Transcript Document

Rotation (III)
• Torque and angular acceleration
• “Moment of inertia”
Text Sections : 10.7, and part of 10.4
Physics 1D03
Angular velocity vector: parallel to the axis
of rotation, following a similar right-hand rule:
w
rotation
direction
w
Angular acceleration vector: parallel to the
angular velocity, if |w| is increasing.
Force causes linear acceleration: Fnet = ma
Torque causes angular acceleration: tnet = Ia
?
Physics 1D03
Q: How much torque does it take to rotate a particular object?
Better: How much torque does it take to change the rate of
rotation?
What property of an object determines the response
(angular acceleration) to an unbalanced external torque?
Force causes linear acceleration: Fnet = ma
Torque causes angular acceleration: tnet = Ia
?
Physics 1D03
The angular acceleration of a particle is proportional to the
net torque applied to it.
Example: A particle accelerates in a circle. Break the net force on it
into radial and tangential components. Only Ft causes tangential
acceleration:
Ft = mat = m(ra ), since at = ra
Multiply by r : rFt = mr
or
torque = (mr 2)
t a
2a
a
Fr
r
Ft
Physics 1D03
For a rigid body, a is the same for all
particles. The net force on each particle is
composed of internal forces f, and external
forces Fext.
-f
The total torque is the sum of the torques on
the individual particles:
2
2
t

m
r
a

(
m
r
 i  ii
 i i )a
i
i
f mi
a
ri
i
Torques due to the internal forces f and –f cancel when we sum over
all particles. The quantity in brackets is the “moment of inertia”:
I   mi ri 2
i
and Newton’s 2nd law for rotation is
t
external
 Ia
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Fext
The Moment of Inertia:
I   mi ri 2
Units: kg m2
i
(r is the distance from a particle to the axis of rotation.)
“I” measures the rotational inertia of an object. It depends on:
1) the total mass. I is proportional to mass.
2) how the mass is distributed (distance from the rotation axis).
I is proportional to (linear size)2, and is larger if the mass is
concentrated farther from the rotaion axis.
3) which axis the object rotates about.
Physics 1D03
Moments of inertia for uniform objects can be calculated using integral
calculus. A few results (see table 10.2 in the text):
I = ½ MR2
I=MR2
L
L
I = 1/12 ML2
I = 1/3 ML2
R
R
2
I  MR 2
5
2
I  MR 2
3
Physics 1D03
Quiz
Three point particles, each of mass m, are arranged in the x-y plane in
an equilateral triangle of side 2b as shown. What is the moment of
inertia:
y
1) about the x axis? - clicker
2) about the y axis?
3) about the z-axis?
a)
b)
c)
d)
e)
mb2
2mb2
3mb2
4mb2
5mb2
2b
2b
x
b
b
Physics 1D03
Quiz
Two spheres are made of the same material. However,
all the dimensions of sphere B are twice as large as
those of sphere A. If the spheres are launched with the
same torques, applied for equal times, sphere A
accelerates faster than B by a factor of :
a) 2
b) 4
c) 8
d) 16
e) 32
Physics 1D03
Quiz
Two spinning tops have equal radius and mass, but
whereas top B is a thin disc, top A has light spokes
connecting the hub to an outer ring. The tops are spun
by applying equal torques to the spindle. Which spins
the fastest?
Top View
a) A
b) B
c) spin at same rate
d) not enough info.
A
B
Physics 1D03
Quiz
Two spinning tops are made from circular disks with a
pointed spindle inserted through the center. The tops
have the same radii and total mass, but since they are
made of different material, top A is thicker. The tops are
launched by equal torques to the spindles. Which spin is
the fastest?
A
B
a) A
b) B
c) spin at the same rate
d) not enough info
Physics 1D03
Quiz
A uniform board is attached to the edge of a table at
one end by a hinge. A coin is placed on the other end,
and the free end of the board is held so that the board
is horizontal.
coin
When the board is released:
a) the board falls faster than the coin
b) the board and coin fall freely together
c) the coin would fall faster, but presses on the board
d) it depends on the length of the board
Physics 1D03
Example
Two (thin, uniform) sticks of different lengths are held nearly upright
(at a slight angle to the vertical) on a table and released
simultaneously. Which hits the ground first? (Try this at home.)
Physics 1D03
Summary
The moment of inertia measures the rotational inertia of a body.
Particle: I = mr2
Extended Body:
I   mi ri 2
i
Newton’s 2nd law for rotation about a fixed axis:
t
external
 Ia
Physics 1D03