# Rotational Kinematics by sanmelody

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```									Rotational Kinematics

Tipler Ch. 9
Position
• In translational motion,               x=3
x
position is represented by a       0          5
point, such as x.
linear
p/2
• In rotational motion,
position is                            r
represented by an                        q
p                  0
angle, such as q, and
angular
3p/2
Displacement
Dx = - 4
• Linear displacement is
represented by the                                x
vector Dx.                    0          5
linear
p/2
• Angular displacement
is represented by Dq,
which is not a vector,        Dq
but behaves like one     p                   0
for small values. The
right hand rule
determines direction.                   angular
3p/2
Tangential and angular
displacement
• A particle that rotates
through an angle Dq                 s
also translates through
a distance s, which is                       r
the length of the arc                   Dq
defining its path.
• This distance s is related to the
angular displacement Dq by the
equation s = rDq.
Speed and velocity
• The instantaneous velocity
has magnitude vT = ds/dt             s    vT
and is tangent to the circle.

• The same particle rotates                       r
with an angular velocity w =              Dq
dq/dt.                           vT   w is outward
• The direction of the angular          according to
velocity is given by the right            RHR
hand rule.
• Tangential and angular
speeds are related by the
equation v = r w.
Acceleration
• Tangential acceleration is
given by aT = dvT/dt.                 s     vT
• This acceleration is parallel
or anti-parallel to the
velocity.                                 Dq     r
• Angular acceleration of this    vT     w is outward
particle is given by a =               according to
dw/dt.                                     RHR
• Angular acceleration is
parallel or anti-parallel to
the angular velocity.                Don’t forget
• Tangential and angular                centripetal
accelerations are related by         acceleration.
the equation a = r a.
Problem: Assume the particle is
speeding up.
a) What is the direction of the           What changes if
instantaneous velocity, v?
the particle is
b) What is the direction of the
angular velocity, w?                   slowing down?
c) What is the direction of the
tangential acceleration, aT?
d) What is the direction of the
angular acceleration a?
e) What is the direction of the
centripetal acceleration, ac?
f) What is the direction of the overall
acceleration, a, of the particle?
First Kinematic Equation
• v = vo + at (linear form)
– Substitute angular velocity for velocity.
– Substitute angular acceleration for
acceleration.
• w = wo + at (angular form)
Second Kinematic Equation
• x = xo + vot + ½ at2 (linear form)
– Substitute angle for position.
– Substitute angular velocity for velocity.
– Substitute angular acceleration for
acceleration.
q = qo + wot + ½ at2 (angular form)
Third Kinematic Equation
• v2 = vo2 + 2a(x - xo)
– Substitute angle for position.
– Substitute angular velocity for velocity.
– Substitute angular acceleration for
acceleration.
• w2 = wo2 + 2a(q - qo)
Torque and Newton’s 2nd Law
• Rewrite SF = ma for rotating systems
– Substitute torque for force.
– Substitute rotational inertia for mass.
– Substitute angular acceleration for
acceleration.
S = I a
: torque
I: rotational inertia
a: angular acceleration
Torque
Torque is the rotational analog of force that
causes rotation to begin.
Consider a force F on the beam that is applied a distance r from
the hinge on a beam. (Define r as a vector having its tail on the
hinge and its head at the point of application of the force.)

Hinge (rotates)
A rotation occurs due to the
combination of r and F. In                           r
this case, the direction is
clockwise.                            Direction of rotation
What do you think is the
F
Direction of torque is
direction of the torque?
INTO THE SCREEN.
Calculating Torque
• The magnitude of the torque is proportional to
that of the force and moment arm, and torque is
at right angles to plane established by the force
and moment arm vectors. What does that sound
like?
• =rF
–  : torque
– r: moment arm (from point of rotation to point of
application of force)
– F: force
Practice Problem
A 1.0-kg wheel of 25-cm radius is at rest on a fixed axis. A force of 0.45 N is applied
tangent to the rim of the wheel for 5 seconds.
a) After this time, what is the angular velocity of the wheel?
b)Through what angle does the wheel rotate during this 5 second period?
Rotational Inertia Calculations

• I = Smr2 (for a system of particles)
• I =  dm r2 (for a solid object)
• I = Icm + m h2 (parallel axis theorem)
– I: rotational inertia about center of mass
– m: mass of body
– h: distance between axis in question and
axis through center of mass
Sample Problem
• A 2.0-kg mass and a 3.0-kg mass are mounted on opposite ends a 2.0-m
long rod of negligible mass. What is the rotational inertia about the center of
the rod and about each mass, assuming the axes of rotation are
perpendicular to the rod?
Practice Problem

