Transcript Physics 207: Lecture 2 Notes

Lecture 9

Today:
 Review session
Assignment: For Thursday, Read Chapter 8, first four sections
Exam Wed., Feb. 18th from 7:15-8:45 PM Chapters 1-7
One 8½ X 11 note sheet and a calculator (for trig.)
Place: Room 2103: All Sections
Physics 207: Lecture 9, Pg 1
Textbook Chapters
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Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Concept of Motion
1D Kinematics
Vector and Coordinate Systems
Dynamics I, Two-dimensional motion
Forces and Free Body Diagrams
Force and Newton’s 1st and 2nd Laws
Newton’s 3rd Law
Exam will reflect most key points (but not all)
~30% of the exam will be more conceptual
~70% of the exam is problem solving
Physics 207: Lecture 9, Pg 2
The flying bird in the cage
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You have a bird in a cage that is resting on your upward
turned palm. The cage is completely sealed to the
outside (at least while we run the experiment!). The bird
is initially sitting at rest on the perch. It decides it needs
a bit of exercise and starts to fly.
Question: How does the weight of the cage plus bird
vary when the bird is flying up, when the bird is flying
sideways, when the bird is flying down?
So, what is holding the airplane up in the sky?
Physics 207: Lecture 9, Pg 3
Example with pulley
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A mass M is held in place by a
force F. Find the tension in each
segment of the massless ropes
and the magnitude of F.
 Assume the pulleys are
massless and frictionless.
The action of a massless
frictionless pulley is to change the
direction of a tension.
This is an example of
static equilibrium.
T4
T1
T3
T2
F
<
T5
M
Physics 207: Lecture 9, Pg 4
Example with pulley
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•
A mass M is held in place by a force F.
Find the tension in each segment of the
rope and the magnitude of F.
T4
 Assume the pulleys are massless and
frictionless.
T1
 Assume the rope is massless.
T3
T2
The action of a massless frictionless
pulley is to change the direction of a
T5
F
tension.
M
Here F = T1 = T2 = T3 = T
•
• Equilibrium means S F = 0 for x, y & z
• For example: y-dir ma = 0 = T2 + T3 – T5
•
<
and ma = 0 = T5 – Mg
So T5 = Mg = T2 + T3 = 2 F  T = Mg/2
Physics 207: Lecture 9, Pg 5
Example
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The velocity of an object as a
function of time is shown in the
graph at right. Which graph
below best represents the net
force vs time relationship for this
object? (E)
Physics 207: Lecture 9, Pg 6
Another Example
What are the coefficients of static
and kinetic friction?
mS=1.0
mk=0.6
acceleration
A 200 kg truck accelerates eastwards on a horizontal road in
response to a gradually increasing frictional force from the
ground. There is an unsecured 50 kg block sitting on the truck
bed liner. There is friction between the block and the bed
liner. An accelerometer is mounted in the truck. The block
accelerates with the truck until the acceleration reaches 10
m/s2. At that instant the block begins to slide and the truck’s
accelerometer now reports a value of 11 m/s2.
11
10
0
time
Physics 207: Lecture 9, Pg 7
Example
Wedge with friction
A mass m slides with friction down a wedge of angle q at
constant velocity. The wedge sits at rest on a frictionless
surface and abuts a wall.
What is the magnitude of the force of the wall on the block?
FBD block
N
fk
q
mg
Physics 207: Lecture 9, Pg 8
Example Wedge with friction
FBD block
A mass m slides with friction down a wedge
of mass M & angle q at constant velocity.
The wedge sits at rest on a frictionless
surface and abuts a wall.
What is the magnitude of the force of the
wall on the block?
fk
3rd Law
N
mg
FBD wedge
-fk
Fw
-N
Mg
q
FF
Physics 207: Lecture 9, Pg 9
Example Wedge with friction
A mass m slides with friction down a wedge
of mass M & angle q at constant velocity.
The wedge sits at rest on a frictionless
surface and abuts a wall.
