UB, Phy101: Chapter 9, Pg 1 Physics 101: Chapter 9 l Today’s lecture will cover Textbook Sections...
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Transcript of UB, Phy101: Chapter 9, Pg 1 Physics 101: Chapter 9 l Today’s lecture will cover Textbook Sections...
UB, Phy101: Chapter 9, Pg 1
Physics 101: Physics 101: Chapter 9 Chapter 9
Today’s lecture will cover Textbook Sections 9.1 - 9.6
UB, Phy101: Chapter 9, Pg 2
Rotation Summary Rotation Summary (with comparison to 1-D kinematics)(with comparison to 1-D kinematics)
Angular Linear
constant
tαωω 0
0 021
2t t
constanta
v v at 0
x x v t at 0 021
2
And for a point at a distance R from the rotation axis:
x = Rv = Ra = R
See text: chapter 8
See Table 8.1
Δθ2 22
0 x2a vv 22
0
UB, Phy101: Chapter 9, Pg 3
New concept: TorqueNew concept: Torque
See text: chapter 9
Rotational analog of force
Torque = (magnitude of force) x (lever arm)
= F l
UB, Phy101: Chapter 9, Pg 4
Comment on axes and signComment on axes and sign(i.e. what is positive and negative)(i.e. what is positive and negative)
Whenever we talk about rotation, it is implied that there is a rotation “axis”.
This is usually called the “z” axis (we usually omit the z subscript for simplicity).
Counter-clockwise (increasing ) is usuallycalled positive.
Clockwise (decreasing ) is usuallycalled negative. z
UB, Phy101: Chapter 9, Pg 5
Chapter 9, PreflightChapter 9, Preflight
The picture below shows three different ways of using a wrench to loosen a stuck nut. Assume the applied force F is the same in each case.
In which of the cases is the torque on the nut the biggest?
1. Case 1 2. Case 2 3. Case 3
CORRECT
UB, Phy101: Chapter 9, Pg 6
Chapter 9, PreflightChapter 9, Preflight
The picture below shows three different ways of using a wrench to loosen a stuck nut. Assume the applied force F is the same in each case.
In which of the cases is the torque on the nut the smallest?
1. Case 1 2. Case 2 3. Case 3
CORRECT
UB, Phy101: Chapter 9, Pg 9
Static EquilibriumStatic Equilibrium
A system is in static equilibrium if and only if:
acm = 0 Fext = 0 = 0 ext = 0 (about any axis)
torque about pivot due to gravity:
g = mgd
(gravity acts at center of mass)
Center of mass
pivotd
W=mg
This object is NOT in static equilibrium
UB, Phy101: Chapter 9, Pg 10
Center of mass
pivotd
W=mg
Torque about pivot 0
Center of mass
pivot
Torque about pivot = 0
Not in equilibrium Equilibrium
UB, Phy101: Chapter 9, Pg 11
Homework HintsHomework Hints
Painter is standing to the right of the support B.
FA FB
Mg mg
What is the maximum distance the painter can move to the right without tipping the board off?
UB, Phy101: Chapter 9, Pg 12
Homework HintsHomework Hints
If its just balancing on “B”, then FA = 0 the only forces on the beam are:
FB
Mg mg
Using FTOT = 0: FB = Mg + mg This does not tell us x
x
UB, Phy101: Chapter 9, Pg 13
Homework HintsHomework Hints
Find net torque around pivot B: (or any other place)
FB
Mg mg
(FB ) = 0 since lever arm is 0
(Mg ) = Mgd1
d1 d2
(mg ) = -mgd2
Total torque = 0 = Mgd1 -mgd2
So d2 = Md1 /m and you can use d1 to find x
UB, Phy101: Chapter 9, Pg 14
Homework HintsHomework Hints
Painter standing at the support B.
FA FB
Mg mg
Find total torqueabout this axisD
d
(FA) = - FAD
(Mg) = Mgd
(FB) = 0 (since distance is 0)
(mg) = 0 (since distance is 0)
Total torque = 0 = Mgd -FAD
So FA = Mgd /D
UB, Phy101: Chapter 9, Pg 15
MORE EXAMPLES (bar and weights suspended by the string):Find net torque around this (or any other) place
(m1g) = 0 since lever arm is 0
x
T
Mgm2g
m1g
UB, Phy101: Chapter 9, Pg 17
(Mg ) = -Mg L/2
(m1g) = 0 since lever arm is 0
x
T
Mgm2g
m1g
(T ) = T x
UB, Phy101: Chapter 9, Pg 18
(Mg ) = -Mg L/2
(m1g) = 0 since lever arm is 0
L
T
Mgm2g
m1g
(T ) = T x
(m2g ) = -m2g L
All torques sum to 0: Tx = MgL/2 + m2gL So x = (MgL/2 + m2gL) / T
UB, Phy101: Chapter 9, Pg 23
Torque and Stability
Center of mass outside of base:
--> unstable
Center of mass over base:
--> stable
UB, Phy101: Chapter 9, Pg 26
Moments of Inertia of Common Objects
Hollow cylinder or hoop about central axis
I = MR2
Solid cylinder or disk about central axis
I = MR2/2
Solid sphere about center
I = 2MR2/5
Uniform rod about center
I = ML2/12
Uniform rod about end
I = ML2/3
UB, Phy101: Chapter 9, Pg 29
Chapter 9, PreflightChapter 9, Preflight
The picture below shows two different dumbbell shaped objects. Object A has two balls of mass m separated by a distance 2L, and object B has two balls of mass 2m separated by a distance L. Which of the objects has the largest moment of inertia for rotations around the x-axis?
