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Chapter 8: Rotational Motion

PHYSICS Principles and Problems. Chapter 8: Rotational Motion. Rotational Motion. CHAPTER 8. BIG IDEA. Applying a torque to an object causes a change in that object’s angular velocity. Table Of Contents. CHAPTER 8. Section 8.1 Describing Rotational Motion Section 8.2 Rotational Dynamics

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Chapter 8: Rotational Motion

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  1. PHYSICS Principles and Problems Chapter 8: Rotational Motion

  2. Rotational Motion CHAPTER8 BIG IDEA Applying a torque to an object causes a change in that object’s angular velocity.

  3. Table Of Contents CHAPTER8 Section 8.1Describing Rotational Motion Section 8.2Rotational Dynamics Section 8.3Equilibrium Click a hyperlink to view the corresponding slides. Exit

  4. Describing Rotational Motion SECTION8.1 MAIN IDEA Angular displacement, angular velocity, and angular acceleration all help describe angular motion. • What is angular displacement? • What is average angular velocity? • What is average angular acceleration, and how is it related to angular velocity? Essential Questions

  5. Describing Rotational Motion New Vocabulary Radian Angular displacement Angular velocity Angular acceleration SECTION8.1 Review Vocabulary • displacement change in position having both magnitude and direction; it is equal to the final position minus the initial position.

  6. Describing Rotational Motion • A grad is of a revolution. • A degree is of a revolution. SECTION8.1 Angular Displacement • A fraction of one revolution can be measured in grads, degrees, or radians.

  7. Describing Rotational Motion The radian is defined as of a revolution. SECTION8.1 Angular Displacement (cont.) One complete revolution is equal to 2radians. The abbreviation of radian is ‘rad’.

  8. Describing Rotational Motion SECTION8.1 Angular Displacement (cont.) • The Greek letter theta, θ, is used to represent the angle of revolution. • The counterclockwise rotation is designated as positive, while clockwise is negative.

  9. Describing Rotational Motion SECTION8.1 Angular Displacement (cont.) • As an object rotates, the change in the angle is called angular displacement. • For rotation through an angle, θ, a point at a distance, r, from the center moves a distance given by d = rθ.

  10. Describing Rotational Motion SECTION8.1 Angular Velocity • Velocity is displacement divided by the time taken to make the displacement. • The angular velocity of an object is angular displacement divided by the time required to make the displacement.

  11. Describing Rotational Motion SECTION8.1 Angular Velocity (cont.) • The angular velocity of an object is given by: Here angular velocity is represented by the Greek letter omega, ω. The angular velocity is equal to the angular displacement divided by the time required to make the rotation.

  12. Describing Rotational Motion SECTION8.1 Angular Velocity (cont.) • If the velocity changes over a time interval, the average velocity is not equal to the instantaneous velocity at any given instant. • Similarly, the angular velocity calculated in this way is actually the average angular velocity over a time interval, t. • Instantaneous angular velocity is equal to the slope of a graph of angular position versus time.

  13. Describing Rotational Motion SECTION8.1 Angular Velocity (cont.) • Angular velocity is measured in rad/s. • For Earth, ωE = (2πrad)/[(24.0 h)(3600 s/h)] = 7.27×10─5 rad/s.

  14. Describing Rotational Motion SECTION8.1 Angular Velocity (cont.) • In the same way that counterclockwise rotation produces positive angular displacement, it also results in positive angular velocity. • If an object’s angular velocity is ω, then the linear velocity of a point at distance, r, from the axis of rotation is given by v = rω. • The speed at which an object on Earth’s equator moves as a result of Earth’s rotation is given by v =r ω = (6.38×106 m) (7.27×10─5 rad/s) = 464 m/s.

  15. Describing Rotational Motion SECTION8.1 Angular Velocity (cont.) • Earth is an example of a rotating, rigid object. Even though different points on Earth rotate different distances in each revolution, all points rotate through the same angle. • The Sun, on the other hand, is not a rigid body. Different parts of the Sun rotate at different rates.

  16. Describing Rotational Motion SECTION8.1 Angular Acceleration • Angular acceleration is defined as the change in angular velocity divided by the time required to make that change. The angular acceleration, α, is represented by the following equation:

  17. Describing Rotational Motion SECTION8.1 Angular Acceleration (cont.) Angular acceleration is measured in rad/s2. If the change in angular velocity is positive, then the angular acceleration is also positive. Angular acceleration defined in this way is also the average angular acceleration over the time interval Δt.

  18. Describing Rotational Motion SECTION8.1 Angular Acceleration (cont.) • One way to find the instantaneous angular acceleration is to find the slope of a graph of angular velocity as a function of time. • The linear acceleration of a point at a distance, r, from the axis of an object with angular acceleration, α, is given by a = r.

  19. Describing Rotational Motion SECTION8.1 Angular Acceleration (cont.) A summary of linear and angular relationships.

  20. Describing Rotational Motion SECTION8.1 Angular Acceleration (cont.) • A rotating object can make many revolutions in a given amount of time. • The number of complete revolutions made by the object in 1 s is called angular frequency. • Angular frequency, f, is given by the equation,

  21. Section Check A. B. C. D. SECTION8.1 What is the angular velocity of the second hand of a clock?

  22. Section Check SECTION8.1 SECTION1.1 Answer Reason:Angular velocity is equal to the angular displacement divided by the time required to complete one rotation.

