Chapter 6 Angular Motion

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Chapter 6 Angular Motion
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Objectives After completing this module, you
should be able to

• Define and apply concepts of angular
  displacement, velocity, and acceleration.
• Draw analogies relating rotational-motion
  parameters (?, ?, ?) to linear (x, v, a) and
  solve rotational problems.
• Write and apply relationships between linear and
  angular parameters.

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Objectives (Continued)

• Define moment of inertia and apply it for several
  regular objects in rotation.
• Apply the following concepts to rotation
• 1. Rotational work, energy, and power
• 2. Rotational kinetic energy and momentum
• 3. Conservation of angular momentum

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Rotation An Introduction

• For some to travel once around a circle, the
  distance formula is 2pr
• The time to go once around is referred to as the
  Period (T)
• V
• In a circle V

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Rotational Displacement, ?
Consider a disk that rotates from A to B
B
Angular displacement q
?
Measured in revolutions, degrees, or radians.
1 rev 360 0 2? rad
The best measure for rotation of rigid bodies is
the radian.
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Definition of the Radian
One radian is the angle ? subtended at the center
of a circle by an arc length s equal to the
radius R of the circle.
360o 2
rad
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Definition of the Radian
rad 57.3o
Converting rads to degrees and degrees to rads
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Example 1 A rope is wrapped many times around a
drum of radius 50 cm. How many revolutions of
the drum are required to raise a bucket to a
height of 20 m?
q 40 rad
Now, 1 rev 2p rad
q 6.37 rev
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Example 2 A bicycle tire has a radius of 25 cm.
If the wheel makes 400 rev, how far will the bike
have traveled?
q 2513 rad
s q R 2513 rad (0.25 m)
s 628 m
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Angular Velocity
Angular velocity,w, is the rate of change in
angular displacement. (radians per second.)
Angular velocity,?, can also be given as the
frequency of revolution, f (rev/s or rpm)
w 2pf rad/s
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Example 3 A rope is wrapped many times around a
drum of radius 20 cm. What is the angular
velocity of the drum if it lifts the bucket to 10
m in 5 s?
q 50 rad
w 10.0 rad/s
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Example 4 In the previous example, what is the
frequency (rev/s) of revolution for the drum?
Recall that w 10.0 rad/s.
Or, since 60 s 1 min
f 95.5 rpm
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Angular Velocity continued
Angular velocity,w, can also be written in terms
of the number of revolutions (2 r) per time of
one turn.
2 rad T
w Angular velocity in rad/s.
Angular velocity,?, can also be given as the
frequency of revolution, f (rev/s or rpm)
w (2p rad) f
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Linear Speed
Linear speed, can be written in terms of the
circumference (2pr) per time of one turn.
v w r speed (m/s)
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Angular and Linear Speed
From the definition of angular displacement
s q R Linear vs. angular displacement
Linear speed angular speed x radius
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Examples
Consider flat rotating disk
wo 0 wf 20 rad/s
R1 20 cm R2 40 cm
What is final linear speed at points A and B?
vAf wAf R1 (20 rad/s)(0.2 m) vAf 4 m/s
vAf wBf R2 (20 rad/s)(0.4 m) vBf 8 m/s
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Angular Acceleration
v2
a

