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  8. 7 Newton's Law of Gravitation
    1. Introduction
    2. 7.1 Kepler's Laws of Planetary Motion
    3. 7.2 Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  9. 8 Momentum
    1. Introduction
    2. 8.1 Linear Momentum, Force, and Impulse
    3. 8.2 Conservation of Momentum
    4. 8.3 Elastic and Inelastic Collisions
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  10. 9 Work, Energy, and Simple Machines
    1. Introduction
    2. 9.1 Work, Power, and the Work–Energy Theorem
    3. 9.2 Mechanical Energy and Conservation of Energy
    4. 9.3 Simple Machines
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  11. 10 Special Relativity
    1. Introduction
    2. 10.1 Postulates of Special Relativity
    3. 10.2 Consequences of Special Relativity
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  12. 11 Thermal Energy, Heat, and Work
    1. Introduction
    2. 11.1 Temperature and Thermal Energy
    3. 11.2 Heat, Specific Heat, and Heat Transfer
    4. 11.3 Phase Change and Latent Heat
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  13. 12 Thermodynamics
    1. Introduction
    2. 12.1 Zeroth Law of Thermodynamics: Thermal Equilibrium
    3. 12.2 First law of Thermodynamics: Thermal Energy and Work
    4. 12.3 Second Law of Thermodynamics: Entropy
    5. 12.4 Applications of Thermodynamics: Heat Engines, Heat Pumps, and Refrigerators
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  14. 13 Waves and Their Properties
    1. Introduction
    2. 13.1 Types of Waves
    3. 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period
    4. 13.3 Wave Interaction: Superposition and Interference
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  15. 14 Sound
    1. Introduction
    2. 14.1 Speed of Sound, Frequency, and Wavelength
    3. 14.2 Sound Intensity and Sound Level
    4. 14.3 Doppler Effect and Sonic Booms
    5. 14.4 Sound Interference and Resonance
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  16. 15 Light
    1. Introduction
    2. 15.1 The Electromagnetic Spectrum
    3. 15.2 The Behavior of Electromagnetic Radiation
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  17. 16 Mirrors and Lenses
    1. Introduction
    2. 16.1 Reflection
    3. 16.2 Refraction
    4. 16.3 Lenses
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  18. 17 Diffraction and Interference
    1. Introduction
    2. 17.1 Understanding Diffraction and Interference
    3. 17.2 Applications of Diffraction, Interference, and Coherence
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  19. 18 Static Electricity
    1. Introduction
    2. 18.1 Electrical Charges, Conservation of Charge, and Transfer of Charge
    3. 18.2 Coulomb's law
    4. 18.3 Electric Field
    5. 18.4 Electric Potential
    6. 18.5 Capacitors and Dielectrics
    7. Key Terms
    8. Section Summary
    9. Key Equations
    10. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    11. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  20. 19 Electrical Circuits
    1. Introduction
    2. 19.1 Ohm's law
    3. 19.2 Series Circuits
    4. 19.3 Parallel Circuits
    5. 19.4 Electric Power
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  21. 20 Magnetism
    1. Introduction
    2. 20.1 Magnetic Fields, Field Lines, and Force
    3. 20.2 Motors, Generators, and Transformers
    4. 20.3 Electromagnetic Induction
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  22. 21 The Quantum Nature of Light
    1. Introduction
    2. 21.1 Planck and Quantum Nature of Light
    3. 21.2 Einstein and the Photoelectric Effect
    4. 21.3 The Dual Nature of Light
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  23. 22 The Atom
    1. Introduction
    2. 22.1 The Structure of the Atom
    3. 22.2 Nuclear Forces and Radioactivity
    4. 22.3 Half Life and Radiometric Dating
    5. 22.4 Nuclear Fission and Fusion
    6. 22.5 Medical Applications of Radioactivity: Diagnostic Imaging and Radiation
    7. Key Terms
    8. Section Summary
    9. Key Equations
    10. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Performance Task
    11. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  24. 23 Particle Physics
    1. Introduction
    2. 23.1 The Four Fundamental Forces
    3. 23.2 Quarks
    4. 23.3 The Unification of Forces
    5. Key Terms
    6. Section Summary
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  25. A | Reference Tables
  26. Index

Section Learning Objectives

By the end of this section, you will be able to do the following:

  • Describe the angle of rotation and relate it to its linear counterpart
  • Describe angular velocity and relate it to its linear counterpart
  • Solve problems involving angle of rotation and angular velocity

Teacher Support

Teacher Support

The learning objectives in this section will help your students master the following standards:

  • (4) Science concepts. The student knows and applies the laws governing motion in a variety of situations. The student is expected to:
    • (C) analyze and describe accelerated motion in two dimensions using equations, including projectile and circular examples.

