Download Rotational Mechanics

Concept Summary
Belton High School Physics
Circular Motion Terms
• The point or line that is the
center of the circle is the
axis of rotation.
• If the axis of rotation is
inside the object, the object
is rotating (spinning).
• If the axis of rotation is
outside the object, the
object is revolving.
Rotational variables
Look at one point P:
Arc length s:
s  r 
Thus, the angular
position is:
s

r
 radian measure 
Planar, rigid object rotating
about origin O.
 is measured in degrees or radians (more common)
r
s=r
One radian is the angle
subtended by an arc length
equal to the radius of the arc.

For full circle:  
Full circle has an angle of 2 radians,
Thus, one radian is 360°/2  57.3
s 2r

 2
r
r
Radian
2

/2
1
degrees
360°
180°
90°
57.3°
2 pod racers complete a 180° turn
and remain neck and neck. Which
had the greater Linear Velocity?
A.
B.
C.
The one closer to the point of rotation.
The one farther from the point of rotation.
Both were equal.
How would you describe the time it
takes for each of the 2 pod racers to
complete the turn?
A. Greater for the one closer to the
point of rotation.
B. Greater for the one farther from the
point of rotation.
C. Both the same.
Angular Velocity
w
• rotational or angular
velocity, which is
the rate angular
position changes.
• Rotational velocity is
measured in
radians/sec,
degrees/second,
rotations/minute
(rpm), etc.
• Common symbol, w
(Greek letter
omega)
Define quantities for circular motion
(note analogies to linear motion!!)
Angular displacement:
    o
Average angular speed: wavg 
  o
t  to


t
 d

t 0 t
dt
Instantaneous angular speed: w  lim
Average angular acceleration:
avg 
Instantaneous angular acceleration:
w  wo
t  to

w
t
w dw
  lim

t 0 t
dt
10-5 Relating the linear and angular
variables
Caution: Measure angular quantities in radians
Arc length s:
s  r 
Tangential speed of a
point P:
v  r w
Tangential acceleration of a
point P:
at  r  
Note, this is not the centripetal
acceleration ar !!
2
ar  v  w2 r
r
Rotational Vs. Tangential
Velocity
• If an object is rotating:
– All points on the object have the same
rotational (angular) velocity.
– All points on the object do not have the
same linear (tangential) velocity
Rotational & Tangential Velocity
• We now see that….
– Tangential (linear) velocity of a
point depends on:
• The rotational velocity of the point.
– More rotational velocity can mean more
linear velocity.
– The distance from the axis of
rotation.
• More distance from the axis means
more linear velocity
Centripetal Acceleration and
Angular Velocity
• The angular velocity and the linear
velocity are related (v = ωr)
• The centripetal acceleration can also
be related to the angular velocity
v
r ω
2
aC  
 rω
r
r
2
2
2
10-3 Are angular quantities
vectors?
Right-hand rule for
determining the direction
of this vector.
Every particle (of a rigid object):
• rotates through the same angle,
• has the same angular velocity,
• has the same angular acceleration.
, w,  characterize rotational motion of entire
object
Linear motion with constant
linear acceleration, a.
10-4 Rotational motion with
constant rotational
acceleration, .
v  v o  at
w  wo  t
x  x o  12 (v  v o )t
  o  12 (w  wo )t
1 2
x  x o  v ot  at
2
1 2
  o  wot  t
2
v 2  v o 2  2a ( x  x o )
w 2  wo 2  2 (  o )
Rotational Inertia
• Equates to “normal” Inertia (mass).
– An object rotating about an axis tends to
keep rotating about that axis.
• Rotational Inertia: resistance to
changes in rotational motion.
Calculation of Rotational inertia
Rotational inertia (or Moment of Inertia) I of an object
depends on:
- the axis about which the object is rotated.
- the mass of the object.
- the distance between the mass(es) and the axis
of rotation.
- Note that w must be in radian unit. The SI
unit for I is kg.m2 and it is a scalar.
I   mi  ri
i
2
I
lim  ri
mi 0 i
2
 mi   r dm   r dV
2
2
Note that the moments of inertia are different for different axes
of rotation (even for the same object)
1
I  ML2
3
1
I  ML2
12
Torque
• Every time you open a door, pull a lever,
or use a wrench you exert a “turning”
force.
• This turning force produces a Torque.
– Forces make things accelerate.
– Torques make things rotate
– Also known as a “couple” or “moment”
Producing Torques
• A torque is produced by
“leverage”
– Greater the “lever” or
length of the lever arm,
greater the torque.
– Greater the force you
apply to the lever arm,
greater the torque.
• AND the angle of the
applied force plays a
part.
– Torque is the Cross
Product of Force X Lever
arm.
The Cross Product
• A way of multiplying
vectors to produce a
different vector.
– NOT the same as the
Dot product
(produced a scalar).
Example:
Torque = r F sin θ
 
A  B  AB sin 
10-8 Torque
  (r)( F sin )  r Ft
  r  F  sin 
 magnitude
• Torque is positive if the direction
of rotation is counterclockwise.
• Torque is negative if the direction
of rotation is clockwise.
• The SI unit of torque is N.m (Note
that the unit of work J is also N.m
. However, the name J is exclusively
reserved for work/energy).
  (r sin )( F )  r F
• It is clear that torque can also be defined as
  r F
torque defined as a vector productof r and F
• We use the right-hand rule, sweeping the fingers of
the right hand from r to F . The outstretched
 then gives the direction of .
right thumb
• When several torques act on a body, the net torque
is the sum of the individual torques, taking into
account of positive and negative torques.
• Newton’s Second law can be applied to Torques!
– An object will rotate in the direction of the net Torque!
– If the Net Torque is zero then no rotation occurs!
Checkpoint: Assuming all of these forces are equal in magnitude, which
direction does the net Torque point?
A: to the right
B: to the left
C: into the board
D: out of the board
If a doorknob were placed in the center of
a door rather than at the edge, how much
more force would be needed to produce the
same torque for opening the door?
a)
b)
c)
d)
Equal amount
Twice as much
Four times as much
None of the above
Suppose that a meterstick is supported at
the center, and a 20 N block is hung at the
80 cm mark. Another block of unknown
weight just balances the system when it is
hung at the 10 cm mark. What is the
weight of the second block?
a)
b)
c)
d)
5 N
10 N
15 N
20 N
Homework (by Wednesday)
• Watch Walter Lewin lecture 19:
Rotating Rigid Bodies
• And lecture 20: Angular momentum
and Torques
– Available on iTunesU or youtube or my
site.