1 / 71

Introduction to Rotational Motion

Learn about the physics of rotating objects, including angular velocity, torque, and Newton's second law for rotation.

tolsen
Télécharger la présentation

Introduction to Rotational Motion

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Suggested Videos for Chapter 7 • PrelectureVideos • Describing Rotational Motion • Moment of Inertia and Center of Gravity • Newton’s Second Law for Rotation • ClassVideos • Torque • Torques and Moment Arms • Walking on a Tightrope • VideoTutorSolutions • Rotational Motion • VideoTutorDemos • Walking the UC Plank • Balancing a Meter Stick

  2. Suggested Simulations for Chapter 7 • ActivPhysics • 7.1, 7.6–7.10 • PhETs • Ladybug Revolution • Torque

  3. Chapter 7 Rotational Motion Chapter Goal: To understand the physics of rotating objects.

  4. Chapter 7 PreviewLooking Ahead: Rotational Kinematics • The spinning roulette wheel isn’t going anywhere, but it is moving. This is rotational motion. • You’ll learn about angular velocity and other quantities we use to describe rotational motion.

  5. Chapter 7 PreviewLooking Ahead: Torque • To start something moving, apply a force. To start something rotating, apply a torque, as the sailor is doing to the wheel. • You’ll see that torque depends on how hard you push and also on where you push. A push far from the axle gives a large torque.

  6. Chapter 7 PreviewLooking Ahead: Rotational Dynamics • The girl pushes on the outside edge of the merry-go-round, gradually increasing its rotation rate. • You’ll learn a version of Newton’s second law for rotational motion and use it to solve problems.

  7. Chapter 7 PreviewLooking Ahead Text p. 189

  8. Chapter 7 PreviewLooking Back: Circular Motion • In Chapter 6, you learned to describe circular motion in terms of period, frequency, velocity, and centripetal acceleration. • In this chapter, you’ll learn to use angular velocity, angular acceleration, and other quantities that describe rotational motion.

  9. Chapter 7 PreviewStop to Think As an audio CD plays, the frequency at which the disk spins changes. At 210 rpm, the speed of a point on the outside edge of the disk is 1.3 m/s. At 420 rpm, the speed of a point on the outside edge is • 1.3 m/s • 2.6 m/s • 3.9 m/s • 5.2 m/s

  10. Reading Question 7.1 If an object is rotating clockwise, this corresponds to a ______ angular velocity. • Positive • Negative

  11. Reading Question 7.1 If an object is rotating clockwise, this corresponds to a ______ angular velocity. • Positive • Negative

  12. Reading Question 7.2 The angular displacement of a rotating object is measured in • Degrees. • Radians. • Degrees per second. • Radians per second.

  13. Reading Question 7.2 The angular displacement of a rotating object is measured in • Degrees. • Radians. • Degrees per second. • Radians per second.

  14. Reading Question 7.3 Moment of inertiais • The rotational equivalent of mass. • The time at which inertia occurs. • The point at which all forces appear to act. • An alternative term for moment arm.

  15. Reading Question 7.3 Moment of inertiais • The rotational equivalent of mass. • The time at which inertia occurs. • The point at which all forces appear to act. • An alternative term for moment arm.

  16. Reading Question 7.4 Which factor does the torque on an object not depend on? • The magnitude of the applied force • The object’s angular velocity • The angle at which the force is applied • The distance from the axis to the point at which the force is applied

  17. Reading Question 7.4 Which factor does the torque on an object not depend on? • The magnitude of the applied force • The object’s angular velocity • The angle at which the force is applied • The distance from the axis to the point at which the force is applied

  18. Reading Question 7.5 A net torque applied to an object causes • A linear acceleration of the object. • The object to rotate at a constant rate. • The angular velocity of the object to change. • The moment of inertia of the object to change.

  19. Reading Question 7.5 A net torque applied to an object causes • A linear acceleration of the object. • The object to rotate at a constant rate. • The angular velocity of the object to change. • The moment of inertia of the object to change.

  20. Section 7.1 Describing Circular and Rotational Motion

  21. Describing Circular and Rotational Motion • Rotational motion is the motion of objects that spin about an axis.

  22. Angular Position • We use the angle θfrom the positive x-axis to describe the particle’s location. • Angle θis the angular position of the particle. • θis positive when measured counterclockwisefrom the positive x-axis. • An angle measured clockwise from the positive x-axis has a negative value.

  23. Angular Position • We measure angleθ in the angular unit of radians, not degrees. • The radian is abbreviated “rad.” • The arc length, s, is the distance that the particle has traveled along its circular path.

  24. Angular Position • We define the particle’s angle θ in terms of arc length and radius of the circle:

  25. Angular Position • One revolution (rev) is when a particle travels all the way around the circle. • The angle of the full circle is

  26. Angular Displacement and Angular Velocity • For linear motion, a particle with a larger velocity undergoes a greater displacement

  27. Angular Displacement and Angular Velocity • For uniform circular motion, a particle with a larger angular velocity will undergo a greater angular displacement Δθ. • Angular velocity is the angular displacement through which the particle moves each second.