Section Key Terms

angle of rotation angular velocity arc length circular motion
radius of curvature rotational motion spin tangential velocity

Angle of Rotation

What exactly do we mean by circular motion or rotation? Rotational motion is the circular motion of an object about an axis of rotation. We will discuss specifically circular motion and spin. Circular motion is when an object moves in a circular path. Examples of circular motion include a race car speeding around a circular curve, a toy attached to a string swinging in a circle around your head, or the circular loop-the-loop on a roller coaster. Spin is rotation about an axis that goes through the center of mass of the object, such as Earth rotating on its axis, a wheel turning on its axle, the spin of a tornado on its path of destruction, or a figure skater spinning during a performance at the Olympics. Sometimes, objects will be spinning while in circular motion, like the Earth spinning on its axis while revolving around the Sun, but we will focus on these two motions separately.

Teacher Support

Teacher Support

[BL][OL] Explain the difference between circular and rotational motions by using the Earth’s rotation about its axis and its revolution about the Sun. Explain that Earth’s rotation is slightly elliptical, although it is very nearly circular.

[OL][AL] Ask students to come up with examples of circular motion.

When solving problems involving rotational motion, we use variables that are similar to linear variables (distance, velocity, acceleration, and force) but take into account the curvature or rotation of the motion. Here, we define the angle of rotation, which is the angular equivalence of distance; and angular velocity, which is the angular equivalence of linear velocity.

When objects rotate about some axis—for example, when the CD in Figure 6.2 rotates about its center—each point in the object follows a circular path.

The diagram shows a picture of a CD with about an eighth of the shiny surface peeled away to display the inside of the CD with pits (dots) arranged in lines from the center. An angle theta is marked from one line of dots to another.
Figure 6.2 All points on a CD travel in circular paths. The pits (dots) along a line from the center to the edge all move through the same angle Δθ Δθ in time Δt Δt .

The arc length, , is the distance traveled along a circular path. The radius of curvature, r, is the radius of the circular path. Both are shown in Figure 6.3.

The diagram shows a circle with the center marked O, a radius r ending on the circumference at A, an arc change of s starting at A and ending at B. The angle for the arc is change of theta. A formula to the right of the circle says change of theta is equal to change of s divided by r.
Figure 6.3 The radius (r) of a circle is rotated through an angle Δθ Δθ . The arc length, Δs Δs , is the distance covered along the circumference.

Consider a line from the center of the CD to its edge. In a given time, each pit (used to record information) on this line moves through the same angle. The angle of rotation is the amount of rotation and is the angular analog of distance. The angle of rotation Δθ Δθ is the arc length divided by the radius of curvature.

Δθ= Δs r Δθ= Δs r

The angle of rotation is often measured by using a unit called the radian. (Radians are actually dimensionless, because a radian is defined as the ratio of two distances, radius and arc length.) A revolution is one complete rotation, where every point on the circle returns to its original position. One revolution covers 2π 2π radians (or 360 degrees), and therefore has an angle of rotation of 2π 2π radians, and an arc length that is the same as the circumference of the circle. We can convert between radians, revolutions, and degrees using the relationship

1 revolution = 2π 2π rad = 360°. See Table 6.1 for the conversion of degrees to radians for some common angles.

2π rad = 360° 1rad = 360° 2π 57.3° 2π rad = 360° 1rad = 360° 2π 57.3°
6.1
Degree Measures Radian Measures
30 30 π 6 π 6
60 60 π 3 π 3
90 90 π 2 π 2
120 120 2π 3 2π 3
135 135 3π 4 3π 4
180 180 π π
Table 6.1 Commonly Used Angles in Terms of Degrees and Radians

Angular Velocity

Teacher Support

Teacher Support

[BL] Review displacement, speed, velocity, acceleration.