1. A 2. B 3. Same
CORRECT
x
2LL
m
m
2m
2m
A B
I = mL2 + mL2
= 2mL2
I = 2m(L/2)2 + 2m(L/2)2
= mL2
UB, Phy101: Chapter 9, Pg 30
Rotational Kinetic Energy
Translational kinetic energy:
KEtrnas = 1/2 MV2cm
Rotational kinetic energy:
KErot = 1/2 I2
Rotation plus translation:
KEtotal = KEtrans + KErot = 1/2 MV2cm + 1/2 I2
UB, Phy101: Chapter 9, Pg 32
See text: chapters 8-9
See Table 8.1
Define Angular MomentumDefine Angular Momentum
MomentumMomentum Angular MomentumAngular Momentum
p = mV L = I
conserved if Fext = 0 conserved if ext =0
Vector Vector!
units: kg-m/s units: kg-m2/s
UB, Phy101: Chapter 9, Pg 33
Chapter 9, Pre-flightsChapter 9, Pre-flights
You are sitting on a freely rotating bar-stool with your arms stretched out and a heavy glass mug in each hand. Your friend gives you a twist and you start rotating around a vertical axis though the center of the stool. You can assume that the bearing the stool turns on is frictionless, and that there is no net external torque present once you have started spinning. You now pull your arms and hands (and mugs) close to your body.
UB, Phy101: Chapter 9, Pg 34
Chapter 9, PreflightChapter 9, Preflight
What happens to your angular momentum as you pull in your arms? 1. it increases 2. it decreases 3. it stays the same
L1 L2
This is like the spinning skater example in the book. Since the net external torque is zero (the movement of the arms and hands involve internal torques), the angular momentum does not change.
CORRECT
UB, Phy101: Chapter 9, Pg 35
Chapter 9, PreflightChapter 9, Preflight
1 2
I2 I1
L L
What happens to your angular velocity as you pull in your arms? 1. it increases 2. it decreases 3. it stays the same
as with the skater example given in the book....as you pull your arms in toward the rotational axis, the moment of inertia decreases, and the angular velocity increases.
CORRECT
My friends and I spent a good half hour doing this once, and I can say...based on a great deal of nausea, that the angular velocity does increase.
UB, Phy101: Chapter 9, Pg 36
Chapter 9, PreflightChapter 9, Preflight
What happens to your kinetic energy as you pull in your arms? 1. it increases 2. it decreases 3. it stays the same
CORRECT
Your angular velocity increases and moment of inertia decreases, but angular velocity is squared, so KE will increase with increasing angular velocity
1 2
I2 I1
L L
K 12
2I 12
2 2
II 1
22
IL (using L = I )
UB, Phy101: Chapter 9, Pg 37
Two different spinning disks have the same angular momentum, but disk 2 has a larger moment of inertia than disk 1. Which one has the biggest kinetic energy ?
(a) disk 1 (b) disk 2
Spinning disksSpinning disks
UB, Phy101: Chapter 9, Pg 38
K 12
2I 12
2 2
II
If they have the same L, the one with the smallest I will have the biggest kinetic energy.
L I1 1
disk 2
L I2 2
disk 1I1 < I2
12
2
IL (using L = I )
UB, Phy101: Chapter 9, Pg 39
Preflights: Turning the bike wheelPreflights: Turning the bike wheel
A student sits on a barstool holding a bike wheel. The wheel is initially spinning CCW in the horizontal plane (as viewed from above). She now turns the bike wheel over. What happens?
1. She starts to spin CCW.2. She starts to spin CW.3. Nothing
CORRECT
UB, Phy101: Chapter 9, Pg 40
Turning the bike wheel...Turning the bike wheel...
Since there is no net external torque acting on the student-stool system, angular momentum is conserved. Remenber, L has a direction as well as a magnitude!
Initially: LLINI = LLW,I
Finally: LLFIN = LLW,F + LLS
LLW,F
LLS
LLW,I LLW,I = LLW,F + LLS
UB, Phy101: Chapter 9, Pg 41
Rotation Summary Rotation Summary (with comparison to 1-d linear motion)(with comparison to 1-d linear motion)
Angular Linear
constant
t 0
0 021
2t t
constanta
v v at 0
x x v t at 0 021
2
See text: chapters 8-9
See Table 8.1
maF I
)2/(2
1 22 mpmvKEtrans ILIKErot 2/
2
1 22
L vm p