1 / 60

Unit 8

Unit 8. Center of Mass. A point that represents the average location for the total mass of a system For symmetric objects, made from uniformly distributed material Center of mass = Geometric center. Center of Mass & Motion. Center of mass follows a projectile path. Rotational Motion.

kibo-lucas
Download Presentation

Unit 8

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. Unit 8

  2. Center of Mass • A point that represents the average location for the total mass of a system • For symmetric objects, made from uniformly distributed material • Center of mass = Geometric center

  3. Center of Mass & Motion • Center of mass follows a projectile path

  4. Rotational Motion • Focuses on pure rotational motion • Motion that consists of rotation about a fixed axis • Points on a rigid object move in circular paths around an axis of rotation • Examples: • Ferris Wheel; CD

  5. Angular Position • In the study of rotational motion, position is described using angles (θ) • Angular Position = The amount of rotation from a reference point • Counterclockwise rotation = positive angle • Clockwise rotation = negative angle • Units = radians

  6. Angular Position For a full revolution:

  7. Angular Displacement • In the study of rotational motion, angular displacement is also described using angles (θ) • Change in angular position • Δθ = θf − θi • Δθ = angular displacement • θf = final angular position • θi = initial angular position • Units = radians (rad) • Counterclockwise rotation = Positive • Clockwise rotation = Negative

  8. Angular Displacement The angle through which the object rotates is called the angular displacement.

  9. PROBLEM: Adjacent Synchronous Satellites Synchronous satellites are put into an orbit whose radius is 4.23×107 m. If the angular separation of the two satellites is 2 degrees, find the arc length that separates them.

  10. ANSWER

  11. Angular Velocity How we describe the rate at which angular displacement is changing…

  12. Average Angular Velocity SI Unit of Angular Velocity: radian per second (rad/s)

  13. Problem Gymnast on a High Bar A gymnast on a high bar swings through two revolutions in a time of 1.90 s. Find the average angular velocity of the gymnast.

  14. Answer

  15. Problem It takes a motorcycle rider 2 seconds to make a counterclockwise lap around a track. What is his average angular velocity?

  16. Angular Acceleration Changing angular velocity means that an angular acceleration is occurring. DEFINITION OF AVERAGE ANGULAR ACCELERATION SI Unit of Angular acceleration: radian per second squared (rad/s2)

  17. Problem A Jet Revving Its Engines As seen from the front of the engine, the fan blades are rotating with an angular speed of -110 rad/s. As the plane takes off, the angular velocity of the blades reaches -330 rad/s in a time of 14 s. Find the angular acceleration, assuming it to be constant.

  18. Answer

  19. Problem Over the course of 1 hour, what is (a) The angular displacement (b) The angular velocity and (c) The angular acceleration of the minute hand?

  20. Answer VARIABLES: Elapsed Time Δt = 1 h  3600 s Angular Displacement Δθ Angular Velocity ω Angular Acceleration α STEP-BY-STEP SOLUTION * Angular Displacement Δθ = -2πrad (Minute hand travels clockwise one revolution) Δt = 1 h  3600 s * Angular Velocity ω = Δθ/Δt = -2πrad/3600s = -((2)(3.14))/3600 = -1.75 x 10-3 rad/s * Since the angular velocity is constant, angular acceleration is zero

  21. Problem At a particular instant, a potter's wheel rotates clockwise at 12.0 rad/s; 2.50 seconds later, it rotates at 8.50 rad/s clockwise. Find its average angular acceleration during the elapsed time.

  22. Answer Average Angular Acceleration: α = Δω / Δt = (-8.5 rad/s) – (-12 rad/s) / 2.5 s = 1.4 rad/s2

  23. Did NOT do slides beyond this point

  24. Angular Velocity Vector Angular Velocity Vector: * Parallel to the axis of rotation * Magnitude of the angular velocity vector is proportional to the angular speed - Faster the object rotates  longer the vector * Direction of the angular velocity vector determined by the right-hand rule

  25. Right-Hand Rule Right-hand Rule: * Used to determine the direction of the angular velocity vector * Fingers of the right hand curl in the direction of rotation * Thumb then points in the direction of the angular velocity vector

  26. Angular Velocity Vector Right-Hand Rule: Grasp the axis of rotation with your right hand, so that your fingers circle the axis in the same sense as the rotation. Your extended thumb points along the axis in the direction of the angular velocity.