[AL] Ask students whether or not velocity changes in uniform circular motion. What about speed? What about acceleration?

How fast is an object rotating? We can answer this question by using the concept of angular velocity. Consider first the angular speed (ω) (ω) is the rate at which the angle of rotation changes. In equation form, the angular speed is

ω= Δθ Δt , ω= Δθ Δt ,
6.2

which means that an angular rotation (Δθ) (Δθ) occurs in a time, Δt Δt . If an object rotates through a greater angle of rotation in a given time, it has a greater angular speed. The units for angular speed are radians per second (rad/s).

Now let’s consider the direction of the angular speed, which means we now must call it the angular velocity. The direction of the angular velocity is along the axis of rotation. For an object rotating clockwise, the angular velocity points away from you along the axis of rotation. For an object rotating counterclockwise, the angular velocity points toward you along the axis of rotation.

Angular velocity (ω) is the angular version of linear velocity v. Tangential velocity is the instantaneous linear velocity of an object in rotational motion. To get the precise relationship between angular velocity and tangential velocity, consider again a pit on the rotating CD. This pit moves through an arc length (Δs) (Δs) in a short time (Δt) (Δt) so its tangential speed is

v= Δs Δt . v= Δs Δt .
6.3

From the definition of the angle of rotation, Δθ= Δs r Δθ= Δs r , we see that Δs=rΔθ Δs=rΔθ . Substituting this into the expression for v gives

v= rΔθ Δt =rω . v= rΔθ Δt =rω .

The equation v=rω v=rω says that the tangential speed v is proportional to the distance r from the center of rotation. Consequently, tangential speed is greater for a point on the outer edge of the CD (with larger r) than for a point closer to the center of the CD (with smaller r). This makes sense because a point farther out from the center has to cover a longer arc length in the same amount of time as a point closer to the center. Note that both points will still have the same angular speed, regardless of their distance from the center of rotation. See Figure 6.4.

The picture shows a circle with a radius r2 going from the center, through a point 1 half-way on the radius and point 2 on the circumference. The distance from center to point 1 is labeled r1. A second radius line is drawn with an angle of change in theta. The arc from point 1 to the second radius is labeled change s1. The arc at point 2 is labeled change s2. There are two formulas to the right of the diagram: theta is equal to change of s1 over r1 and theta is equal to change of s2 over r2.
Figure 6.4 Points 1 and 2 rotate through the same angle ( Δθ Δθ ), but point 2 moves through a greater arc length ( Δ s 2 Δ s 2 ) because it is farther from the center of rotation.

Teacher Support

Teacher Support

[AL] Explain that the time period Δt Δt in the equation that defines tangential velocity ( v= Δs Δt v= Δs Δt ) must be short so that the arc described by the moving object can be approximated as a straight line. This allows us to define the direction of the tangential velocity as being tangent to the circle. This approximation becomes increasingly accurate as Δt Δt becomes increasingly small.

Now, consider another example: the tire of a moving car (see Figure 6.5). The faster the tire spins, the faster the car moves—large ω ω means large v because v=rω v=rω . Similarly, a larger-radius tire rotating at the same angular velocity, ω ω, will produce a greater linear (tangential) velocity, v, for the car. This is because a larger radius means a longer arc length must contact the road, so the car must move farther in the same amount of time.

The diagram shows an illustration of the front part of a car. The diagram shows an arrow on the wheel pointing clockwise and labeled omega (angular velocity). There is a green arrow pointing toward the front of the car labeled v (velocity). The radius of the wheel is labeled r. To the right of the wheel is an equation v equals r times omega.
Figure 6.5 A car moving at a velocity, v, to the right has a tire rotating with angular velocity ω ω . The speed of the tread of the tire relative to the axle is v, the same as if the car were jacked up and the wheels spinning without touching the road. Directly below the axle, where the tire touches the road, the tire tread moves backward with respect to the axle with tangential velocity v=rω v=rω , where r is the tire radius. Because the road is stationary with respect to this point of the tire, the car must move forward at the linear velocity v. A larger angular velocity for the tire means a greater linear velocity for the car.