What is the magnitude of the force of the
wall on the block?
x-dir:
FBD block
N
fk
y
q
mg
S Fx = 0 = -fk + mg sin q
fk = mg sin q
y-dir:
S Fy = 0 =
N - mg cos q
N = mg cos q
Physics 207: Lecture 9, Pg 10
x
Example Wedge with friction
A mass m slides with friction down a wedge
of mass M & angle q at constant velocity.
The wedge sits at rest on a frictionless
surface and abuts a wall.
FBD wedge
What is the magnitude of the force of the
wall on the block?
mg cos q sin q
mg cos q
Notice that
mg cos q sin q - mg cos q sin q = 0 !
Fw
q
Force wall = 0
But there are faster ways.
q
q
mg cos q sin q
Mg
FF
Physics 207: Lecture 9, Pg 11
Example
Another setting
Three blocks are connected on the table as shown. The
table has a coefficient of kinetic friction of mK=0.40, the
masses are m1 = 4.0 kg, m2 = 1.0 kg and m3 = 2.0 kg.
m2
m1
T1
m3
(A) What is the magnitude and direction of acceleration on the
three blocks ?
(B) What is the tension on the two cords ?
Physics 207: Lecture 9, Pg 12
Another example with a pulley
Three blocks are connected on the table as shown. The
table has a coefficient of kinetic friction of mK=0.40, the
masses are m1 = 4.0 kg, m2 = 1.0 kg and m3 = 2.0 kg.
N
m2
T1
T1
m1g
m1
T3
m2g
m3
m3g
(A) FBD (except for friction)
(B) So what about friction ?
Physics 207: Lecture 9, Pg 13
Problem recast as 1D motion
Three blocks are connected on the table as shown. The
center table has a coefficient of kinetic friction of mK=0.40,
the masses are m1 = 4.0 kg, m2 = 1.0 kg and m3 = 2.0 kg.
m1g
m1
T1
N
T3
m2
ff
frictionless
m3g
m3
frictionless
m2g
m1g > m3g and m1g > (mkm2g + m3g)
and friction opposes motion (starting with v = 0)
so ff is to the right and a is to the left (negative)
Physics 207: Lecture 9, Pg 14
Problem recast as 1D motion
Three blocks are connected on the table as shown. The
center table has a coefficient of kinetic friction of mK=0.40,
the masses are m1 = 4.0 kg, m2 = 1.0 kg and m3 = 2.0 kg.
m1g
m1
T1
T1
N
T3
T3
m2
ff
frictionless
m3g
m3
frictionless
m2g
x-dir: 1.
S Fx = m2a = mk m2g
- T1 + T3
m3a = m3g - T3
m1a = - m1g + T1
Add all three: (m1 + m2 + m3) a = mk m2g+ m3g – m1g
Physics 207: Lecture 9, Pg 15
Chapter 2
Physics 207: Lecture 9, Pg 17
Chapter 2
Also average speed and average velocity
Physics 207: Lecture 9, Pg 18
Chapter 3
Physics 207: Lecture 9, Pg 19
Chapter 3
Physics 207: Lecture 9, Pg 20
Chapter 4
Physics 207: Lecture 9, Pg 21
Chapter 4
Physics 207: Lecture 9, Pg 22
Chapter 5
Physics 207: Lecture 9, Pg 23
Chapter 5 & 6
Physics 207: Lecture 9, Pg 24
Chapter 6
Physics 207: Lecture 9, Pg 25
Chapter 7
Physics 207: Lecture 9, Pg 26
Chapter 7
Physics 207: Lecture 9, Pg 27
Short word problems
After breakfast, I weighed myself and the scale read 588 N.
On my way out, I decide to take my bathroom scale in the
elevator with me.
What does the scale read as the elevator accelerates
downwards with an acceleration of 1.5 m/s2 ?
W= (1.0-1.5/9.8) 588 N
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A bear starts out and walks 1st with a velocity of
0.60 j m/s for 10 seconds and then walks at
0.40 i m/s for 20 seconds.
What was the bear’s average velocity on the walk?
What was the bear’s average speed on the walk (with
respect to the total distance travelled) ?
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Physics 207: Lecture 9, Pg 28
Conceptual Problem
The pictures below depict cannonballs of identical mass which are
launched upwards and forward. The cannonballs are launched
at various angles above the horizontal, and with various
velocities, but all have the same vertical component of velocity.
