Download Angular Displacements

Chapter 8
Rotational Kinematics – Angular
displacement, velocity,
acceleration
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Rotational motion – the motion of a
spinning object
• Any point on a spinning object undergoes
circular motion
• The motion of a point can be described
by the angle the point moves in a certain
time
Angular Displacements
l
l
θ
r
θ
r
r = radius of circle or
distance from axis
l = arc length or
distance travelled by
the point
θ = the angle travelled
in radians
Not on equation sheet
Radians & Degrees
• Any angle can be converted from radians to
degrees & vice versa
• θ(rad) = π θ(deg)
180
• 1 revolution = 2π rad = 360º
Not on equation sheet
Angular Kinematics
Angular displacement:
Δθ = θ2 – θ1
Not on equation sheet
Angular Velocity (units of rad/s):
Not on equation sheet
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Angular Acceleration (units of rad/s/s):
Different form on equation sheet
All points on a rotating object have the same angular speed and angular
acceleration regardless of their location. However, their linear (tangential)
speed and acceleration are different.
The tangential speed of the outermost point of the object is also the
translational speed of the whole object
v = rω
Not on equation sheet
You must memorize!!
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a = rα
Objects farther from the axis of rotation will move faster.
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Centripetal Acceleration
Not on equation sheet – you can derive from what
is on the equation sheet and what you must
memorize
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Angular & Linear Kinematic Equations
• Angular Equations:
ω = ω0 + αt
θ = θ0 + ω0t + ½αt2
ω2 = ω02 +2α(θ-θ0)
Not on equation sheet
• Linear Equations:
v = v0 + at
x = x0 + v0t + ½ at2
v2 = v02 +2a(x-x0)
Frequency and Period
The frequency is the number of complete revolutions
per second:
f = ω/2π
Frequencies are measured in hertz.
1 Hz = 1 s−1
The period is the time one revolution takes:
=2 π/ ω
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On equation sheet
Rolling Motion (Without Slipping)
In (a), a wheel is rolling without slipping. The
point P, touching the ground, is instantaneously
at rest, and the center moves with velocity v.
(Remember static friction for rolling wheels!)
In (b) the same wheel is seen from
a reference frame where C is at
rest. Now point P is moving with
velocity –v.
Relationship between linear and angular speeds:
v = rω
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Practice Problems
8-1. A bike wheel rotates 4.50 revolutions. How many radians has it
rotated?
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Practice Problems
8-2. A particular bird’s eye can just distinguish objects that move forming
an angle no smaller than about 3x10-4 rad. A) How many degrees is this?
B)How small can the length of the object be so the bird can just distinguish
it when flying at a height of 100m?
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Practice Problems
8-3 (Modified). On a rotating carousel or merry-go-round, one child sits on
a horse near the outer edge and another child sits on a lion halfway out from
the center. A) Which child has the greatest linear velocity? B) Which child
has the greater angular velocity? C) Each child drops a piece of popcorn.
Which piece of popcorn is most likely to move from where it is dropped?
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Practice Problems
8-4. A carousel is initially at rest. At t=0, it is given a constant angular
acceleration of α=0.060rad/s/s which increases its angular velocity for 8.0s.
A) What is the angular velocity of the carousel? B) What is the linear
velocity of a child that is located 2.5m away from the center?
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Practice Problems
8-6 (Modified). A centrifuge rotor is accelerated from rest for 30s to
20,000rpm (rev/min). A) What is its average angular acceleration?
B)Through how many radians worth of revolutions does the rotor turn
during this time? How many revolutions is this?
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Practice Problems
8-7. A bike slows down from 8.40m/s to rest over a distance of 115m. Each
wheel has a diameter of 68.0cm. A)What is the angular velocity of the
wheels initially? B)How many revolutions do the wheels rotate while
coming to a stop? C)what is the angular acceleration of the wheels?
D)How long does it take to stop?
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Chapter 8
Rotational Dynamics – Torque,
Momentum, KE
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Forces Causing Rotational Motion
To make an object start rotating, a force is needed
The position and direction of the force matter
The perpendicular distance from the axis of rotation to the line along which
the force acts is called the lever arm (r) or moment arm.
The longer the lever arm, the easier the rotation
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Which force will be more successful?
