rotational mechanics ap physics c: mechanics. enough with the particles… do you ever get tired of...
TRANSCRIPT
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Rotational Mechanics
AP Physics C: Mechanics
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Enough with the particles…
• Do you ever get tired of being treated like a particle?
• We can not continue to lump all objects together and pretend that they undergo the same motion when acted upon by the same force…
• We will now study the rotation of rigid bodies.
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What do we already know?
Objects rotate about their
center of mass.
For rotational motion it is useful to consider
tangential and radial
components instead of x
and yIdeas of circular
motion and centripetal
force.
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What is a rigid body?• An extended object whose size
and shape do not change as it moves.
• This size and shape can not be neglected when modeling its motion.
• Objects that are held together by “massless rods” of molecular bonds.
• Some objects can be modeled as rigid during parts of their motion.
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Three types of motion of a rigid
body:
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There is an rotational analogy to
every concept of linear motion.• We have looked at the basics when studying
circular motion:
s r
ddt
ddt
vt ds
dt
a t dvt
dt
vt d r
dtr
a t d r
dtr
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Tangential Components:
vt r
vt
r
a t
r
a t rω
vt
r
Rotational Velocity
Rotational Acceleration
Tangential Velocity
Tangential Acceleration
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Radial (centripetal) Component
ar vt
2
r
ar r 2
r
vr 0
ar 2r
ω
vt Radial Velocity
Radial Acceleration
ar
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Rotational Kinematic Equations:
1
2t 2 0t 0
0 t
2 02 2
v v0 at
d 1
2at 2 v0t d0
v2 v02 2ad
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Sign Conventions:Counter-clockwise is positive direction
for ω and vt. So positive α can be speeding up in the ccw direction or slowing down in the cw direction.
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A Rotating Crankshaft
A car engine is idling at 500rpm. When the light turns green, the crankshaft rotation speeds up at a constant rate
to 2500rpm over an interval of 3 seconds. How many revolutions does the crankshaft make during this time
interval?This is a rotating rigid body with constant angular
acceleration
Imagine painting a dot on the crankshaft. IF the dot is at θ=0 and t=0, at a
later time, the dot will be at:
1
2t 2 0t
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A Rotating Crankshaft
A car engine is idling at 500rpm. When the light turns green, the crankshaft rotation speeds up at a constant rate
to 2500rpm over an interval of 3 seconds. How many revolutions does the crankshaft make during this time
interval?
0
t
0 500rev
min
1min
60sec
2rad
rev
52.4rad /s
2500rev
min50 262rad /s
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A Rotating Crankshaft
A car engine is idling at 500rpm. When the light turns green, the crankshaft
rotation speeds up at a constant rate to 2500rpm over an interval of 3 seconds.
How many revolutions does the crankshaft make during this time interval?
1
2t 2 0t
1
269.9 32 52.4 3 472rad
0
t
262 52.4 rad /s
3s69.9rad /s2
472rad1rev
2rad
75revs
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The Center of MassA 500g ball and a 2kg ball are
connected by a massless 50cm long rod.
Where is the center of mass?
What is the speed of each ball if they rotate about the center of mass at
40rpm?
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The Center of MassA 500g ball and a 2kg ball are
connected by a massless 50cm long rod.
Where is the center of mass?
xcm 2kg 0 0.5kg 0.5
2.5kg0.10m
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The Center of MassA 500g ball and a 2kg ball are
connected by a massless 50cm long rod.
What is the speed of each ball if they rotate about the center of mass at
40rpm?
xcm 0.10m
40rev
min
1min
60s
2rad
1rev
4.16rad /s
vt1 r1 0.10m 4.16rad /s 0.42m/s
vt 2 r2 0.40m 4.16rad /s 1.68m/s
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What can we measure?
• What can we measure and analyze with a bike in order to further understand rotational mechanics?
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Other Rotational Analogs?
Rotational Force?
Rotational Energy?
Rotational Momentum?
Rotational Mass?
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Rotational Energy?
• Why does the Sun rise in the morning? Why do magnets stick together? Because everybody says so. Everybody.
• - Michael Scott
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Rotational Energy?
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Rotational EnergyWhy must there be such a thing?
