Download Ch. 9 Rotational Kinematics

Ch. 9 Rotational
Kinematics
AP Physics C

Rotation of a rigid
body about a fixed
axis

A rigid body is not
deformable; that is,
the separations
between particles
remain constant.
Introduction

Measured in radians

As a particle moves
from A to B, it
moves through an
arc length of s.

s = rθ

What is a radian?
Angular position, θ


1 revolution =
_____ rad
1 revolution =
_____ deg
Conversions:

Convert:
1. 30o = ______ rad

2. 36 rad = _____ rev

3. 10 rev = _____ rad

4. 120o = _____ rev



As the particle moves
from A to B, its
angular
displacement,
Δθ = θf - θi.
Its angular average
velocity is

avg 
t

Its instantaneous angular
velocity is
  lim
t 0

t
d

dt
Angular Displacement & Velocity:
Right-hand rule

If the instantaneous angular velocity of
an object changes from ωi to ωf over a
time interval of Δt, then the object has
an average angular acceleration of

 avg 
t

The instantaneous angular acceleration is
 d 
  lim

t 0
t
dt
Angular Acceleration:

The angular position θ, in radians, of a rotating object
is given by the following equation:
3
2
  2t  t  3t
a.
b.
c.
d.
e.
f.
Determine the object’s average angular speed from
1 s to 5 s.
Determine the object’s instantaneous angular speed
as a function of t.
What is the object’s instantaneous speed at 3 s?
What is the object’s average angular acceleration
from 1 s to 5 s?
Determine the object’s instantaneous angular
acceleration as a function of time.
What is the object’s instantaneous angular
acceleration at 3 s?
Sample Problem:
Linear Variables
Rotational Variables
x
v
a
Rotational Motion is analogous to
linear motion.
Linear Motion
Rotational Motion
Constant Accelerated Motion:
A wheel rotates with a constant angular
acceleration of 2.5 rad/s/s. If the initial
angular speed is 2.0 rad/s,
a. What is its final angular speed after 10.0
s?
b. What is its angular displacement in

i. Radians?
ii. Revolutions?
iii. Degrees?
Sample Problem:

What is the time
derivative of s = rθ?
ds
d
r
dt
dt
vtan  r

dvtan
d
What is the
r
dt
dt
tangential
acceleration? atan  r
Relating Linear and Angular
Kinematics:

Recall:
2
v
ac 
r
2
(r )
ac 
r
2
ac  r
Centripetal Acceleration:

Information is stored on a CD or DVD in a
coded pattern of tiny pits. The pits are
arranged in a track that spirals outward
toward the rim of the disc. As the disc
spins inside a player, the track is scanned
at a constant linear speed. How must the
rotation speed of the disc change as the
player’s scanning head moves over the
track?
Test Your Understanding:

A discuss thrower moves the
discus in a circle of radius
80.0 cm. At a certain instant,
the thrower is spinning at an
angular speed of 10.0 rad/s
and the angular speed is
increasing at 50.0 rad/s/s. At
this instant, find the
tangential and centripetal
acceleration of the discus and
the magnitude of the
acceleration.
Sample Problem:

The smaller gear
shown to the right
has a radius of 5.0
cm and the larger
one has a radius of
10.0 cm. If the
angular speed of
the smaller gear is
25 rad/s, what is
the angular speed
of the larger gear?
Sample Problem:

Consider a rigid body that is made up of
an infinite number of infinitesimal
particles and rotating about a fixed axis,
the kinetic energy of each particle is:
1
K i  mi vi 2
2

How would you write this kinetic energy
expression in terms of angular speed?
Rotational Kinetic Energy:

What would be the
total kinetic energy
of the rigid body?
Rotational Kinetic Energy
MOI is a property of physics that indicates
the relative difference in how easy or
difficult it will be to set any object in
motion about a defined axis of rotation.
 MOI is always measured relative to a
point of reference.
 MOI depends on an object’s mass and on
its shape.
 MOI depends on the distribution of mass.

Moment of Inertia (MOI)
Newton’s first law of motion states, “A
body maintains the current state of
motion unless acted upon by an external
force.”
 In linear motion, the measure of inertia
refers to the mass of the system.
 In rotational motion, the measure of
inertia refers to the moment of inertia of
the system.

Moment of Inertia (MOI)

MOI for a system of
particles: I   mi ri 2

Determine MOI of
this system of 4
points masses, as
they rotate about
the:
◦ X-axis
◦ Y-axis
◦ Axis perpendicular to
the origin
Moment of Inertia (MOI)

MOI
for a
rigid
body:
I   r 2 dm

Pg.
342
Moment of Inertia (MOI)

The rim has a mass
M and radius R.
Each spoke has
mass m and length
L. Assume that the
rim can be
considered a thinwalled cylinder and
each spoke as a
thin rod.
Find the total moment of inertia.

What is the total kinetic energy of a rigid
body?

What is the MOI for a system of particles?

What is the kinetic energy of a rotating
rigid body in terms of its MOI?
Rotational Kinetic Energy

Given the
equilateral triangle
to the right with
equal masses of m
at each vertex,
determine the:
◦ MOI about an axis
perpendicular to the
plane at the x and
◦ The kinetic energy of
the system about this
axis.
Sample Problem:

What is the
gravitational
potential energy for
each particle of a
rigid body?

What is the total
gravitational
potential energy of
a rigid body?
Gravitational Potential Energy

Determine the
speed of the disk
and the hoop,
shown to the right,
when they reach
the bottom of the
ramp. Let h
represent the
height that their
CMs start at.
Sample Problem

The mass of the
pulley is 2.5 kg and
its radius is 0.25 m.
If the mass on the
left is 6.0 kg and the
one on the right is
4.5 kg, what is the
speed at which the
6.0 kg block hits the
floor if it moved a
distance of 1.0 m?
Sample Problem

What is the speed
of the block when it
has traveled a
distance of h?

What is the angular
speed of the pulley
when the block has
traveled a distance
of h?
Sample Problem

Determine the MOI
of the rigid body
about an axis that
is perpendicular to
the origin.
Parallel-Axis Theorem

Use the ParallelAxis Theorem to
determine the MOI
about an axis that
is perpendicular to
point O.
Sample Problem

Use the ParallelAxis Theorem, to
determine the MOI
about an axis that
is parallel to the zaxis an a distance
of r/2 from the zaxis.
Sample Problem

Calculate the MOI of a uniform thin rod
with an axis of rotation that is
perpendicular to its length.
Moment of Inertia Calculation

Calculate the MOI of a uniform hollow or
uniform solid cylinder that is rotating
about its axis of symmetry.
Moment of Inertia Calculation