  28. Angular Displacement and Angular Velocity • The angular velocity ω = Δθ/Δt is constant for a particle moving with uniform circular motion.

  29. Example 7.1 Comparing angular velocities Find the angular velocities of the two particles in Figure 7.2b. prepareFor uniform circular motion, we can use any angular displacement ∆θ, as long as we use the corresponding time interval ∆t. For each particle, we’ll choose the angular displacement corresponding to the motion from t= 0 s to t= 5 s.

  30. Example 7.1 Comparing angular velocities (cont.) solve The particle on the left travels one-quarter of a full circle during the 5 s time interval. We learned earlier that a full circle corresponds to an angle of 2πrad, so the angular displacement for this particle is ∆θ= (2πrad)/4 =π/2 rad. Thus its angular velocity is

  31. Example 7.1 Comparing angular velocities (cont.) The particle on the right travels halfway around the circle, or πrad, in the 5 s interval. Its angular velocity is

  32. Example 7.1 Comparing angular velocities (cont.) assessThe speed of the second particle is double that of the first, as it should be. We should also check the scale of the answers. The angular velocity of the particle on the right is 0.628 rad/s, meaning that the particle travels through an angle of 0.628 rad each second. Because 1 rad ≈ 60°, 0.628 rad is roughly 35°. In Figure 7.2b, the particle on the right appears to move through an angle of about this size during each 1 s time interval, so our answer is reasonable.

  33. Angular Displacement and Angular Velocity • The linear displacement during a time interval is • Similarly, the angular displacement for uniform circular motion is

  34. Angular Displacement and Angular Velocity • Angular speed is the absolute value of the angular velocity. • The angular speed is related to the period T: • Frequency (in rev/s) f= 1/T:

  35. QuickCheck 7.1 A ball rolls around a circular track with an angular velocity of 4 rad/s. What is the period of the motion? • s • 1 s • 2 s • s • s

  36. QuickCheck 7.1 A ball rolls around a circular track with an angular velocity of 4 rad/s. What is the period of the motion? • s • 1 s • 2 s • s • s

  37. Example 7.3 Rotations in a car engine The crankshaft in your car engine is turning at 3000 rpm. What is the shaft’s angular speed? prepare We’ll need to convert rpm to rev/s and then use Equation 7.6. solveWe convert rpm to rev/s by Thus the crankshaft’s angular speed is

  38. Angular-Position and Angular-Velocity Graphs • We construct angular position-versus-time graphs using the change in angular position for each second. • Angular velocity-versus-time graphs can be created by finding the slope of the corresponding angular position-versus-time graph.

  39. Relating Speed and Angular Speed • Speed v and angular speed ωare related by • Angular speed ωmust be in units of rad/s.

  40. Example 7.5 Finding the speed at two points on a CD The diameter of an audio compact disk is 12.0 cm. When the disk is spinning at its maximum rate of 540 rpm, what is the speed of a point (a) at a distance 3.0 cm from the center and (b) at the outside edge of the disk, 6.0 cm from the center? prepareConsider two points A and B on the rotating compact disk in FIGURE 7.7. During one period T, the disk rotates once, and both points rotate through the same angle, 2πrad. Thus the angular speed, ω = 2π/T, is the same for these two points; in fact, it is the same for all points on the disk.

  41. Example 7.5 Finding the speed at two points on a CD (cont.) But as they go around one time, the two points move different distances. The outer point B goes around a larger circle. The two points thus have different speeds. We can solve this problem by first finding the angular speed of the disk and then computing the speeds at the two points.

  42. Example 7.5 Finding the speed at two points on a CD (cont.) solveWe first convert the frequency of the disk to rev/s: We then compute the angular speed using Equation 7.6:

  43. Example 7.5 Finding the speed at two points on a CD (cont.) We can now use Equation 7.7 to compute the speeds of points on the disk. At point A, r= 3.0 cm = 0.030 m, so the speed is At point B, r= 6.0 cm = 0.060 m, so the speed at the outside edge is

  44. Example 7.5 Finding the speed at two points on a CD (cont.) assessThe speeds are a few meters per second, which seems reasonable. The point farther from the center is moving at a higher speed, as we expected.

  45. QuickCheck 7.7 This is the angular velocity graph of a wheel. How many revolutions does the wheel make in the first 4 s? • 1 • 2 • 4 • 6 • 8

  46. QuickCheck 7.7 This is the angular velocity graph of a wheel. How many revolutions does the wheel make in the first 4 s? • 1 • 2 • 4 • 6 • 8  = area under the angular velocity curve

  47. QuickCheck 7.9 Starting from rest, a wheel with constant angular acceleration turns through an angle of 25 rad in a time t. Through what angle will it have turned after time 2t? • 25 rad • 50 rad • 75 rad • 100 rad • 200 rad

  48. QuickCheck 7.9 Starting from rest, a wheel with constant angular acceleration turns through an angle of 25 rad in a time t. Through what angle will it have turned after time 2t? • 25 rad • 50 rad • 75 rad • 100 rad • 200 rad

  49. Section 7.2 The Rotation of a Rigid Body

More Related