Transcript Chapter 10b

Chapter 10:Rotation of a rigid object about a fixed axis
Part 2
Reading assignment:
Chapter 11.1-11.3
Homework : (due Wednesday, Oct. 12, 2005):
Problems:
Q4, 2, 5, 18, 21, 23, 24,
• Rotational motion,
• Angular displacement, angular velocity, angular acceleration
• Rotational energy
• Moment of Inertia (Rotational inertia)
• Torque
• For every rotational quantity, there is a linear analog.
Black board example 11.3
HW 27
(a) What is the angular speed w
about the polar axis of a
point on Earth’s surface at a
latitude of 40°N
(b) What is the linear speed v of
that point?
(c) What are w and v for a point
on the equator?
Radius of earth: 6370 km
Rotational
energy
A rotating object (collection
of i points with mass mi) has
a rotational ___________
energy of
1
K R  I  ____
2
Where:
I   mi  ____
i
Rotational inertia
Demo:
Both sticks have the same weight.
Why is it so much more difficult to
rotate the blue stick?
Black board example 11.4
2
What is the
rotational inertia?
3
1
4
Four small spheres are mounted on the corners of a frame as shown.
a) What is the rotational energy of the system if it is rotated about
the z-axis (out of page) with an angular velocity of 5 rad/s
b) What is the rotational energy if the system is rotated about the yaxis?
(M = 5 kg; m = 2 kg; a = 1.5 m; b = 1 m).
Rotational inertia of an object depends on:
- the ________ about which the object is rotated.
- the __________ of the object.
- the __________ between the mass(es) and the
axis of rotation.
I   mi  ri
i
2
Calculation of Rotational inertia for
____________ ________________ objects
I
lim  ri
mi 0 i
2
 mi   r dm   r dV
2
2
Refer to Table11-2
Note that the moments of inertia are different for different
________ of rotation (even for the same object)
1
I  ML2
3
I
1
ML2
12
1
I  MR 2
2
Rotational inertia for some objects
Page 227
Parallel axis
theorem
 Rotational inertia for a rotation about an axis that is
____________ to an axis through the center of mass
I CM
I  I CM  _____
h
Blackboard example 11.4
What is the rotational energy of a sphere (mass m = 1 kg, radius R = 1m) that is
rotating about an axis 0.5 away from the center with w = 2 rad/sec?
Conservation of energy (including rotational energy):
Again:
If there are no ___________________ forces: Energy is
conserved.
Rotational _____________ energy must be included in
energy considerations!
Ei  E f
U i  K linear,initial  K rotational,initial  U f  Klinear, final  K rotational, final
Black board example 11.5
Connected cylinders.
Two masses m1 (5 kg) and m2 (10
kg) are hanging from a pulley of
mass M (3 kg) and radius R (0.1
m), as shown. There is no slip
between the rope and the pulleys.
(a) What will happen when the
masses are released?
(b) Find the velocity of the masses after they have fallen a
distance of 0.5 m.
(c) What is the angular velocity of the pulley at that moment?
Torque
F  sin f
r
F
f
F  cos f
A force F is acting at an angle f on a lever that is rotating around
a pivot point. r is the ______________ between F and the pivot
point.
This __________________ pair results in a torque t on the lever
t  r  F  sin f
Black board example 11.6
Two mechanics are trying to
open a rusty screw on a ship
with a big ol’ wrench. One
pulls at the end of the wrench
(r = 1 m) with a force F = 500
N at an angle F1 = 80 °; the
other pulls at the middle of
wrench with the same force
and at an angle F2 = 90 °.
What is the net torque the two mechanics are applying to the screw?
Torque t and
angular acceleration a.
Newton’s __________ law for rotation.
Particle of mass m rotating in a
circle with radius r.
force Fr to keep particle
on circular path.
force Ft accelerates
particle along tangent.
Ft  mat
Torque acting on particle is ________________
to angular acceleration a:
t  Ia
 
dW  F  ds
 
W  F s
Definition of work:
Work in linear motion:
 
dW  F  ds
 
W  F  s  F  s  cos 
Component of force F along
displacement s. Angle 
between F and s.
Work in rotational motion:
 
dW  F  ds
Torque t and angular
dW  t  ___
W  t  ___
displacement q.
Work and Energy in rotational motion
Remember work-kinetic energy theorem for linear motion:
1
1
2
2
W

mv

mv

f
i
2
2
External work done on an object changes its __________ energy
There is an equivalent work-rotational kinetic energy theorem:
W 
1
1
2
2
___ w f  ___ wi
2
2
External, rotational work done on an object changes its _______________energy
Linear motion with constant
linear acceleration, a.
Rotational motion with constant
rotational acceleration, a.
v xf  v xi  a x t
w f  _________
x f  xi  12 (vxi  vxf )t
q f  ________________
1 2
x f  xi  v xi t  a x t
2
q f  ____________________
vxf  vxi  2ax ( x f  xi )
w f  ___________________
2
2
2
Summary: Angular and linear quantities
Linear motion
1
2
K

m

v
Kinetic Energy:
2
Force:
F  ma
Momentum:
p  mv
Work:
 
W  F s
Rotational motion
Kinetic Energy: K R  _________
Torque:
t  ______
Angular Momentum:
Work:
L  __
W  _____