(d)
Physics 207: Lecture 9, Pg 29
Conceptual Problem
A bird sits in a birdfeeder suspended from a
tree by a wire, as shown in the diagram at
left. (f)
Let WB and WF be the weight of the
bird and the feeder respectively.
Let T be the tension in the wire and
N be the normal force of the feeder
on the bird. Which of the following
free-body diagrams best
represents the birdfeeder? (The
force vectors are not drawn to
scale and are only meant to show
the direction, not the magnitude, of
each force.)
Physics 207: Lecture 9, Pg 30
Graphing problem
The figure shows a plot of velocity vs. time for an object
moving along the x-axis. Which of the following statements
is true? (C)
(A) The average
acceleration over the 11.0
second interval is -0.36
m/s2
(B) The instantaneous
acceleration at t = 5.0 s is
-4.0 m/s2
(C) Both A and B are
correct.
(D) Neither A nor B are
correct.
Physics 207: Lecture 9, Pg 31
Conceptual Problem
A block is pushed up a 20º ramp by a 15 N force which may be
applied either horizontally (P1) or parallel to the ramp (P2).
How does the magnitude of the normal force N depend on the
direction of P? (B)
(A) N will be smaller if P is
horizontal than if it is parallel the
ramp.
(B) N will be larger if P is
horizontal than if it is parallel to
the ramp.
(C) N will be the same in both
cases.
(D) The answer will depend on the
coefficient of friction.
20°
Physics 207: Lecture 9, Pg 32
Conceptual Problem
A cart on a roller-coaster rolls down the track shown below.
As the cart rolls beyond the point shown, what happens to
its speed and acceleration in the direction of motion (D)?
A. Both decrease.
B. The speed decreases, but
the acceleration increases.
C. Both remain constant.
D. The speed increases, but
acceleration decreases.
E. Both increase.
F. Other
Physics 207: Lecture 9, Pg 33
Conceptual Problem
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A person initially at point P in the illustration stays there a
moment and then moves along the axis to Q and stays there a
moment. She then runs quickly to R, stays there a moment, and
then strolls slowly back to P. Which of the position vs. time
graphs below correctly represents this motion? (2)
Physics 207: Lecture 9, Pg 34
Sample Problem
A 200 kg wood crate sits in the back of a truck. The
coefficients of friction between the crate and the truck are
μs = 0.9 and μk = 0.5.
The truck starts moving up a 20° slope. What is the
maximum acceleration the truck can have without the crate
slipping out the back?
 Solving:
 Visualize the problem, Draw a picture if necessary
 Identify the system and make a Free Body Diagram
 Choose an appropriate coordinate system
 Apply Newton’s Laws with conditional constraints
(friction)
 Solve
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Physics 207: Lecture 9, Pg 36
Sample Problem

A physics student on Planet Exidor throws a ball that
follows the parabolic trajectory shown. The ball’s position is
shown at one-second intervals until t = 3 s. At t = 1 s, the
ball’s velocity is v = (2 i + 2 j) m/s.
a. Determine the ball’s velocity at t = 0 s, 2 s, and 3 s.
b. What is the value of g on Planet Exidor? -2 m/s2
Physics 207: Lecture 9, Pg 37
Another question to ponder
How high will it go?
 One day you are sitting somewhat pensively in an
airplane seat and notice, looking out the window, one of
the jet engines running at full throttle. From the pitch of
the engine you estimate that the turbine is rotating at
3000 rpm and, give or take, the turbine blade has a radius
of 1.00 m. If the tip of the blade were to suddenly break
off (it occasionally does happen with negative
consequences) and fly directly upwards, then how high
would it go (assuming no air resistance and ignoring the
fact that it would have to penetrate the metal cowling of
the engine.)
Physics 207: Lecture 9, Pg 38
Lecture 9
Assignment: For Thursday, Read Chapter 8, first four sections
Exam Wed., Feb. 18th from 7:15-8:45 PM Chapters 1-7
One 8½ X 11 note sheet and a calculator (for trig.)
Place: Room 2103: All Sections
Physics 207: Lecture 9, Pg 39