midterm 3 review (ch 9-14) - physics and astronomy at...
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Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
PowerPoint® Lectures forUniversity Physics, Twelfth Edition
– Hugh D. Young and Roger A. Freedman
Lectures by James Pazun
Midterm 3 Review (Ch 9-14)
Ch 9 Overview: Rotational Motion
• Take our results from “linear” physics and do the same for “angular” physics
• Analogue of – Position ←– Velocity ←– Acceleration ←– Force ←– Mass ←– Momentum ←– Energy ←
Cha
pter
s 1-3
Cha
pter
s 4-7
Ch 9: Rotational Kinematics
l R atan vtan
t R arad
v2tan
R R 2vtan
lt
R
Relation between linear quantities and angular quantities (by geometry)
Kinematic equation just like before:θ like displacement xω like velocity vα like acceleration a
Kinetic energy of a rotating object about a fixed (stationary) axis:
KErot 12
I 2 I miri2
i Moment of inertia or “angular mass”
(DEPENDS ON THE AXIS CHOSEN)
KE 12
mvcm2
12
Icm2 Sum of translational and rotational energies
Rolling without Slipping• In reality, car tires both rotate and
translate• They are a good example of
something which rolls (translates, moves forward, rotates) without slipping
• Is there friction? What kind?SM Flag: 30%Relation between translational and rotational velocities in no slip condition
A Rolling Wheel
• A wheel rolls on the surface without slipping with velocity V (your speedometer)
• What is the velocity of the center of the wheel (point C)?
• What is the velocity of the lowest point (point P) w.r.t. the ground?– Does it make sense to you?
Try Differently: Paper Roll• A paper towel unrolls
with velocity V– Conceptually same thing
as the wheel– What’s the velocity of
points: – A? B? C? D? ABC
C B A
D• Point C is where rolling
part separates from the unrolled portion– Both have same velocity
there
Bicycle comes to RestA bicycle with initial linear velocity V0 decelerates
uniformly (without slipping) to rest over a distance d. For a wheel of radius R:
a) What is the angular velocity at t0=0?b) Total revolutions before it stops?c) Total angular distance traversed
by wheel?d) The angular acceleration?e) The total time until it stops?
An athlete throwing the discusA discus thrower moves the discus in a circle of radius 80.0 cm. At a certain instant the thrower is spinning at an angular speed of 10.0 rad/s and the angular speed is increasing at 50.0 rad/s2. At this instant find the acceleration of the discus and its magnitude
atan r (0.800 m)(50.0 rad/s2) = 40.0 m/s2
arad r 2 (10.0 rad/s)2(0.800 m) = 80.0 m/s2
a atan2 arad
2 89.4 m/s2
Finding the moment of inertia for common shapes: Moment of Inertia will be provided for the exam
What is the velocity of the block when it hits the ground?
The work done by the cable is zero since the two tension forces cancel each other out so energy is conserved
KEi PEi KE f PE f
0 mgh 12
mv2 12
I 2 0
0 mgh 12
mv2 12
12
MR2
vR
2
0
v =2mgh
(m +M/2)
Rolling without slipping
KE 12
mvcm2
12
Icm2
12
mvcm2
12
Icm
R2 v2
When rolling without slipping then vcm R This is the condition to roll without slipping.
Then, if you are rolling without slipping the kinetic energy is
Also note that when one is rolling without slipping (i.e. rolling down an incline) the friction force is static so no work is done by it and energy is conserved in this case.
Consider the speed of a yo-yo toyWhat is the speed of the Yo-yo at the bottom (use conservation of energy)
Why conservation of energy: the hand is not moving so it does no work on the system. You may be confused about the tension but keep in mind that it is an internal force so the sum of the upper and lower tension is zero.
Ei E f
0 Mgh 12
Mvcm2
12
Icm2
0 Mgh 12
Mvcm2
12
12
MR2
vcm
R
2
Mgh 34
Mvcm2
vcm 43
gh
Ch 10: Rotational Dynamics
The direction of the torque is given by the RHR
Torque results from force applied at a distance from a pivot point
sinrFr r r
r F
Torque• Write Torque as
• To find the direction of the torque, wrap your fingers in the direction the torque makes the object twist
• If the axis is fixed, what is the net Torque on the wheel?
Fr
Fr
sin||||||
Torque and Moment of Inertia• Force vs. Torque
F=ma and = I
• Mass vs. Moment of Inertia
dmrI
mrIm2
2
or
Flywheel problem from Ch 9(using work energy theorem)
The cable is wrapped around a cylinder. If it unwinds 2.0 m by pulling it with a force of 9.0 N and it starts at rest, what is its final angular velocity and velocity of the cable? (use work energy theorem)
Wtotal KE f KEi
Fx 12
I 2 0
2Fx
I
4FxmR2 20 rad/s
v R (20 rad/s)(0.060 m) =1.2 m/s
Flywheel problem using torque(using work energy theorem)
The cable is wrapped around a cylinder. If it unwinds 2.0 m by pulling it with a force of 9.0 N and it starts at rest, what is its final angular velocity and velocity of the cable? (use work energy theorem)
FR (9.0 N)(0.06 m)
I 12
MR2
I I
2FRMR2
2FMR
6.0 rad/s2
Use α to get acceleration of the cable:
atan R (0.06m)(6.0 rad/s2) 0.36 m/s2
v2 v02 2atan (x x0)Then use kinematics
v 0 2(0.36 m/s2)(2 m) 1.2 m/s
What is the velocity of the block when it hits the ground?