K t 1
2mv2
All of the atoms in the object are moving so they must have kinetic energy!
Translational Kinetic Energy
Can we use our analogies to find an expression for Rotational
Kinetic Energy?
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Rotational KE for a particle traveling in a
circle:
K t K r
v r
K t 1
2mv2
m ?
K r 1
2m r 2
K r 1
2mr22
m I
K r 1
2mr2 2
K r 1
2I2
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Moment of Inertia• Has nothing to do with a moment in time. The
word comes from the Latin momentum which means motion.
• The rotational analog to mass.
• Describes the distribution of mass relative to the axis of rotation.
• Is different for each shape and orientation.
• An object with a large mass is hard to accelerate, an object with a large moment of inertia is difficult to rotate.
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Moment of Inertia (Rotational Laziness)
• Inertia is the resistance to changes in motion
• Moment of inertia is the resistance to changes in rotation.
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Moment of InertiaConsider an irregular shape that is rotating:
The object’s rotational energy is the sum of the kinetic energies of each
particle
K r 1
2m1 r 1
2 1
2m2 r 2
2 ...
K r 1
2miri
2 2
I miri2Moment of Inertia can be calculated as the
sum of the contributions from each particle in an object
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Moment of Inertia
Calculating moment of
inertia can be very difficult for odd shapes. I for
many shapes has been
tabulated and printed in
handbooks for scientists and
engineers.
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Note about the axis…
If the rotation axis is not through the center of mass, then rotation may cause the center of mass to move up
or down in a gravitational field. The gravitational potential energy of the object will change as it spins.
With no friction in the axle or other dissipative forces, the mechanical
energy can be described as:
E mech K r Ug 1
2I2 Mgycm
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The three masses, held together by lightweight plastic rods, rotate about an axle passing through the right–
angle corner. At what angular velocity does the triangle have 100mJ of rotational energy?
150g
250g300g
axleω
6cm
8cm
I miri2
K r 1
2I2
I 0.15kg 0.06m 2 0.25kg 0.08m 2 0.3kg 0 2
I 2.14 10 3kgm2
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The three masses, held together by lightweight plastic rods, rotate about an axle passing through the right–
angle corner. At what angular velocity does the triangle have 100mJ of rotational energy?
150g
250g300g
axleω
6cm
8cm
K r 1
2I2
2K r
I
I 2.14 10 3kgm2
2 100 10 3 J
2.14 10 3kgm2 9.67rad /s
9.67rad /s60s
1min
1rev
2
92rpm
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A 1m long, 0.2kg rod is hinged at one end and connected to a wall. It is held out horizontally, then released. What is the speed of the tip of
the rod as it hits the wall?Conservation of
Energy:
E i E f
Mgycm 1
2I2
K r Ug i K r Ug
f
MgL
2
1
2
1
3ML2
2
g
2
L
62
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A 1m long, 0.2kg rod is hinged at one end and connected to a wall. It is held out horizontally, then released. What is the speed of the tip of
the rod as it hits the wall?Conservation of
Energy:
g
2
L
62
3g
L
vt
r
r L
L3g
Lvt
vt 3gL 5.4m/s
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Calculating Moment of Inertia:
Like finding inertia, we can not simply place the object on a scale to find its
moment of inertia. We must go through the calculation.
I ri2miRecall: Moment of Inertia can be calculated as
the sum of the contributions from each particle in an object
as Δm approaches zero it can be replaced with the differential dm.
I r2dm
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Calculating Moment of Inertia: Tips
I r2dmBreak the object into elements that you will sum
together. Do this in a way that keeps the same distance from the axis for all particles in each element.
You will sum the elements over a range of distances so you must find an expression to substitute dm with a
differential dx, dy, or dz.
M
L
dm
dx
Densities are helpful but not necessarily the only way to solve:
M
A
dm
dA
M
V
dm
dVFor a complex object made up of parts with known
moments of inertia, sum the terms of each to find part to find the moment of inertia of the object:
Iobject I1 I2 I3 ...
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Find the moment of inertia of a circular disk of radius R and mass M that rotates
on an axis passing through its center.
I r2dm0
R
M
A
dm
dA
A R2
dA 2rdr
I r2 M
R22rdr
0
R
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Find the moment of inertia of a circular disk of radius R and mass M that rotates
on an axis passing through its center.