However, there are cases where linear velocity and tangential velocity are not equivalent, such as a car spinning its tires on ice. In this case, the linear velocity will be less than the tangential velocity. Due to the lack of friction under the tires of a car on ice, the arc length through which the tire treads move is greater than the linear distance through which the car moves. It’s similar to running on a treadmill or pedaling a stationary bike; you are literally going nowhere fast.

Tips For Success

Angular velocity ω and tangential velocity v are vectors, so we must include magnitude and direction. The direction of the angular velocity is along the axis of rotation, and points away from you for an object rotating clockwise, and toward you for an object rotating counterclockwise. In mathematics this is described by the right-hand rule. Tangential velocity is usually described as up, down, left, right, north, south, east, or west, as shown in Figure 6.6.

The figure shows an illustration of a vinyl record with an arrow omega (angular velocity) pointing in a clockwise direction. There are two lines for the radius, marked r, one going from the center up and the other going from the center to the right. There are three flies on the record. One is positioned at the top of the record on the vertical radius. A v (velocity) arrow points to the right. A second fly is half-way around the circumference toward the horizontal radius and an arrow v is pointing tangential to the fly. The third fly is on the circumference at the horizontal radius and an arrow v is pointing down.
Figure 6.6 As the fly on the edge of an old-fashioned vinyl record moves in a circle, its instantaneous velocity is always at a tangent to the circle. The direction of the angular velocity is into the page this case.

Watch Physics

Relationship between Angular Velocity and Speed

This video reviews the definition and units of angular velocity and relates it to linear speed. It also shows how to convert between revolutions and radians.

Grasp Check

For an object traveling in a circular path at a constant angular speed, would the linear speed of the object change if the radius of the path increases?
  1. Yes, because tangential speed is independent of the radius.
  2. Yes, because tangential speed depends on the radius.
  3. No, because tangential speed is independent of the radius.
  4. No, because tangential speed depends on the radius.

Solving Problems Involving Angle of Rotation and Angular Velocity

Snap Lab

Measuring Angular Speed

In this activity, you will create and measure uniform circular motion and then contrast it with circular motions with different radii.

  • One string (1 m long)
  • One object (two-hole rubber stopper) to tie to the end
  • One timer
Procedure
  1. Tie an object to the end of a string.
  2. Swing the object around in a horizontal circle above your head (swing from your wrist). It is important that the circle be horizontal!
  3. Maintain the object at uniform speed as it swings.
  4. Measure the angular speed of the object in this manner. Measure the time it takes in seconds for the object to travel 10 revolutions. Divide that time by 10 to get the angular speed in revolutions per second, which you can convert to radians per second.
  5. What is the approximate linear speed of the object?
  6. Move your hand up the string so that the length of the string is 90 cm. Repeat steps 2–5.
  7. Move your hand up the string so that its length is 80 cm. Repeat steps 2–5.
  8. Move your hand up the string so that its length is 70 cm. Repeat steps 2–5.
  9. Move your hand up the string so that its length is 60 cm. Repeat steps 2–5
  10. Move your hand up the string so that its length is 50 cm. Repeat steps 2–5
  11. Make graphs of angular speed vs. radius (i.e. string length) and linear speed vs. radius. Describe what each graph looks like.

Grasp Check

If you swing an object slowly, it may rotate at less than one revolution per second. What would be the revolutions per second for an object that makes one revolution in five seconds? What would be its angular speed in radians per second?
  1. The object would spin at 1 5 rev/s . The angular speed of the object would be 2 π 5 rad/s .
  2. The object would spin at 1 5 rev/s . The angular speed of the object would be π 5 rad/s .
  3. The object would spin at 5 rev/s . The angular speed of the object would be 10 π rad/s .
  4. The object would spin at 5 rev/s . The angular speed of the object would be 5 π rad/s .

Now that we have an understanding of the concepts of angle of rotation and angular velocity, we’ll apply them to the real-world situations of a clock tower and a spinning tire.

Worked Example

Angle of rotation at a Clock Tower

The clock on a clock tower has a radius of 1.0 m. (a) What angle of rotation does the hour hand of the clock travel through when it moves from 12 p.m. to 3 p.m.? (b) What’s the arc length along the outermost edge of the clock between the hour hand at these two times?