Derive the rotational inertia of a long thin rod of length L and mass M
Practice Problem

Derive the rotational inertia of a long thin rod of length L and mass M
about a point 1/3 from one end
a)   using integration of I =  r2 dm
b)   using the parallel axis theorem and the rotational inertia of a rod
Practice Problem
Derive the rotational inertia of a ring of mass M and radius R about the
center using the formula I =  r2 dm.
Angular Momentum
• Angular momentum is a quantity that tells
us how hard it is to change the rotational
motion of a particular spinning body.
• Objects with lots of angular momentum
are hard to stop spinning, or to turn.
• Objects with lots of angular momentum
have great orientational stability.
Angular Momentum of a particle

• For a single particle with known
momentum, the angular momentum can
be calculated with this relationship:
• L=rp
– L: angular momentum for a single particle
– r: distance from particle to point of rotation
– p: linear momentum
Practice Problem
Determine the angular momentum for the revolution of
a)  the earth about the sun.
b)  the moon about the earth.
Practice Problem
Determine the angular momentum for the 2 kg particle shown
a) about the origin.                              y (m)
5.0

5.0   x (m)

-5.0
v = 3.0 m/s
Angular Momentum - solid
object
• For a solid object, angular momentum is
analogous to linear momentum of a solid object.
• P = mv (linear momentum)
– Replace momentum with angular momentum.
– Replace mass with rotational inertia.
– Replace velocity with angular velocity.
• L = I w (angular momentum)
– L: angular momentum
– I: rotational inertia
– w: angular velocity
Practice Problem
Set up the calculation of the angular momentum for the rotation of the
earth on its axis.
Law of Conservation of Angular Momentum
• The Law of Conservation of Momentum states
that the momentum of a system will not change
unless an external force is applied. How would
you change this statement to create the Law of
Conservation of Angular Momentum?
• Angular momentum of a system will not change
unless an external torque is applied to the
system.
• LB = LA (momentum before = momentum after)
Practice Problem

A figure skater is spinning at angular velocity wo. He brings his arms and legs
closer to his body and reduces his rotational inertia to ½ its original value.
What happens to his angular velocity?
Practice Problem
A planet of mass m revolves around a star of mass M in a highly elliptical orbit. At
point A, the planet is 3 times farther away from the star than it is at point B. How does
the speed v of the planet at point A compare to the speed at point B?
Practice Problem
A 50.0 kg child runs toward a 150-kg merry-go-round of radius 1.5 m, and jumps
aboard such that the child’s velocity prior to landing is 3.0 m/s directed tangent to the
circumference of the merry-go-round. What will be the angular velocity of the merry-
go-round if the child lands right on its edge?
Angular momentum and torque
• In translational systems, remember that
Newton’s 2nd Law can be written in terms of
momentum.
• F = dP/dt
– Substitute force for torque.
– Substitute angular momentum for momentum.
 = dL/dt
– t: torque
– L: angular momentum
– t: time
So how does torque affect angular
momentum?
• If  = dL/dt, then torque changes L with respect
to time.
• Torque increases angular momentum when the
two vectors are parallel.
• Torque decreases angular momentum when the
two vectors are anti-parallel.
• Torque changes the direction of the angular
momentum vector in all other situations. This
results in what is called the precession of
spinning tops.
If torque and angular momentum are
parallel…

Consider a disk rotating
as shown. In what
direction is the angular
momentum?

Consider a force applied
as shown. In what                   r
direction is the torque?
F
The torque vector is parallel to
the angular momentum vector.        L is out
Since  = dL/dt, L will increase
with time as the rotation speeds.
 is out
If torque and angular momentum are
anti-parallel…

Consider a disk rotating
as shown. In what
direction is the angular
momentum?

Consider a force applied
as shown. In what                    r
direction is the torque?
F
The torque vector is anti-parallel
to the angular momentum vector.      L is in
Since  = dL/dt, L will decrease
with time as the rotation slows.
 is out
If the torque and angular momentum
are not aligned…

• For this spinning
top, angular
momentum and
torque interact in a
more complex way.      r   L
• Torque changes                   = r  Fg
the direction of the
angular                              
momentum.                  Fg
• This causes the                  = dL/dt
characteristic
precession of a                      DL
spinning top.

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