Announcements

1
Announcements

• Homework 8 is due 1100 Wednesday, November 5
• Midterm 2 will be on Friday, Nov. 14
• 830 950 AM in BH 166
• Will cover chapters 7 10
• Similar format as Midterm 1

2
Rotations

• So far, we have mostly ignored effects of
  rotations
• Motion of a point-like particle
• Rotation of a point-like particle has no effect
• Systems of particles
• Focused on center of mass motion
• COM behaves like a point-like particle acted on
  by net force on object
• Uniform circular motion
• Focused on centripetal acceleration and motion of
  center of mass
• Ignored rotation of extended object as it turns
• Example Merry-go-round if I initially face
  east, when the merry-go-round rotates by 180, I
  am facing west (thus, not only did my COM move in
  a circle, but I rotated about my COM)
• Motion that simply displaces an object without
  changing its orientation is called translation
• Motion that changes the orientation of the object
  includes Rotation
• No discussion of mechanics is complete without
  rotations, so here we go

3
Describing Rotational Motion

• We will focus on motion of a rigid body about a
  fixed axis
• Example fan axis is through the center of the
  fan
• Example stationary bicycle axis is along the
  bicycles axis
• Axis of rotation may or may not go through the
  COM
• Example Tarzan swinging on a vine axis of
  rotation is the branch the vine is attached to
• For an object rotating about a fixed axis, each
  point on the object makes a circle about the axis
  of rotation
• Radius of circle depends on distance of point to
  the axis of rotation
• Example Rotating rod
• Axis of rod is different from axis of rotation
• Any point on the rod makes a circle about the
  axis
• Radius is distance to vertical axis
• All points of the rod move together
• This is what we mean by a rigid body

4
Describing Rotations

• We need some way to quantify the orientation of a
  rotating object
• Points not on the axis of rotation are moving as
  the object rotates
• After one complete revolution, each point on the
  object is exactly where it was before the
  revolution
• At any given time, can specify orientation of
  entire object by specifying position of any point
  not on the axis of rotation
• We will often describe an objects orientation by
  the location of a specified reference point on
  the object
• Reference point measured relative to
  (non-rotating) coordinate axes aligned with axis
  of rotation
• Orientation specified by angle of reference point
  relative to coordinate axes

z
y
y
q
r
r
x
x
5
Angular Position and Displacement

• We will always specify angular position q in
  units of radians
• 1 revolution 360 2p radians
• We want q to be continuous, so after two
  revolutions, q 4p radians
• In terms of distance s travelled by the reference
  point, we have
• Specifying q(t) specifies how object rotates as a
  function of time
• Analogous to specifying x(t) for translational
  motion
• An angular displacement Dq specifies a change in
  the angular position

6
Angular Velocity

• A rotating object will undergo an angular
  displacement Dq during a time interval Dt
• Can define an angular velocity w as
• For a rigid body, all points on the body have the
  same angular velocity
• Example seconds hand on a clock has the angular
  velocity
• Angular velocity in rotations is analogous to
  velocity in translations

7
Angular Acceleration

• An object whose rate of rotation is changing will
  have a change in angular velocity Dw during a
  time interval Dt
• Can define an angular acceleration a as
• For a rigid body, all points on the body have the
  same angular acceleration
• Example An airplane propeller spins down from
  100 revolutions/s to stop in 1 minute
• Angular acceleration in rotations is analogous to
  acceleration in translations

8
Constant Angular Acceleration

• A number of useful relationships can be worked
  out for the case of constant angular acceleration
• Just as was possible for acceleration, can use
  these two equations to eliminate any variable
  that appears in both
• Example eliminate t

9
Constant Acceleration Example

• A screw-gun uniformly accelerates a screw from
  rest to an angular velocity of 10 rad/s in one
  turn. What is the angular acceleration?
• How long does it take to make the first rotation?

10
Angular Velocity Vector

• Rotations about the x, y, and z coordinate axes
  are distinct
• Example Rotation of blocks about x, y, z axis
• In each case, object rotates in plane
  perpendicular to axis of rotation
• To describe the angular velocity of a rigid
  object, we need to specify both the magnitude and
  direction of the angular velocity
• We will use axis of rotation to specify the
  direction of the angular velocity using the
  right-hand rule
• Let your fingers curl in the direction of the
  rotation, your thumb will point in the direction
  of the angular velocity
• It turns out that angular velocity is a true
  vector, just like translational velocity
• Not so easy to prove - adding two angular
  velocity vectors means you need to deal with
  rotating coordinate systems

11
Angular Displacements are not Vectors!

• While its beyond the scope of Ph5 to prove
  angular velocity is a vector, its easy to show
  that angular displacements are not vectors
• Addition of two vectors commutes A B B A
• Consider two successive rotations, one about the
  x axis and the other about the y axis
• The result depends of such a rotation depends on
  which rotation you do first
• Example Rotations of blocks of wood
• It turns out that angular acceleration is also a
  vector

12
Angular Velocity and Frequency/Period

• Period T Time it takes for an object to make one
  complete revolution
• Frequency f Number of revolutions per second
• For an object moving with a constant angular
  velocity, the object makes an angular
  displacement of 2p during the period T

13
Linear Velocity and Acceleration

• Any given point on the object is moving in
  circular motion
• Can measure position s along the circular path
• Like q, take s to be continuous (even when you
  complete a full circle)
• Magnitude of velocity given by rate of change in
  s
• Direction of velocity is always tangential to
  circular path
• Change in speed gives tangential acceleration
• Change in direction gives radial acceleration
• Minus sign used to denote acceleration is inward
  towards the axis of rotation

14
Kinetic Energy of Rotating Object

• If we have a system of objects, the total kinetic
  energy is given by
• For a rotating rigid body, all elements have the
  same angular velocity w
• The moment of inertia I depends only on the
  geometry of the object

15
Moment of Inertia Example

• Consider a dumb-bell of length l rotated about
  its center
• We will ignore finite size of weights at ends
• What happens if we instead rotate it about the
  end?
• Moment of inertia depends on axis of rotation
• Example weights on a bar
• Larger moment of inertia, kinetic energy

16
Moment of Inertia of a Solid Object

• For a system of objects, moment of inertia
  obtained by summing over objects
• For a solid object, take the limit m ? dm, sum ?
  integral
• If the object has density r, moment of inertia
  becomes

17
Example Moment of Inertia for a Disk

• Consider a disk of thickness L, radius R
• Consider a thin cylinder that makes up part of
  the disk
• Cylinder has an inner radius of r, outer radius
  of rdr
• Volume of cylinder given by area x thickness
• Now, calculate moment of inertia

18
Example Objects Rolling Down Plane

• Which will reach the bottom faster?
• All objects have approximately equal masses
• Solid disk
• Thin disk
• Sphere