Combination of Translation and Rotation
Phys 407 7/18/05
Combination of Translation and Rotation
Rolling Motion
• Rolling: there is no slipping at the
point of contact where the rotating
object touches the ground.
• Relations between the magnitude of
linear velocity (acceleration) and
angular velocity (acceleration):
v = v T = r ! , a = aT = r "
• Sign is the opposite: forward
moving has a clockwise rotation.
• Relative to the ground, the velocity of the point of contact is
zero. The velocity of the axis is v, moving forward. The velocity
at the top of the tire is 2 v, moving forward.
Phys 407 7/18/05
Combination of Translation and Rotation
Rolling Motion (Example)
• a car accelerates with acceleration
a = 0.8 m/s2 for t = 20. 0 s.
• r = 0.330 m
• Find the angular displacement.
• First, find the angular acceleration:
a 0.8 m/s 2
a = r! , or, ! = =
= 2.42 rad/s 2
r 0.33 m
• Then find the angular displacement:
0
(
)
! = " 0t + 21 #t 2 = 12 $2.42 rad/s 2 (20.0 s)2 = $484 rad
Phys 407 7/18/05
Combination of Translation and Rotation
Rolling Motion as Pure Rotation
• same !
• different I
• parallel-axis theorem
I = " cm + MR 2
!
• kinetic energy
1
1
1
KE = I" 2 = # cm" 2 + MR 2" 2
2
2
2
1
1
2
= # cm" 2 + Mv cm
2
2
Phys 407 7/18/05
!
Combination of Translation and Rotation
Example
Take two quarters and lay them on a table. Press down on one quarter so it cannot move. Then,
starting at the 12:00 position, roll the other quarter along the edge of the stationary quarter, as
the drawing suggests. How many revolutions does the rolling quarter make when it travels
once around the circumference of the stationary quarter?
• Total angular displacement = angular displacement of the rolling quarter
relative to the circumference of the stationary quarter + angular
displacement of going around the stationary quarter without rolling = 1 rev
+ 1 rev = 2 revolutions
Phys 407 7/18/05
Combination of Translation and Rotation
Conservation of Total Mechanical Energy (Example)
• Two cylinders, one solid (radius rs, mass ms), one hollow (radius rh, mass
mh) start at rest at the top of an incline and roll down a vertical height of h0.
Determine which one has a greater speed at the bottom (and thus reaching
there faster) is there is no energy loss due to friction.
• Total mechanical energy is conserved if there is no external work done: Ef = E0
Ef = 12 mvf2 + 12 I! f2 = E0 = mgh0
mvf2
vf = r! f
Ivf2
2mgh0
+ 2 = 2mgh0 vf =
r
m + I / r2
• For hollow cylinder:
• For solid cylinder:
I = mh rh2 , vfh = gh0
I = 12 ms rs2 , vfs =
4 gh
3 0
• Solid cylinder moves faster, independent
of mass and radius.
Phys 407 7/18/05
Combination of Translation and Rotation
Conservation of Total Mechanical Energy (Example)
Phys 407 7/18/05
Combination of Translation and Rotation
Forces of Rolling
• Newton’s 2nd law for the acceleration
along the ramp: f " Mgsin # = Ma
s
cm
• Newton’s 2nd law for the angular
acceleration: f sR = Icm"
! rolling condition:
• smooth
acm = "R#
gsin #
acm = "
!
1+ Icm / MR 2
!
v 2f = 2acm x =
!
!
2gh
1+ Icm / MR 2
• For hollow cylinder:
• For solid cylinder:
!
!
I = MR 2 , v fh = gh
4
I = 12 MR 2 ,Phys
v fs 407
= 7/18/05
3 gh
Combination of Translation and Rotation
Example: Yo-Yo
acm = "
g
1+ Icm / MR02
!
Concept questions
Phys 407 7/18/05
Rotational Dynamics
Torque in vector form
• magnitude of the torque around an axis is: " = Fr sin #
• the direction of the torque vector is along the axis
r
• vector torque is given by: "r = rr # F (cross product)
!
" = rF# = r# F
!
!
• one is free to choose the origin in calculating torque, as long
as the same coordinate system is used throughout a problem
Phys 407 7/18/05
1. Two wheels roll side-by-side without sliding, at the
same speed. The radius of wheel 2 is twice the radius of
wheel 1. The angular velocity of wheel 2 is:
!
A)
twice the angular velocity of wheel 1
B)
the same as the angular velocity of wheel 1
C)
half the angular velocity of wheel 1
D)
more than twice the angular velocity of wheel 1
E)
less than half the angular velocity of wheel 1
2. A forward force acting on the axle accelerates a rolling
wheel on a horizontal surface. If the wheel does not
slide the frictional force of the surface on the wheel is:
!
A)
zero
B)
in the forward direction and does zero work on
the wheel
C)
in the forward direction and does positive work
on the wheel
D)
in the backward direction and does zero work on
the wheel
E)
in the backward direction and does positive
work on the wheel
3. When the speed of a rear-drive car is increasing on a
horizontal road the direction of the frictional force on
the tires is:
!
A)
forward for all tires
B)
backward for all tires
C)
forward for the front tires and backward for the
rear tires
D)
backward for the front tires and forward for the
rear tires
E)
zero
4. A hoop, a uniform disk, and a uniform sphere, all with
the same mass and outer radius, start with the same
speed and roll without sliding up identical inclines.
Rank the objects according to how high they go, least to
greatest.
!
A) hoop, disk, sphere
B)
disk, hoop, sphere
C)
sphere, hoop, disk
D) sphere, disk, hoop'
E)
hoop, sphere, disk
5. Two uniform cylinders have different masses and
different rotational inertias. They simultaneously start
from rest at the top of an inclined plane and roll
without sliding down the plane. The cylinder that gets
to the bottom first is:
!
A)
the one with the larger mass
B)
the one with the smaller mass
C)
the one with the larger rotational inertia
D)
the one with the smaller rotational inertia
E)
neither (they arrive together)
Rotational Dynamics
Newton’s 2nd Law for a Particle in Angular Form
• Newton’s 2nd Law for a Particle: Fnet = dp = d(mv)
dt
dt
r
d (r # p) d ( mr # v) d l
• calculate torque: " net = r # Fnet =
=
$
dt
dt
dt
!
dr
" p = mv " v = 0
note that:
dt
!
r
• l = r " p = mr " v is the angular momentum of a particle
!
!
Concept questions
Phys 407 7/18/05
1. A single force acts on a particle situated on the positive
x axis. The torque about the origin is in the negative z
direction. The force might be:
!
A)
in the positive y direction
B)
in the negative y direction
C)
in the positive x direction
D)
in the negative x direction
E)
in the positive z direction
2. A rod rests on frictionless ice. Forces that are equal in magnitude
and opposite in direction are simultaneously applied to its ends
as shown. The quantity that vanishes is its:
!
A)
angular momentum
B)
angular acceleration
C)
total linear momentum
D)
kinetic energy
E)
rotational inertia