Download Rotational Motrion and Torque - Parkway C-2

Rotational Motrion and Torque
We define
angular displacement, θ, and
angular velocity, ω
Units: θ = rad
ω = rad/s
What's a radian?
Radian is the ratio between the length of an arc and its radius
note: counterclockwise is +
clockwise is 1
Rotational Motrion and Torque
Consider purely angular measures
ω = Δθ
Δt
for 1 revolution
recall
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Rotational Motrion and Torque
"Suggested " problems for Thursday . . . .
page 101: 34, 37
page 202: 1, 8, 11, 13, 14, 18
Problems if you want extra practice . . . .
page 202: 3, 10, 15, 16, 19, 21
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Rotational Motrion and Torque
Rotation of a Rigid Body
Translation - the object as a whole moves along a
trajectory, but does not rotate.
Rotation - the object rotates about a fixed point.
Every point on the object
moves in a circle.
During uniform rotation of a rigid body every point on the body has the
same angular velocity, ω , and the
same angular displacement, θ
∆θ= ω∆ t
true for both the
red and green
points
v = ωr & v = ωr
but v≠v
linear velocities are not the same
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Rotational Motrion and Torque
Angular Acceleration - if you push on the edge of a bicycle
wheel it begins to rotate, if you
continue to push is rotates faster.
recall linear acceleration was defined
By analogy we define angular
acceleration, α, as
Linear and Circular Motion Variables
standard units: m, m/s, m/s2
standard units: rad, rad/s, rad/s2
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Rotational Motrion and Torque
Linear and Circular Motion Equations
v2 - vo2 = 2a(x-xo)
ω 2 − ω ο 2 = 2α(θ−θ ο )
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The disk in a computer disk drive spins up to5400 rpm in
2.00 seconds. What is the angularacceleration, α, of the
disk? (give answer to 3sig figs and in standard units)
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Rotational Motrion and Torque
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For the same computer disk, at the end of 2 seconds
what angle, θ, has the disk turned through? (The
disk in a computer disk drive spins up to 5400 rpm in
2.00 seconds.) (standard units and 3 sig figs)
ac = v2/r -- the acceleration an object in uniform
circular motion undergoes --- and ac is always directed
toward the center of the circle.
Uniform Circular Motion implies
∆ω = 0,
so α = 0.
But we have just discussed a case
where α ≠ 0 --- what does this mean
for ac?
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Rotational Motrion and Torque
the acceleration has 2 components: the ac from before and at ,
tangential acceleration
at is in the same direction as the velocity and iswhat caused v to
increase.
at = ∆v/∆t
= (∆ω /∆t)r
at = αr
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Rotational Motrion and Torque
x = rθ
v = rω
at = rα ac = 0α = 0
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A ball on the end of string swings in a horizontal circle
once every second. The magnitude of which of the
following is zero. Choose all the correct quantities.
A Velocity
B
Angular Velocity
C Centripetal Acceleration
D Angular Acceleration
E Tangential Acceleration
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Rotational Motrion and Torque
4 A ball on the end of string swings in a horizontal circle
once every second. The magnitude of which of the
following is constant, but not zero. Choose all the correct
quantities.
A
Velocity
B
Angular Velocity
C Centripetal Acceleration
D
Angular Acceleration
E Tangential Acceleration
Torque -- the rotational analog of force
the ability of a force to cause
rotation depends on 3 factors:
1. the magnitude of the force
2. the distance, r, from the pivot point that the force is
applied
3. the angle, φ, at which the force is applied
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Rotational Motrion and Torque
τ=rxF
τ = rFsinθ, with the direction given by the RHR
ixi=jxj=kxk=0
ixj=k
jxk=i
kxi=j
j x i = -k
k x j = -i
i x k = -j
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Rotational Motrion and Torque
5
6
In trying to open a door, Ryan pushes perpendicular to the
door's surface with a force of 240N at a distance of 75 cm
from the hinges. What torque does Ryan exert on the door?
(standard units and 2 sig figs)
Lulu uses a 20 cm long wrench to turn a nut. The wrench
handle is tilted 30 o above the horizontal and Lulu pulls
straight down on the end with a force of 100N. How much
torque does Lulu exert on the nut? (standard units, 2 sig
figs)
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Rotational Motrion and Torque
Which has the largest torque? The rods all have the same length and are pivoted at the dot.
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A B
2N
C
2N
D
2N
E
4N
o
45
4N
Which has the smallest torque? The rods all have the same length and are pivoted at the dot.
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A B
2N
C
D
2N
E
2N
4N
o
45
4N
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Rotational Motrion and Torque
Net Torque
r = 10 cm
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Two forces act on the wheel shown. What third force, acting at point P, will make the net torque on the wheel zero?
P
A B C D Pivot
E 14
Rotational Motrion and Torque
Page 232: 3, 5, 6, 7, 8, 10, 13, 14
F
m
T
A tangential force, F, exerts a torque
on the particle of mass, m, and causes
a tangential acceleration, at
the torque is given by:
how is this related to at?
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Rotational Motrion and Torque
What if there is more than on particle in motion?
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Rotational Motrion and Torque
We define the moment of inertia, I, as
I = Σmiri2
so that
Στ = Iα
I = Σmiri2
Στ = Iα
Recall:
Σ F = ma
What is the significance of
the moment of inertia?
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Rotational Motrion and Torque
An object with a large moment of inertia is hard to start
rotating (and to stop) and vice versa.
Depends not only on how much mass, but how
the mass is distributed .
The further the mass is from the axis of rotation the larger I is
and the more torque is needed to make the object rotate.
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Rotational Motrion and Torque
Moments for common shapes
1
(calculated from ∫r2dm)
Which will roll faster down a hill?
A
cylindrical hoop
B
cylindrical disk
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Rotational Motrion and Torque
An engine on a small airplane is specified to
have a torque of 500N .m. This engine drives a
2.0 m, 40 kg single blade propeller. On
start up, how long does it take the
propeller to reach 2000 rpm?
Conditions of Equilibrium
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Rotational Motrion and Torque
Consider a 100 N, 3.0 m long ladder supported by two
sawhorse, placed at one end and 1 meter from the other end.
What forces do the sawhorses exert on the ladder?
What is the minimum value of the coefficient of friction
so that a 3.0 m long ladder, inclined at an angle of 60 o
will not slip?
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Rotational Motrion and Torque
Try this with a soda can.
Center of Gravity - the point where the force
of gravity exerts no torque
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Rotational Motrion and Torque
A 10 kg mass hangs from a pulley on a rope. The pulley is
2.00kg and has a radius of 10.0 cm. Find the tension in the
rope and the acceleration of the mass.
Angular Momentum
for a single mass
recall p = mv
l=rxp=mrxv
for a rigid body
L = Iω
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Rotational Motrion and Torque
An ice skater spins around on the tips of her blades while
holding a 5.0 kg mass in each hand. She begins with her hand
outstretched and her hands 140 cm apart. While spinning at
2.0 rev/s she pulls the masses in a holds them 50 cm apart
against his shoulders. If we neglect the mass of the skater
how fast is she spinning after pulling the masses in?
Angular Momentum
recall: p = mv
by direct analogy
L = Iω
Conservation of Angular Momentum
L is conserved if τnet = 0
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Rotational Motrion and Torque
Page 234: 26, 30, 32, 34
Page 291: 30, 32
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