Download Torque, Moment of Inertia and angular motion

November 13, 2014
Teacher
note: do torque first - in torque folder
Demonstrations
·
· torque and moment of inertia using meter stick with masses
· gyroscope - angular momentum - why doesn't it tip?
· bicycle wheel - angular momentum and torque
Angular Motion Introduction
exploratory
· Angular motion Phet
Torque Exploratory
· Phet
I stands for Moment of Inertia
τ = Iα
Torque is the rotational equivalent to the linear force.
Torque causes a rotational or angular acceleration
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November 13, 2014
What is the moment of inertia around a new axis?
How are these two things different?
Example 8.9, Exercise 8.9
What is the moment of inertia around a new axis?
November 13, 2014
Net Torque causes an object to angularly accelerate
Net Torque causes an object to angularly accelerate
** In an equation, if the symbol is for a rotational quantity, it
must be in radians. Sometimes radians "just appears" or
simply "disappears" do not be alarmed by this. This is due to
the definition of radians.
Which would win a race down an inclined
plane, both starting from rest?
Which is traveling the fastest?
Explain?
November 13, 2014
Force causing a torque on the object
Force causing a torque on the object
radius
radius
Fg(perp)
Fg(perp)*r = τ = I*α
Fg(perp)
Fg
Fg
Fg(parallel)
Fg(parallel)
2
mg*cosθ *r = ∑mr *α
Notice the m's cancel - so the mass of the object is irrelevant
If at= α*r Then, while on the ramp we could find the
tangential acceleration
The angular velocity will be greater if
the _______________ is smaller?
The angular velocity will be greater if
the _______________ is smaller?
If the ball gets to the bottom of the ramp and does not
accelerate any more, could we find the speed of the ball?
If the object gets to the bottom of the ramp
and does not accelerate any more, could we
find the speed of the object?
** for a hoop or cylindrical shell
PEg = KE linear + KE rotational
mgh = 1/2mv2 + 1/2Iω2
14 meters tall
vt = ωr so ω = vt/r and I = ∑mr2
mgh = 1/2mvt2 + 1/2*∑mr2*(vt / r )2
What cancels?
At the bottom there is no net force on the object
What kind of energy does it have? be careful
Where does it come from?
What would the energy conservation equation look like?
Hoop
Find speed of hoop at the bottom of a 14m high ramp?
Find speed of cylinder, sphere
check answers
try problem 8.34
November 13, 2014
8.34
8.34
Fchain = force of chain causing forward motion
Fchain = force of chain causing forward motion
Fr = resistive force slowing the wheel down
Fr = resistive force slowing the wheel down
Both forces cause a torque on the wheel.
Both forces cause a torque on the wheel.
What is the net torque?
What is the net torque?
Fchain
* write the net torque equation
Fchain
τchain - τr = Iα
Fr
Fr
8.34
**Why is one subtracted? it causes a torque in the other
direction
8.34
τchain - τr = Iα
τchain - τr = Iα
(Fchain*.045m) - (120N*.32m) = Iwheel*4.5 rad/s2
(Fchain*.045m) - (120N*.32m) = Iwheel*4.5 rad/s2
* .045m and .32m, why? book gave us diameters, we want radii
Fchain
* .045m and .32m, why? book gave us diameters, we want radii
Fchain
*assume wheel is a hoop
Fr
Iwheel
= MR2 = 1.8kg*(.32m)2 = .184 kgm2 *chart page 241
substitute and solve for Fchain
*assume wheel is a hoop
Fr
Iwheel = MR2 = 1.8kg*(.32m)2 = .184 kgm2
substitute and solve for Fchain
a. 872 N
b. 1400 N
November 13, 2014
AP 8.36 8.40
solve first with forces
then with energy
spool is a solid disk
use table on page 241
moment of inertia
**so that the accelerations
are in the same direction
-
+
at = r*α
November 13, 2014
try 8.31, 8.33, 8.36
Conservation of angular momentum and Energy
concepts
November 13, 2014
Conservation of Angular Momentum, applying ratios
length (L) meters
vt (m/s)
If the objects are moved to half their distance from the axis of
rotation, what happens to the following?
angular momentum?
angular speed?
tangential speed?
rotational kinetic energy?
try 8.48 and 8.49
length = 1.0 meters
vt = 5.0 m/s
L =?
length = 1.0 meters
vt = 5.0 m/s
L =?
If the objects are moved to half their distance from the axis of rotation, what
happens to its angular momentum? angular speed? tangential speed?
If the objects are moved to half their distance from the axis of rotation, what
happens to its angular momentum? angular speed? tangential speed?
vt = rω ; ω = 5.00m/s / .5m = 10 rad/s
vt = rω ; ω = 5.00m/s / .5m = 10 rad/s
L (angular momentum) remains the same due to
conservation of mometum
Ii = 4kg*(.5m)2 + 3kg*(.5m)2 = 1.75 kgm2
L (angular momentum) remains the same due to
conservation of mometum
Iω =I ω
i i f f
Iω =I ω
i i f f
1.75kgm2*10 rad/s = .438kgm2*ωf
1.75kgm2*10 rad/s = .438kgm2*ωf
ωf = 40 rad /s
vt = rω ; .25m * 40 rad/s = 10 m/s
Ii = 4kg*(.5m)2 + 3kg*(.5m)2 = 1.75 kgm2
ωf = 40 rad /s
vt = rω ; .25m * 40 rad/s = 10 m/s
If = 4kg*(.25m)2 + 3kg*(.25m)2 = .438 kgm2
half of the radius --- _____* the angular velocity
If = 4kg*(.25m)2 + 3kg*(.25m)2 = .438 kgm2
half of the radius ---4* the angular velocity
half of the radius -- ____ the moment of Inertia
half of the radius --- _____* the tangential speed
half of the radius -- 1/4 the moment of Inertia
half of the radius ---2* the tangential speed
KE's ?
November 13, 2014
What is the form or forms of energy, while a rolling along
horizontally? *be careful
What is the form or forms of energy, while a rolling object is
a rest on top of a hill?
zero level
zero level
Apply conservation of energy formula
AP 8.40 Conservation of Energy Example
todo
Write the conservation of energy formula for the system
practice ratio problems - in general - see universal G
M1
M1
at rest
Ratios with Kepler's third law
M2
moment before
the object hits the
ground
height
M2
Ground
November 13, 2014