Classical Mechanics Lecture 9: Rotational Kinematics and Moment of Inertia: Classical Mechanics Lecture 9: Rotational Kinematics and Moment of Inertia Today’s Concepts: Rotation of Rigid bodies Moment of Inertia Rotational Kinetic energy Mechanics Lecture 14, Slide 1
A ladybug is on a platform spinning with constant speed, CCW. What is the direction of her velocity at the instant shown?: A ladybug is on a platform spinning with constant speed, CCW. What is the direction of her velocity at the instant shown? E it is zero D A C B Recap: Uniform C ircular Motion
PowerPoint Presentation: T: period [s] Recap: Uniform Circular Motion [Hz] [m/s] [ rads /s] Seconds per revolution Revolutions per second
Question: Question A disk spins at 2 revolutions/sec. What is its period? A) T = 2 sec B) T = 2 p sec C) T = ½ sec Mechanics Lecture 14, Slide 4 What is its angular velocity?
A ladybug and a beetle are on a platform spinning with constant speed, CCW. Which one has a greater tangential speed v at the instant shown?: A ladybug and a beetle are on a platform spinning with constant speed, CCW. Which one has a greater tangential speed v at the instant shown? Ladybug Beetle Same Can’t tell, not enough info http://phet.colorado.edu/en/simulation/rotation
Mathematical description of rotation (general): Mathematical description of rotation (general)
Kinematic Equations: Kinematic Equations
PowerPoint Presentation: A student sees the following question on an exam: A flywheel with mass 120 kg, and radius 0.6 m, starting at rest, has an angular acceleration of 0.1 rad/s 2 . How many revolutions has the wheel undergone after 10 s? Which formula should the student use to answer the question? A B C
CheckPoint: Spinning wheel: A wheel which is initially at rest starts to turn with a constant angular acceleration. After 4 seconds it has made 4 complete revolutions. How many revolutions has it made after 8 seconds? A) 8 B) 12 C) 16 CheckPoint : Spinning wheel a Mechanics Lecture 14, Slide 9
PowerPoint Presentation: A ladybug is on a platform spinning CCW and speeding up. What is the direction of her acceleration at the instant shown?
Relating linear to angular variables: Relating linear to angular variables Angular displacement Angular velocity Angular acceleration For ladybug at point P P She has travelled a distance Centripetal acceleration t
Kinetic energy of a rotating rigid object: Kinetic energy of a rotating rigid object I: tendency of an object to resist any change in its ( rotational) motion I is moment of inertia Depends on distribution of mass, rotation axis
CheckPoint: Triangle: A triangular shape is made from identical balls and identical rigid, massless rods as shown. The moment of inertia about the a , b , and c axes is I a , I b , and I c respectively. Which of the following orderings is correct? CheckPoint : Triangle a b c A) I a > I b > I c B) I a > I c > I b C) I b > I a > I c Mechanics Lecture 14, Slide 13
Calculation Moment of Inertia: Bigger when the mass is further out Calculation Moment of Inertia Mechanics Lecture 14, Slide 14 Rod of length L Rotation axis at center: Rotation axis at edge: disk Page 291
PowerPoint Presentation: Example A wagon wheel is constructed as shown in the figure . The radius of the wheel is R , and the rim has mass M kg. Each of the eight spokes, have mass m. What is the moment of inertia of the wheel about an axis through its center and perpendicular to the plane of the wheel? 2R
Parallel Axis Theorem: Parallel Axis Theorem Mechanics Lecture 15, Slide 16 Rod example
Question: A solid ball of mass M and radius is connected to a rod of mass m and length L as shown. What is the moment of inertia of this system about an axis perpendicular to the other end of the rod? M L m R axis Question A) B) C) D ) Mechanics Lecture 15, Slide 17
CheckPoint: Dumbbell: CheckPoint : Dumbbell A ball of mass 3 M at x = 0 is connected to a ball of mass M at x = L by a massless rod. Consider the three rotation axes A , B , and C as shown, all parallel to the y axis . For which rotation axis is the moment of inertia of the object smallest? (It may help you to figure out where the center of mass of the object is.) A B C B 3 M M C A L /2 L /4 0 x y L
PowerPoint Presentation: Mechanics Lecture 14, Slide 19 In both cases shown below a hula hoop with mass M and radius R is spun with the same angular velocity about a vertical axis through its center. In Case 1 the plane of the hoop is parallel to the floor and in Case 2 it is perpendicular. In which case does the spinning hoop have the most kinetic energy? A) Case 1 B) Case 2 C) Same CheckPoint: Rotating Hoop w R R w Case 2 Case 1
Right Hand Rule for finding direction of angular velocity, ω: Right Hand Rule for finding d irection of angular velocity, ω Why use this rule? Mechanics Lecture 15, Slide 20
Question: Question A ball rolls across the floor, and then starts up a ramp as shown below. In what direction does the angular velocity vector point when the ball is rolling up the ramp? A) Into the page B) Out of the page C) Up D) Down Mechanics Lecture 15, Slide 21 ⊗ ⊙ ↑ ↓
Question: Question A ball rolls across the floor, and then starts up a ramp as shown below. In what direction does the angular acceleration vector point when the ball is rolling up the ramp? A) Into the page Out of the page C) Up D) Down Mechanics Lecture 15, Slide 22 ⊗ ⊙ ↑ ↓
Question: Question A ball rolls across the floor, and then starts up a ramp as shown below. In what direction does the angular acceleration vector point when the ball is rolling back down the ramp? A) into the page out of the page C) Up D) Down Mechanics Lecture 15, Slide 23 ⊗ ⊙ ↓ ↑
PowerPoint Presentation: Kinematic Equations: Example Dario , a prep cook at an Italian restaurant, spins a salad spinner 20.0 times in 5.00 seconds and then stops spinning it. The salad spinner rotates 6.00 more times before it comes to rest. Assume that the spinner slows down with constant angular acceleration . What is the angular velocity of the salad spinner while Dario is spinning it? What is the angular acceleration of the salad spinner as it slows down? How long does it take for the salad spinner to come to rest?
PowerPoint Presentation: Relating linear and angular variables: Example In a charming 19th-century hotel, an old-style elevator is connected to a counterweight by a cable that passes over a rotating disk 3.0 m in diameter (the figure ). The elevator is raised and lowered by turning the disk, and the cable does not slip on the rim of the disk but turns with it . At how many rpm must the disk turn to raise the elevator at 30.0 cm/s? To start the elevator moving, it must be accelerated at g/8 m/s 2 . What must be the angular acceleration of the disk? Through what angle (in radians ) has the disk turned when it has raised the elevator 2.95 m between floors?
PowerPoint Presentation: Rotational kinetic energy: Example A frictionless pulley has the shape of a uniform solid disk of mass 4 kg and radius 0.5 m. A 2 kg stone is attached to a very light wire that is wrapped around the rim of the pulley (the figure ), and the system is released from rest. How far must the stone fall so that the pulley has 25 J of kinetic energy?