The work done by the cable is zero since the two tension forces cancel each other out so energy is conserved
KEi PEi KE f PE f
0 mgh 12
mv2 12
I 2 0
0 mgh 12
mv2 12
12
MR2
vR
2
0
v =2mgh
(m +M/2)
Another look at the unwinding cableWhat is the linear acceleration of the block?
These are two coupled objects; one rotates and the other moves linearly
For the rotating wheel we have:
I
TR 12
MR2 aR
T
12
Ma
For the block we have:
mg T ma
mg 12
Ma ma
a mg
m M /2
Combine the two equations to get
SM Flag: 55%Connection between rotational and translational accln of coupled objects
SM Flag: 50%Angular momentum both rxp and Iw
Conservation of angular momentum
Before we saw that if the external forces on a system are zero then linear momentum is conserved. Similar for angular momentum.
If the external torques on a system are zero then the TOTAL angular momentum of the systems is conserved.
Note that in this case the TORQUE has to be zero. There can be still forces acting on the system but they do not generate any torque (e.g. force due to gravity on the cat which acts at the center of mass)
I11 I2 2
r r r
r F Recall that torque was defined as
Similarly the angular momentum of a particle is
r L r r r p r r m r v
How a car’s clutch work
The clutch disk and the gear disk is pushed into each other by two forces that do not impart any torque, what is the final angular velocity when they come together?
Lzbefore Lzafter
IAA IBB (IA IB ) final
final IAA IBB
(IA IB )
Ch 11: Conditions for equilibrium
• Net Force is zero (x,y,z)• Net torque is zero.
Strategy
Fx 0
Fy 0 0
1st Draw the free body diagram and the location where the forces are applied
2nd Choose a Pivot Point which will delete the largest amount of torques
3rd Compute the torques from each force
4th Solve the equations
Problem 11.27The horizontal beam in the figure weighs 150 N, and its center of gravity is at its center. Find the tension in the cable and on the beam at the hinge.
SM Flag: 30%Contact forces
Ch 13: Newton’s Law of Gravitation• The gravitational force
is always attractive and depends on both the masses of the bodies involved and their separations.
Fg Gm1m2
r2
G 6.6742 10-11 N m2
kg2
Weight (skip Weight Watchers, just climb upward)
• Gravity (and hence, weight) decreases as altitude rises.
Gravitational potential energy• Objects changing their distance from earth are also
changing their potential energy with respect to earth.
PE Gm1m2
r12
This is the true potential energy. The zero level is set when they are very far apart
Satellite circular motion
The force is radial so Newton’s 2nd law reads:
r : G ME ms
R2 msv2
R ms
4 2RT 2
v GME
R
T 4 2R3
GME
T R3 / 2
SM Flag: 40%Gravitational force provides centripetal accln
Escaping from the EarthWhat is the velocity you need to shoot straight up from the surface of the earth an not come back (conservation of energy)?
KE1 PE1 KE PE
12
m v2GmME
RE
0 0
v 2GME
RE
1.12 104 m/s = 25,000 mph
SM Flag: 40%Energy conservation
Ch 14: Simple Harmonic Motion• The spring drives the glider back
and forth on the air-track and you can observe the changes in the free-body diagram as the motion proceeds from –A to A and back.
The pendulum is another example
Elements of harmonic motion:
Frequency: f=1/TAngular frequency: ω=2πfAmplitude: maximum deflection/stretch/compression
Simple harmonic motion• An ideal spring
responds to stretch and compression linearly, obeying Hooke’s Law.
Fx kxThe restoring force towards the equilibrium point is LINEAR from the displacement from equilibrium
Let’s look at Newton’s 2nd law:
Fx max kx max md2xdt 2
d2xdt 2
km
x
d2xdt 2 2x
Watch variables change for a glider example • As the glider undergoes
SHM, you can track changes in velocity and acceleration as the position changes between the turning points.
x(t) Acos(t )v(t) Asin(t )a(t) A 2 cos(t )
Energy in SHM• Energy is conserved during SHM and the forms
(potential and kinetic) interconvert as the position of the object in motion changes.
E 12
mvx2
12
kx 2 12
kA2 12
mvmax2
SM Flag: 37%Energy at different points
Energy in SHM II
The simple pendulum
Ftan mgsin mg
Ok when Θ<<1
Ftan matan mLd2dt 2
mLd2dt 2 mg
d2dt 2
gL
gL
f 1
2gL
T 2Lg
The physical pendulum• A physical pendulum
is any real pendulum that uses an extended body in motion. This illustrates a physical pendulum.
T 2I
mgd