CHAPTER 8 NOTES - ROTATIONAL KINEMATICS
Axis of rotation is the center of motion for all parts of a
rotating body. It is a line segment within the body about which
all parts rotate.
Angular displacement is an expression of the angle through
which an object has rotated or revolved. Rotate means to spin
about an internal axis, revolve means to move around some
external point or axis. The skater's body is rotating. The points
A, B, and C are revolving.
Angular displacement can be expressed in revolutions, degrees
or radians. The SI unit is the radian.
In the diagram, Ө, Өo, and ΔӨ are examples of angular
displacement.
In scientific applications, the radian is often used because this
makes the relationship between arc length, radius of the circle,
and angular displacement easier to work with.
The equation for this is:
S = RӨ
Where S is arc length, R is the radius of the circle and Ө is the
angular displacement when expressed in radians.
If S is the entire circumference of a circle, Ө must be 2Π since
C = 2ΠR
This means that 2Π radians is the same angular measure as
360° or one revolution. We may then convert between units of
angular displacement using these relationships.(1 rad = 57.3°)
The diagram above illustrates the relationship between arc
length and radius. An observer on earth sees the moon and the
sun subtending nearly the same angle even though the sun has
a much larger diameter than the moon. Since the sun is about
375 times farther from the earth than the moon, its arc length
must be about 375 times larger to subtend the same angle.
Since these conditions are met, the moon can almost completely
cover the sun by occupying the same area of the sky producing
a solar eclipse.
Angular velocity is defined as the time rate of change of the
angular displacement. The equation is:
ω = ΔӨ/Δt
where ω is the average angular velocity, ΔӨ is the angular
displacement, and Δt is the time interval during which the
displacement occurs. Notice that this equation is very similar to
the equation for linear velocity.(V = ΔS/Δt)
The SI unit is rad/s.
Angular velocity is a vector and has a direction as any other
vector does. When working in two dimensions, the direction
may be clockwise or counterclockwise. If the fingers of the
right hand are curled in the direction of rotation, the right
thumb points in the direction of the angular velocity vector.
Example
A diver completes 3.5 somersaults in 1.7 s. Find the average
angular speed in rad/s of the diver.
Example
A device used to measure the speed of a bullet consists of two
rotating disks separated by a distance d = 0.85m and having a
known angular velocity of 95.0 rad/s. A bullet is fired through
the disks and the angular displacement between the two holes
is 0.240 rad. Find the speed of the bullet.
Angular acceleration is defined as the time rate of change of
the angular velocity. Average angular acceleration is calculated
using the equation:
α = Δω/Δt
where α is the angular acceleration, Δω is the change in
angular velocity, and Δt is the change in time. Notice that this
equation is very similar to the equation for linear
acceleration.(a = ΔV/Δt)
The SI unit for angular acceleration is the rad/s2.
We will be working with angular acceleration that is constant
or average.
Example
A CD has a playing time of 74 min. When the music starts, the
CD rotates at 480 rpm and at the end it is rotating at 210 rpm.
Find the average angular acceleration in rad/s2.
The equations of rotational kinematics are very similar to the
equations for linear kinematics. Since we have seen that linear
displacement along an arc equals the product of angular
displacement and the radius of the circle(S = RӨ), linear
velocity along the arc equals the product of angular velocity
and the radius of the circle(V = Rω), and linear acceleration
equals the product of the angular acceleration and the radius
of the circle(a = Rα), it follows that the equations of linear
kinematics may be used for circular motion. If we convert the
linear variables into rotary variables using the three
relationships given above, we can use the same equations.
V = Vo + at
ω = ωo + αt
S = ½( Vo + V)t
Ө = ½(ωo + ω)t
S = Vot + ½at2
Ө = ωot + ½αt2
V2 = Vo2 + 2aS
ω2 = ωo2 + 2αӨ
These equations are used in the same way that their linear
counterparts are used. Remember these are valid only if the
angular acceleration is constant.
The units used may be radians, degrees or revolutions if they
are consistent throughout the equation. Unless the answer is
specified to be in degrees or revolutions, the radian is the unit
of choice.
Example
A toy top at rest has a 64 cm string wrapped around it at a
place where the radius of the top is 2.0 cm. The string's
thickness is negligible. Find the final angular velocity of the top
when the string is pulled completely off with an angular
acceleration of +12 rad/s2.
Tangential velocity and acceleration are linear vectors directed
along the path of the arc followed by the object as it revolves.
Their directions are constantly changing. Their magnitudes
can be found using the relationship between each of them and
their rotary counterparts.
As you can see from the diagram, the larger R is, the larger the
tangential velocity of a skater must be if the angular velocity is
the same for all of them. This is also true for the relationship
between linear and angular acceleration.
Example
The earth has a radius of 6.38 x 106 m and turns once on its
axis every 23.9 hours. (a)Find the tangential speed of a person
on the equator in m/s. (b) Find the angle θ in the diagram
where the tangential speed is 1/3 of that at the equator.
While studying uniform circular motion we condidered the
linear speed of the whirling object to be constant. During the
time it is reaching that speed or slowing down, the speed is not
constant and a tangential acceleration exists. This type of
motion is called nonuniform circular motion and the net
acceleration and net force must include both tangential and
centripetal components.
Since they are at right angles, the Pythagorean Theorem can
be used to determine the net acceleration and then the net
force.
Example
A rectangular plate is rotating about an axis that passes
perpendicularly through one corner as in the diagram. The
tangential acceleration at corner A has twice the magnitude of
that at corner B. Find the ratio of the lengths of the two sides
of the rectangle.
When an object is rolling we can consider the linear velocity of
its center of rotation to have the same magnitude as the
tangential velocity of a point on its rim. This is true if there is
no slipping.
We can also conclude that the linear distance traveled by its
center of rotation is the same as the linear distance traveled by
a point on the rim. The equations V = Rω and S = RӨ allow us
to relate the linear speed of the rolling object to its angular
speed.
Example
A car is traveling at a speed of 20.0 m/s on a straight, level
road. The radius of each tire is 0.300 m. If the car speeds up
with a linear acceleration of 1.5 m/s2 for 8.0 s, find the angular
displacement of each tire(no slipping).
Since rotary motion does not occur in one direction as linear
motion can, the direction of rotary motion is expressed
differently. If we consider the rotation to be in a plane, the
direction of the rotary motion vector can be taken to be at
right angles to the plane of rotation.
The two possibilities are clockwise and counterclockwise.
Counterclockwise is generally considered to be positiveand
clockwise is considered to be negative.
Use your right hand to determine the direction of the vector. If
you curl the fingers of your right hand in the direction of
rotation and extend your thumb, your thumb will point in the
direction of the rotary motion vector.
These vectors may be added and subtracted in the same
manner as the linear vectors already discussed.
Page 241, Questions 3, 4, 6, 7, 12, 13
Page 242, Problems 4, 5, 7, 9, 11, 13, 16, 19, 21, 22, 31, 33, 35,
39, 43, 47, 48, 49, 56, 61