  23. Section Check SECTION8.1 Answer Reason:In one minute, the second hand of a clock completes one rotation. Therefore, = 2π rad. Therefore,

  24. Section Check SECTION8.1 When a machine is switched on, the angular velocity of the motor increases by a total of 10 rad/s for the first 10 seconds before it starts rotating with full speed. What is the angular acceleration of the machine in the first 10 seconds? A.  rad/s2 B.1 rad/s2 C. 100 rad/s2 D. 100 rad/s2

  25. Section Check SECTION8.1 Answer Reason:Angular acceleration is equal to the change in angular velocity divided by the time required to make that change.

  26. Section Check SECTION8.1 When a fan performing 10 revolutions per second is switched off, it comes to rest after 10 seconds. Calculate the magnitude of the average angular acceleration of the fan after it was switched off. A. 1 rad/s2 B. 2 rad/s2 C.  rad/s2 D. 10 rad/s2

  27. Section Check SECTION8.1 Answer Reason:Angular displacement of any rotating object in one revolution is 2 rad. Since the fan is performing 10 revolutions per second, its angular velocity = 2×10 = 20 rad/s.

  28. Section Check SECTION8.1 Answer Reason:Angular acceleration is equal to the change in angular velocity divided by the time required to make that change.

  29. Rotational Dynamics SECTION8.2 MAINIDEA Torques cause changes in angular velocity. • What is torque? • How is the moment of inertial related to rotational motion? • How are torque, the moment of inertia, and Newton’s second law for rotational motion related? Essential Questions

  30. New Vocabulary Lever arm Torque Moment of inertia Newton’s second law for rotational motion Rotational Dynamics SECTION8.2 Review Vocabulary • magnitude a measure of size

  31. Rotational Dynamics SECTION8.2 Force and Angular Velocity • The change in angular velocity depends on the magnitude of the force, the distance from the axis to the point where the force is exerted, and the direction of the force.

  32. Rotational Dynamics SECTION8.2 Force and Angular Velocity (cont.) • To swing open a door, you exert a force. • The doorknob is near the outer edge of the door. You exert the force on the doorknob at right angles to the door, away from the hinges.

  33. Rotational Dynamics SECTION8.2 Force and Angular Velocity (cont.) • To get the most effect from the least force, you exert the force as far from the axis of rotation (imaginary line through the hinges) as possible.

  34. Rotational Dynamics SECTION8.2 Force and Angular Velocity (cont.) • Thus, the magnitude of the force, the distance from the axis to the point where the force is exerted, and the direction of the force determine the change in angular velocity.

  35. Rotational Dynamics SECTION8.2 Force and Angular Velocity (cont.) • For a given applied force, the change in angular velocity depends on the lever arm,which is the perpendicular distance from the axis of rotation to the point where the force is exerted.

  36. Rotational Dynamics SECTION8.2 Force and Angular Velocity (cont.) • For the door, it is the distance from the hinges to the point where you exert the force. • If the force is perpendicular to the radius of rotation then the lever arm is the distance from the axis, r.

  37. Rotational Dynamics SECTION8.2 Force and Angular Velocity (cont.) • If a force is not exerted perpendicular to the radius, however, the lever arm is reduced.

  38. Rotational Dynamics SECTION8.2 Force and Angular Velocity (cont.) The lever arm, L, can be calculated by the equation, L = r sin θ, where θ is the angle between the force and the radius from the axis of rotation to the point where the force is applied.

  39. Rotational Dynamics SECTION8.2 Force and Angular Velocity (cont.) • Torqueis a measure of how effectively a force causes rotation. The magnitude of torque is the product of the force and the lever arm. Because force is measured in newtons, and distance is measured in meters, torque is measured in newton-meters (N·m).

  40. Rotational Dynamics SECTION8.2 Force and Angular Velocity (cont.) Torque is represented by the Greek letter tau, τ.

  41. Rotational Dynamics SECTION8.2 Lever Arm A bolt on a car engine needs to be tightened with a torque of 35 N·m. You use a 25-cm long wrench and pull on the end of the wrench at an angle of 60.0° to the handle of the wrench. How long is the lever arm, and how much force do you have to exert?

  42. Rotational Dynamics SECTION8.2 Lever Arm (cont.) Step 1: Analyze and Sketch the Problem Sketch the situation. Find the lever arm by extending the force vector backward until a line that is perpendicular to it intersects the axis of rotation.

  43. Rotational Dynamics SECTION8.2 Lever Arm (cont.) Identify the known and unknown variables. Known: r = 0.25 m θ = 60.0º Unknown: L = ? F = ?

  44. Rotational Dynamics SECTION8.2 Lever Arm (cont.) Step 2: Solve for the Unknown

  45. Rotational Dynamics SECTION8.2 Lever Arm (cont.) Solve for the length of the lever arm. L = r sin 

  46. Rotational Dynamics SECTION8.2 Lever Arm (cont.) Substitute r = 0.25 m, θ = 60.0º L = (0.25 m)(sin 60.0°) = 0.22 m

  47. Rotational Dynamics SECTION8.2 Lever Arm (cont.) Solve for the force.

  48. Rotational Dynamics SECTION8.2 Lever Arm (cont.) Substitute τ = 35 N·m, r = 0.25 m, θ = 60.0º

  49. Rotational Dynamics SECTION8.2 Lever Arm (cont.) Step 3: Evaluate the Answer

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