Rank the lever arms: rA>rC>rD=0
So, FA has greatest torque and FD has zero
Angular acceleration is proportional to the force and the lever arm
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Torque – the product of force and the lever arm
a measure of how much a force acting on an object causes that object to
rotate (units mN)
=rFsinθ
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Net torque is the sum of all torques
Torque from gravity is negative,
torque from person is positive
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Rotational equilibrium
• If an object is in angular equilibrium, then it is either at rest or else it is
rotating with a constant angular velocity
• The net torque is zero
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Torque and Rotational Inertia
The amount of torque required to get an
object rotating depends on the object’s
rotational inertia (moment of inertia)
I = Σmr2 is the rotational inertia of an object.
I and mass (inertia) are related to each other.
One for rotation and the other for
translation.
The distribution of mass matters—these two
objects have the same mass, but the one on
the top has a greater rotational inertia (I), as
so much of its mass is far from the axis of
rotation. More torque is required for the top
one.
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Torque and Rotational Inertia
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Different shapes all have their own
unique rotational inertia
You are not responsible for knowing
how to calculate I for different
objects – you must understand the
concept and apply it in equations
when given to you
Know this:
I long rod< I sphere < I solid
cylinder (disc) < I hoop
https://www.youtube.com/watch?v=yAWLLo
5cyfE
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Rotational Kinetic Energy
A object that has both translational and rotational motion also has both
translational and rotational kinetic energy:
Not on equation sheet
Conservation of energy must include both rotational and translational
kinetic energy
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Which makes it to the bottom first (same mass)?
Solid wins because lower I, so more E goes into translational KE
Turns out this is independent of mass and radius
Speed at bottom only depends on gravitational PE and shape
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Which makes it to the bottom first (same mass and frictionless
ramp)?
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Your car sliding on ice and anti-lock brakes
What kind of friction stronger?
How does kinetic energy also now explain this:
Sliding: only K=1/2mv2
Rolling: K=1/2mv2 + 1/2Iω2
So translational motion will be faster (and more out of control) when
sliding on the ice.
Take your foot of the break!
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Angular Momentum and Its Conservation
Angular momentum L:
The total torque is the rate of change of angular momentum. No net torque
from outside forces means angular momentum is conserved.
ΔL = τΔt
If the net torque on an object is zero, the total angular momentum is
constant.
Iω = I0ω0 = constant
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Iω = I0ω0 = constant
Systems/objects that can change their rotational inertia through internal
forces will also change their rate of rotation:
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Right Hand Rule of Angular Momentum
The angular velocity vector points along the axis of rotation; its direction is
found using a right hand rule
-curl fingers of right hand in direction of rotation, thumb points in direction
of vector
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Angular acceleration and angular
momentum vectors also point
along the axis of rotation.
Use the right hand rule – in what
direction is the vector of the
person’s angular velocity and
momentum? How about the
platform?
• https://www.youtube.com/watch?v=_XgYTP0kB7A
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Angular motion demo
• https://www.youtube.com/watch?v=NeXIV-wMVUk
momentum and torque
are perpendicular, so
momentum is chasing
torque like video says
and change in
momentum is same
direction as torque
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Practice Problems
8-8. The biceps muscle exerts a vertical force of 700N on the lower arm.
Calculate the torque about the axis of rotation through the elbow joint
assuming the muscle is attached 5.0cm from the axis of rotation.
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Practice Problems
8-10. A 15.0N force is applied to a cord wrapped around a pulley of mass
4.00kg and a radius of 33.0cm. The pulley accelerates from rest to an
angular speed of 30.0rad/s in 3.00s. A) What is the angular acceleration of
the pulley? B) If there is a frictional torque of 1.10 mN at the axle, what is
the moment of inertia of the pulley?
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Practice Problems
8-12 Modified. What is the speed of a 1.0kg solid sphere that has an
I=0.4kgm2 that rolls down a ramp from a height of 2.5m with an angular
velocity of 5.5rad/s?
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Practice Problems
8-13. Several objects roll without slipping down an incline with a vertical
height, h. They all start from rest at the exact same time. The objects are a
thin metal hoop, a marble, a solid D-cell battery, and an empty soup can.
Additionally, a greased box slides down without friction. In what order do
they reach the bottom of the incline or do they all reach it at the same time?
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Practice Problems
8-14 Modified. A simple clutch consists of 2 cylindrical plates that can be
pressed together to connect two sections of an axle. Plate A has an
I=1.08kgm2 and plate B has an I=1.62kgm2. A) If the plates are initially
separated and Plate A is accelerated from rest to an angular velocity of
7.2rad/s in 2.0s, what is the final angular momentum of Plate A? B)What
was the torque required to cause this acceleration? C) If Plate B is initially
at rest and then placed into firm contact with Plate A, what is the new
angular velocity of the two plates?
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