Rotational Motion
Rotational Motion
Rotational motion is the motion of a
body about an internal axis. In
rotational motion the axis of motion
is part of the moving object.
All of the properties of linear motion
which we have discussed so far this
year have corresponding rotational
(angular) properties.
Rotational Motion
linear property
angular property
distance(d) = angular displacement (θ)
velocity(v)
= angular velocity (ω)
acceleration(a) =
ang. accel. (α)
inertia (m) = rotational inertia (I)
force (F)
=
torque (τ)
Rotational Motion
The motion of an object which moves
in a straight line can only be
described in terms of linear
properties. The motion of an object
which rotates can be described in
terms of linear or rotational
properties.
Angular Displacement
Since all rotational quantities have linear
equivalents, we can convert between them.
Angular displacement is the rotational
equivalent of distance. To find the distance
a point on a rotating object has traveled
(its arc length) we need to multiply the
angular displacement by the radius.
d  ()(r)
t
Angular Velocity
where angular displacement is
measured in “radians”:
angular velocity= (angular
displacement /time)
ω = (Δθ)/t
where angular velocity can be
measured in radians/sec, or
revolutions/sec.
Angular Velocity
Angular speed is the rotational
equivalent of linear speed. To find
the linear speed of a rotating object
(its tangential speed) we need to
multiply the angular speed by the
radius.
v  (r)( )
t
Angular Acceleration
Angular acceleration is also similar to
linear acceleration.
Angular acceleration=angular speed/
time
α = ω/t
Angular Acceleration
Angular acceleration is the rotational
equivalent of linear acceleration. To
find the linear acceleration of a
rotating object (its tangential
acceleration) we need to multiply the
angular acceleration by the radius.
a  (r)( )
t
Rotational Motion
While an object rotates, every point
will have different velocities, but
they will all have identical angular
velocities.
All of the equations of linear motion
which we have discussed so far this
year have corresponding rotational
(angular) equations.
Rotational Motion
linear equation
angular equation
v = ∆x/∆t
=>
ω =∆ θ /∆t
a = ∆v/∆t
=>
α=∆ ω /∆t
vf = vi + a∆t => ω f = ω i +α∆t
∆d = vi∆t + 1/2a(∆t)2
=>
∆ θ = ω i∆t + 1/2α(∆t) 2
vf = √(vi2 + 2a∆x)
=>
ω f = √(ω i2 + 2a∆d)
F=ma
=> τ=I α
Rotational Motion
Practice problems p. 145-147
An object moving at constant speed in a
circular path will have a zero change in
angular speed, and therefore a zero
angular acceleration.
That object is changing its direction,
however, and therefore has a changing
linear velocity and a non-zero linear
acceleration. It has an acceleration
directed toward the center of the circle
causing it not to move in a straight line.
This is a centripetal (center seeking)
acceleration.
v
a   (r)( )
r
2
t
c
2
Centripetal Acceleration
This acceleration is perpendicular to
the tangential (linear) acceleration.
All accelerations are caused by forces
and centripetal acceleration is caused
by centripetal force. A force directed
towards the center of a circle which
causes an object to move in a
circular path.
v
F  ma  m(
)  mr 
r
2
t
c
c
2
Centrifugal Force
A centripetal force pulls an object
towards the center of a circle while
its inertia (not a force) tries to
maintain straight line motion.
This interaction is felt by a rotating
object to be a force pulling it
outward. This "centrifugal" force
does not exist as there is nothing to
provide it. It is merely a sensation
felt by the inertia of a rotating object
Rotational Motion
Practice problems p. 149-150
Rotational Inertia
Force causes acceleration.
Inertia resists acceleration.
Torque causes rotational
acceleration.
Rotational inertia resists rotational
acceleration.
τ=I α = r F
I = m r2
Rotational Inertia
Inertia is measured in terms of
mass. Rotational inertia is measured
in terms of mass and how far that
mass is located from the axis.
The greater the mass or the greater
the distance of that mass from the
axis, the greater the rotational
inertia, and therefore the greater the
resistance to rotational acceleration.