Strategy

We can figure out the angle of rotation by multiplying a full revolution ( 2π 2π radians) by the fraction of the 12 hours covered by the hour hand in going from 12 to 3. Once we have the angle of rotation, we can solve for the arc length by rearranging the equation Δθ= Δs r Δθ= Δs r since the radius is given.

Discussion

We were able to drop the radians from the final solution to part (b) because radians are actually dimensionless. This is because the radian is defined as the ratio of two distances (radius and arc length). Thus, the formula gives an answer in units of meters, as expected for an arc length.

Worked Example

How Fast Does a Car Tire Spin?

Calculate the angular speed of a 0.300 m radius car tire when the car travels at 15.0 m/s (about 54 km/h). See Figure 6.5.

Strategy

In this case, the speed of the tire tread with respect to the tire axle is the same as the speed of the car with respect to the road, so we have v = 15.0 m/s. The radius of the tire is r = 0.300 m. Since we know v and r, we can rearrange the equation v=rω v=rω , to get ω= v r ω= v r and find the angular speed.

Discussion

When we cancel units in the above calculation, we get 50.0/s (i.e., 50.0 per second, which is usually written as 50.0 s−1). But the angular speed must have units of rad/s. Because radians are dimensionless, we can insert them into the answer for the angular speed because we know that the motion is circular. Also note that, if an earth mover with much larger tires, say 1.20 m in radius, were moving at the same speed of 15.0 m/s, its tires would rotate more slowly. They would have an angular speed of

ω = 15.0m/s 1.20m = 12.5rad/s ω = 15.0m/s 1.20m = 12.5rad/s
6.8

Practice Problems

1.

What is the angle in degrees between the hour hand and the minute hand of a clock showing 9:00 a.m.?

  1. 90°
  2. 180°
  3. 360°
2.

What is the approximate value of the arc length between the hour hand and the minute hand of a clock showing 10:00 a.m if the radius of the clock is 0.2 m?

  1. 0.1 m
  2. 0.2 m
  3. 0.3 m
  4. 0.6 m

Check Your Understanding

3.
What is circular motion?
  1. Circular motion is the motion of an object when it follows a linear path.
  2. Circular motion is the motion of an object when it follows a zigzag path.
  3. Circular motion is the motion of an object when it follows a circular path.
  4. Circular motion is the movement of an object along the circumference of a circle or rotation along a circular path.
4.

What is meant by radius of curvature when describing rotational motion?

  1. The radius of curvature is the radius of a circular path.
  2. The radius of curvature is the diameter of a circular path.
  3. The radius of curvature is the circumference of a circular path.
  4. The radius of curvature is the area of a circular path.
5.
What is angular velocity?
  1. Angular velocity is the rate of change of the diameter of the circular path.
  2. Angular velocity is the rate of change of the angle subtended by the circular path.
  3. Angular velocity is the rate of change of the area of the circular path.
  4. Angular velocity is the rate of change of the radius of the circular path.
6.
What equation defines angular velocity, ω ? Take that r is the radius of curvature, θ is the angle, and t is time.
  1. ω = Δ θ Δ t
  2. ω = Δ t Δ θ
  3. ω = Δ r Δ t
  4. ω = Δ t Δ r
7.
Identify three examples of an object in circular motion.
  1. an artificial satellite orbiting the Earth, a race car moving in the circular race track, and a top spinning on its axis
  2. an artificial satellite orbiting the Earth, a race car moving in the circular race track, and a ball tied to a string being swung in a circle around a person's head
  3. Earth spinning on its own axis, a race car moving in the circular race track, and a ball tied to a string being swung in a circle around a person's head
  4. Earth spinning on its own axis, blades of a working ceiling fan, and a top spinning on its own axis
8.
What is the relative orientation of the radius and tangential velocity vectors of an object in uniform circular motion?
  1. Tangential velocity vector is always parallel to the radius of the circular path along which the object moves.
  2. Tangential velocity vector is always perpendicular to the radius of the circular path along which the object moves.
  3. Tangential velocity vector is always at an acute angle to the radius of the circular path along which the object moves.
  4. Tangential velocity vector is always at an obtuse angle to the radius of the circular path along which the object moves.

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