Rotation

1
CHAPTER-10

• Rotation

2
Ch 10-2 Rotation

• Rotation of a rigid body about a fixed axis
• Every point of the body moves in a circle, whose
  center lies on the axis of rotation
• Every point of the body moves through the same
  angle during a particular interval of time
• Angular position ? Angle of reference line
  (fixed on rigid body and ? rotational axis)
  relative to Zero angular position
• ? (rad) s/r 1 rad 57.3?

3
Ch 10-2 Rotational Variable

• Linear displacement
• ?x xf - xi
• Average linear velocity
• vavg?x/?t(xf - xi)/?t
• Instantaneous linear velocity
• v lim ?v/?t dv/dt
• Average linear acceleration
• aavg ?v /?t(vf - vi )/(tf ti)
• Instantaneous linear acceleration
• a lim ?v/?t
• dv/dt d2?/dt2

• Angular displacement
• ?? ?f - ?i
• Average angular velocity
• ?avg??/?t(?f - ?i)/?t
• Instantaneous angular velocity
• ? lim ??/?t d?/?t
• Average angular acceleration
• ?avg ?? /?t(?f - ?i )/(tf ti)
• Instantaneous angular acceleration
• ? lim ??/?t
• d?/dt d2?/dt2

4
Ch 10-3 Are Angular quantities Vectors?

• Yes, they are
• Right hand curl rule

5
Ch 10-Check Point 1

• A disk can rotate about its central axis like the
  one . Which of the following pairs of values for
  its initial and final angular position ,
  respectively, give a negative angular
  displacement?
• A) -3 rad, 5 rad
• B) -3 rad, -7 rad
• C) 7 rad, -3 rad

• ?? ?f-?i 5-(-3)8 rad
• ?? ?f -?i -7-(-3)-4 rad
• ?? ?f - ?i -3-7 -10 rad
• Ans b abd c

6
Ch 10-4 Rotation with Constant Angular
Acceleration- Equations of Motion

• Linear Motion
• ?xtvavg t(vfvi)/2
• vf viat
• vf2 vi22ax
• x vitat2/2

• Rotational Motion
• ?? t?avgt(?f?i)/2
• ?f ?i?t
• ?f2 ?i22??
• ? ?it ?t2/2

7
Ch 10 Check Point 2

• In four situations, a rotating body has an
  angular position ?(t) given by
• a) ? 3t-4
• 2) ? -5t34t26
• 3) ? 2/t2-4/t
• 4) ? 5t2-3
• To which of these situations do the equations of
  Table 2-1 apply?

• Ans Table 10-1 deals with constant angular
  acceleration case hence calculate acceleration
  for each equation
• 1) ? d2 ?/dt20
• 2) ? d2 ?/dt2-30t8
• 3) ? d2 ?/dt2 12/t4-8/t2
• 4) ? d2 ? /dt2 10
• Ans 1 and 4 ( constant angular acceleration case)

8
Ch 10-5 Relating the Linear and Angular Variables

• Position sr?
• Speed ds/dtr d?/dt
• v r?
• Period T 2?r/v 2?/?
• Acceleration
• Tangential acceleration atdv/dtr d?/dt r?
• Radial acceleration aRv2/r r ?2

9
Ch 10 Check Point 3

• A cockroach rides the rim of a rotating
  merry-go-round . If the angular speed of the
  sytem ( merry-o-round cockroach) is constant ,
  does the cockroach have
• a) radial acceleration
• b) tangential acceleration
• If ? is decreasing , does the cockroach have
• radial acceleration
• b) tangential acceleration

• at ? r
• aR ?2 r
• Then
• Yes aR b) No at
• If ? is decreasing then
• a) yes b) yes

10
Ch 10-6 Kinetic Energy of Rotation

• Kinetic energy of a rapidly rotating body sum of
  particles kinetic energies (vcom0)
• K ?Kparticle ½(?miv2i) but viri?I
• Then K?Ki½ ? mi(ri?i)2
• ½ ?(mri)2 ?2where ?i2 ?2
• I ?(mri)2 I is rotational inertia or
  moment of inertia
• Then rotational kinetic energy K ½ I?2
• Rotational analogue of m is I
• A rod can be rotated easily about an axis through
  its central axis (longitudinal) case a than an
  axis ? to its length case b

11
Ch 10-7 Calculating Rotational Inertia

• I ?(mri)2 ?mdr2
• Parallel-Axis Theorem
• IIcomMh2
• Example (a) For rod IcomML2/12
• And for two masses m , each has moment of inertia
  ImmL2/4 and then ItotIrod2Im
• Itot ML2/12 2(mL2/4)
• L2(M/12 m/2)
• For case (b)
• Then IrodIcomMh2 ML2/12M(L/2)2
• ML2/3
• Itot Irod Im ML2/3 mL2
• L2(M/3 m)

12
(No Transcript)
13
Ch 10 Check Point 4

• The figure shows three small spheres that rotates
  about a vertical axis. The perpendicular distance
  between the axis and the center of each sphere is
  given. Rank the three spheres according to their
  rotational inertia about that axis, greatest
  first.

• Imr2
• 1) I36 x 1236 kg.m2
• 2) I9 x 2236 kg.m2
• 3) I4 x 3236 kg.m2
• Ans All tie

14
Ch 10 Check Point 5

• The figure shows a book-like object (one side is
  longer than the other) and four choices of
  rotation axis, all perpendicular to the face of
  the object. Rank the choices according to the
  rotational inertia of the object about the axis,
  greatest first.

• Parallel Axis Theorem
• IIcomMh2
• Moment of inertia in decreasing order
• I1 I2 I4 and I3

15
Ch 10-8 Torque

• Torque is turning or twisting action of a body
  due to a force F
• If a force F acts at a point having relative
  position r from axis of rotation , then
• Torque ? r F sin?rFt r?F, where (? is
  angle between r and F)
• Ft is component of F ? to r, while r? is ?
  distance between the rotation axis and extended
  line running through F.
• r?is called moment arm of F.
• Unit of torque (N.m)
• Sign of ? Positive torque for counterclockwise
  rotation
• Negative torque for clockwise
  rotation

16
Ch 10-9 Newtons Second Law for Rotation

• Newtons Second Law for linear motion
• Fnet ma
• Newtons Second Law for linear motion
• ?net I?
• Proof
• ?netFtrmatrm(r?)rmr2 ?
• where Ftmat atr?
• ?netFtrmatrmr2?I?
• ? expressed in radian/s2

17
Ch 10 Check Point 6

• ? rt x F
• ?F2 0 ?F5
• ?F3 ?F1 maximum
• ?F4 next to maximum
• Ans F1 and F3 (tie), F4, then F1 and F5( Zero,
  tie)

• The figure show an overhead view of a meter stick
  that can pivot about the dot at the position
  marked 20 (20 cm). All five forces on the stick
  are horizontal and have the same magnitude. Rank
  the forces according to magnitude of the torque
  they produce, greatest first

?
18
Ch 10-10 Work and Rotational Kinetic Energy

• Linear Motion
• Work-Kinetic Energy theorem
• ?KKf-Kim(vf2-vi2)/2W
• Work in one dimension motion W?F.dx
• Work in one dimension motion under constant force
• WF?dx Fx ?X
• Power (one dimension motion)
• P dW/dt F.v

• Rotation
• Work-Kinetic Energy theorem
• ?KKf-KiI(?f2-?i2)/2W
• Work in rotation about fixed axis W??.d?
• Work in rotation about fixed axis under constant
  torque ?
• W??d? ???
• Power(rotation about fixed axis )
• P dW/dt ?.?