Download rotational motion

ROTATIONAL MOTION
Student is expected to understand
the physics of rotating objects.
TORQUE
• Torque is the rotational equivalent of force.
• It measures the “effectiveness” of a force
at causing an object to rotate about a
pivot.
• A torque causes angular acceleration.
• Problem 9.13, 9.19
FIND THE CENTER OF GRAVITY
• For symmetrical
objects, CG lies at its
geometrical center.
• For irregular-shaped
objects,
x1m1  x2 m2  x3m3  ...
xcg 
m1  m2  m3  ...
y1m1  y2 m2  y3 m3  ...
ycg 
m1  m2  m3  ...
CENTER OF GRAVITY
• A uniform carpenter's square has the shape of
an L, as shown in the figure. Locate the center
of mass relative to the origin of the coordinate
system.
CAN DO!
• Take a soda can that
is full and try and
balance it on its edge.
What happened?
• This time, have the
soda can only about
1/3 of the way full (3.5
oz or 100 ml), and try
it again. What
happened?
CENTER OF GRAVITY
CHALLENGE
• Balance the nails
ROTATION OF A RIGID BODY
• Rigid body is an extended object whose
size and shape do not change as it moves.
• Every point in a rotating rigid body has the
same angular velocity.
Linear & Circular Motion
• Position, x
• Angular position, θ radian
• Velocity, v
• Angular velocity, ω rad/s
x
v
t
• Acceleration, a
v
a
t


t
• Angular acceleration, α


t
Spinning up a computer disk
• The disk in a computer disk drive spins up
to 5,400 revolutions per minute (rpm) in
2.00 s. What is the angular acceleration of
the disk? At the end of 2.00 s, how many
revolutions has the disk made?
Useful conversion:
1 revolution = 2π radians
Tangential Acceleration & Angular
Acceleration in Uniform Circular Motion
• Tangential
acceleration
measures the rate at
which the particle’s
speed around the
circle increases.
• If the angular velocity
of a bicycle wheel is
changing, then the
wheel has an angular
acceleration.
Relationship between Tangential
Acceleration & Angular Acceleration
Linear & Circular Motion
• Displacement at constant
speed
x  v(t )
• Change in velocity at constant
acceleration
v  a(t )
• Displacement at constant
acceleration
1
x  vo (t )  a(t ) 2
2
• Angular displacement at
constant angular speed
   ( t )
• Change in angular velocity at
constant angular acceleration
   (t )
• Angular displacement at
constant angular acceleration
1
  o (t )   (t ) 2
2
THINK!
• A ball on the end of a
string swings in a
horizontal circle once
every second. State
whether the
magnitude of each of
the following
quantities is ZERO,
CONSTANT (BUT
NOT ZERO), or
CHANGING:
• A) velocity
• B) angular velocity
• C) centripetal
acceleration
• D) angular
acceleration
• E) tangential
acceleration
MOMENT OF INERTIA
• It is the rotational equivalent of mass.
• It is a measure of object’s resistance to
angular acceleration about an axis.
ROTATIONAL DYNAMICS
Newton’s second law for rotation

 net
I
where, I is the moment of inertia.
Demo: Hammering home inertia!
MOMENT OF INERTIA
I   mi ri  m r  m r  m r  ...
2
2
1 1
2
2 2
2
3 3
MOMENT OF INERTIA
Table 9.1, page 260
• Problem 9.39
ROTATIONAL KINETIC ENERGY
1 2
KE  I
2
Demo: Rolling cylinders, gyroscope
Solve Problem 9.49
ANGULAR MOMENTUM
•
•
•
•
•
Linear dynamics
Force, F
Mass, m
Velocity, v
Momentum, p
•
•
•
•
•
Rotational dynamics
Torque, τ
Moment of Inertia, I
Angular velocity, ω
Angular momentum, L
Conservation of Angular Momentum
• Joey, whose mass is 36 kg, stands at the
center of a 200 kg merry-go-round that is
rotating once every 2.5 s. While it is
rotating, Joey walks out to the edge of the
merry-go-round, 2.0 m from its center.
What is the rotational period of the merrygo-round when Joey gets to the edge?
Conservation of Angular Momentum
• An ice skater spins around on the tips of his
blades while holding a 5.0 kg weight in each
hand. He begins with his arms straight out from
his body and his hands 140 cm apart. While
spinning at 2.0 rev/s, he pulls the weights in and
holds them 50 cm apart against his shoulders. If
we neglect the mass of the skater, how fast is he
spinning after pulling the weights in?