I r2 M
R2 2rdr0
R
I 2M
R2 r3dr0
R
I 2M
R2
r4
4
0
R
I MR2
2
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The four T’s in the diagram are made from identical rods. Rank in order, from
largest to smallest, the moments of inertia for rotation about each dashed
line.
Ia>Id>Ib>I
c
Which has the most mass distributed farthest from the axis???
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Parallel-Axis Theorem
We have been calculating the moment of inertia for rotational axes that run through the center of mass. This
theorem helps us if we wish to use an off-center axis but know where it is in relation to a parallel, on-center axis.
I Icm Md2
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Parallel-Axis Theorem
Find the moment of inertia of a thin rod with mass M and length L about an axis
1/3rd of the length from one end.
I Icm Md2
Icm 1
12ML2 Moment of inertia through the
center of mass of a thin rod from table.
I ML2
12
M
L
6
2
ML2
12
ML2
36
ML2
9
cm1/3rd
d=L/6
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Rotational Force?Why must there be such a thing?
Because all net forces do not cause rotation!
How can rotational force be maximized?
How do we calculate it?
Apply a force at the proper location.
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Torque
• Torque is to rotational motion as force is to linear motion.
• Torque is given the Greek symbol capital tau (τ)
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Torque• Is greater with a greater
force
• Is greater if the force is applied farther from the axis or rotation.
• Is greater if the angle of application of the force is perpendicular to the radial line.
F
r
sin
Frsin
F r
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Torque
F r
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Torque and Lever arm or Moment Arm
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How much Torque does Luis Provide?
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Rank the Torques
τe>τa=τd>τb
>τc
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No Change in rotation
Analog to Newton’s 2nd Law
0No Change in
motion (no acceleration)
F 0
EQUILIBRIUM
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Torque due to gravity
grav MgxcmCenter of mass
is relative to the axis of
rotation
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Torque due to gravityThe gravitational torque is found by treating the object as if all its mass were concentrated at the center of
mass.
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What is the torque on the 500kg steel
beam?
grav Mgxcm 500kg 9.8m/s2 0.8 3920Nm
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Torque
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Equilibrium Lab Challenge:
• Please DO NOT touch or alter any set-up.
• Report the unknown mass in each. I will collect one per group for score based on amount correct.
• 7 Minutes per table remaining time may be used to reconsider past tables but not revisit them while other groups are working.
• Bonus goes to group with most correct.
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Rotational DynamicsA net centripetal force will cause an object’s path to change direction.
Fc mac mvt
2
r m2r
A net tangential force will cause a rotating object to speed up or slow
down.
Ft mat mr at ac
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Rotational DynamicsOnly component of force tangent to
a circle causes rotation change:
Ft mat mr
at ac
F r Ftr
Ftr mr r mr2 I
Surprised?
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Analog to Newton’s 2nd Law
I
F maUnbalanced
ForcesUnbalanced
Torques
A net torque causes an angular acceleration!
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Rank the angular accelerations!
αb > αa > αc = αd = αe
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Angular Momentum (L)
L I
p m
v
Linear Momentum Angular Momentum
m I
v
p
F t
L
t
L
F
r t
L
p
r
F
Angular Momentum for a particle in circular motion:
L I
mr2 mr2
v
r
mr
v
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Angular Momentum
net
dL
dt
F net
dp
dt
Newton’s 2nd Law:
A net torque causes a particles angular
momentum to change.
The rate of change of a systems
angular momentum is the net torque on
the system.
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Conservation of Angular Momentum:The angular momentum of an isolated (no
external torques) system is conserved.
L i
L f
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A note on direction:Why is it easier to balance a bike
when the wheels are spinning than when they are not?
An object with a large linear momentum is difficult to slow down or to knock it off of its
straight line path.An object with a large angular momentum is
difficult to slow down or to change the direction of its axis
L
p
r
F
r
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General Direction of Cross Product: The
RHR1. Point the fingers of your right hand in the direction of
vector A. 2. Close your fingers into the direction of vector B.3. Your thumb points in the direction of vector C.
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Gyroscope!
PrecessionUse of a gyroscope