  27. Angular Acceleration Vector Angular Acceleration Vector: * If the object is speeding up  its angular velocity vector is increasing in magnitude  Angular acceleration vector points in the same direction as the angular velocity vector * If the object is slowing down  its angular velocity vector is decreasing in magnitude  Angular acceleration vector points in the opposite direction as the angular velocity vector Preview Kinetic Books: 10.15

  28. Recall… The Kinematic Equations For Constant Acceleration Five kinematic variables: 1. displacement, x 2. acceleration (constant), a 3. final velocity (at time t), v 4. initial velocity, vo 5. elapsed time, t

  29. Rotational Kinematic Equations: For Constant Angular Acceleration ANGULAR ACCELERATION ANGULAR VELOCITY TIME ANGULAR DISPLACEMENT

  30. Rotational Kinematic Equations

  31. Reasoning Strategy 1. Make a drawing. 2. Decide which directions are to be called positive (+) and negative (-). 3. Write down the values that are given for any of the five kinematic variables. 4. Verify that the information contains values for at least three of the five kinematic variables. Select the appropriate equation. 5. When the motion is divided into segments, remember that the final angular velocity of one segment is the initial velocity for the next. 6. Keep in mind that there may be two possible answers to a kinematics problem.

  32. PROBLEM: Blending with a Blender The blades are whirling with an angular velocity of +375 rad/s when the “puree” button is pushed in. When the “blend” button is pushed, the blades accelerate and reach a greater angular velocity after the blades have rotated through an angular displacement of +44.0 rad. The angular acceleration has a constant value of +1740 rad/s2. Find the final angular velocity of the blades.

  33. Tangential Velocity Tangential Velocity: * Linear velocity at an instant - Magnitude = magnitude of linear velocity - Direction: Tangent to circle * vT = rω vT = tangential speed r = distance to axis ω = angular velocity Direction: tangent to circle Preview Kinetic Books: 10.11

  34. Tangential Velocity

  35. Tangential Velocity

  36. Tangential Acceleration Tangential Acceleration: - Rate of change of tangential speed - Increases with distance from center - Direction of vector is tangent to circle - aT = rα aT = tangential acceleration r = distance to axis α = angular acceleration Preview Kinetic Books: 10.12

  37. Tangential Acceleration

  38. PROBLEM: A Helicopter Blade A helicopter blade has an angular speed of 6.50 rev/s and an angular acceleration of 1.30 rev/s2. For point 1 on the blade, find the magnitude of (a) The tangential speed (b) The tangential acceleration.

  39. Centripetal Acceleration

  40. Problem: A Discus Thrower Starting from rest, the thrower accelerates the discus to a final angular speed of +15.0 rad/s in a time of 0.270 s before releasing it. During the acceleration, the discus moves in a circular arc of radius 0.810 m. Find the magnitude of the total acceleration.

  41. Centripetal Acceleration & Tangential Acceleration

  42. Rotational Motion Review • UNIFORM CIRCULAR MOTION: • Example- Spinning Ferris wheel or an orbiting satellite • Object moves in a circular path and at a constant speed • The object is accelerating, however, because the direction of the object’s velocity is constantly changing • Centripetal acceleration  Directed toward the center of the circle • Net force causing the acceleration is a centripetal force  Also directed toward the center of the circle • In this section, we will examine a related type of motion  The motion of a ROTATING RIGID OBJECT

  43. The Motion of a Rotating Rigid Object • A football spins as it flies through the air • If gravity is the only force acting on the football  Football spins around a point called its center of mass • As the football moves through the air, its center of mass follows a parabolic path

  44. Rotational Dynamics • Study of Rotational Kinematics • Focuses on analyzing the motion of a rotating object • Determining such properties as its angular displacement, angular velocity or angular acceleration • Study of Rotational Dynamics • Explores the origins of rotational motion * Preview Kinetic Books 11.1

  45. Torque • Every time you open a door, turn of a water faucet, or tighten a nut with a wrench  you are exerting a TURNING FORCE • Turning force produces a TORQUE • Review… • If you want to make an object move  Apply a force • Forces make things accelerate • If you want to make an object turn or rotate  Apply a torque • Torques produce rotation

  46. Torque • Torque • A force that causes or opposes rotation • The amount of torque depends on… • The amount of force • When more force is applied  there is more torque • The distance from the axis of rotation to the point of force

  47. The amount of torque depends on where and in what direction the force is applied, as well as the location of the axis of rotation.

More Related