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Regular acceleration m/s
r
Angular acceleration is the rate of change in
angular velocity. (Radians per sec per sec.)
The angular acceleration can by subsituting wr
for v and you would get (w2r2)/r OR
a w2r m/s2
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Angular Acceleration
Angular acceleration is the rate of change in
angular velocity. (Radians per sec per sec.)
The angular acceleration can also be found from
the change in frequency, as follows
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Acceleration Example
What is the average angular and linear
acceleration at B?
a 5.00 rad/s2
a 2.00 m/s2
a aR (5 rad/s2)(0.4 m)
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Example 5 The block is lifted from rest until
the angular velocity of the drum is 16 rad/s
after a time of 4 s. What is the average angular
acceleration?
a 4.00 rad/s2
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Angular and Linear Acceleration
From the velocity relationship we have
v wR Linear vs. angular velocity
Linear accel. angular accel. x radius
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Angular vs. Linear Parameters
But, a aR and v wR, so that we may write
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6.3 Dynamics (Forces) and Circular Motion
Known Equations Fnet m x a
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6.3 Dynamics (Forces) and Circular Motion
An 80 kg person steps into the graviton 2000 that
has a radius of 8 m. Graviton 2000 makes 15
turns per minute. What linear velocity does the
person have? With what force does the wall push
on the person
2pr T
2p 8m 4 s
v
v
12.57 m/s
v
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6.3 Dynamics (Forces) and Circular Motion
An 80 kg person steps into the graviton 2000 that
has a radius of 8 m. Graviton 2000 makes 30
turns per minute. What linear velocity does the
person have? With what force does the wall push
on the person
mv2 r
(80kg) (12.57 m/s)2 8m
Fnet
Fnet
Fnet
1579N
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6.4 Apparent Forces during Circular Motion
The circular Fnet equation works well to
determine the net force, but it doesnt determine
how much force the wall or string applies if
there are outside forces. If you ever gone
through a loop in a roller coaster, you know that
you feel heavier at the bottom than at the top
when you are upside down.
mv2 r
n
Fnet
w
Fnet n - w
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6.5 Orbits
Earth
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6.5 Orbits
A look back at Projectile Motion objects are
given only horizontal velocity
Earth
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6.5 Orbits
Zoom out and move the projectile up 3,000 or so
miles
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6.5 Orbits
Zoom out again about 25,000 miles
Earth
Which way is gravity when the object is here?
Which way is gravity when the object is here?
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6.5 Orbits
So how fast would an object need to go to orbit??
v2 r
v2/r since orbit is circular
a g
v2 r g
At the surface of the earth, g 9.8 m/s/s
At the surface of the earth, radius 6.37 x 106 m
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6.5 Orbits
So how fast would an object need to go to orbit??
v
v (6.37 x 106 m)( 9.8m/s2)
v
v (6.24 x 107 m)
v 7900 m/s about 5 miles/ second!!!
Problem the acceleration of gravity from the
earth is not always 9.8m/s2 it decreases as you
get farther away
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6.6 Newtons Law of Gravity

• For every force there is an equal and opposite
  force
• Forces is are inversely proportional to the
  square of the distance between them
• The force is directly proportional to the
  product of the masses of the two objects.

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6.6 Newtons Law of Gravity
Fgravity E on M
Fgravity M on E
Earth
Moon
G m1 m2 r2
F(M on E) F(E on M)
Fgravity
G (Gravitational Constant) 6.67 x 10-11 N
m2/kg2
r is the radius or distance between the centers
of the objects in this case the moon and the
earth.
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Linear Example A car traveling initially at 20
m/s comes to a stop in a distance of 100 m. What
was the acceleration?
a -2.00 m/s2
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Angular analogy A disk (R 50 cm), rotating at
600 rev/min comes to a stop after making 50 rev.
What is the acceleration?
wo 600 rpm wf 0 rpm q 50 rev
50 rev 314 rad
a -6.29 m/s2
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Problem Solving Strategy

• Draw and label sketch of problem.
• Indicate direction of rotation.
• List givens and state what is to be found.

Given ____, _____, _____ (q,wo,wf,a,t) Find
____, _____

• Select equation containing one and not the other
  of the unknown quantities, and solve for the
  unknown.

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Example 6 A drum is rotating clockwise
initially at 100 rpm and undergoes a constant
counterclockwise acceleration of 3 rad/s2 for
2 s. What is the angular displacement?
Given wo -100 rpm t 2 s a 2 rad/s2
q -14.9 rad
q -20.9 rad 6 rad
Net displacement is clockwise (-)
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Summary of Formulas for Rotation
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CONCLUSION Chapter 11A Angular Motion