January 27, 2012
John G. Cramer
Professor Emeritus, Department of Physics
Office: B451 PAB jcramer@uw.edu

 Homework Assignment #3 is now due at 11:59 PM on January 24
(tonight). Homework Assignment #4 is due at 11:59 PM on
Thursday, February 2.
 There was an answer-key typo on Exam 1 Question 6 (D, not B).
This has been corrected, and your WebAssign Exam1-MC score may have changed from yesterday. 145 students gained 5 points and 6 students lost 5 points.
 The Exam 1 solutions are available on the web from the “Exam 1” link on the Physics 114A Lecture Schedule. Compare your answers with the solutions before asking any questions about the grading.
 If you have a question or complaint about the grading of Exam 1, attach a blank sheet of paper to the exam page at issue, write your complaint/question on the blank page, and turn it in to Susan
Miller by Monday.
January 27, 2012 Physics 114A - Lecture 12 2/23

Out[12]=
15
10
25
High = 100
Median = 76
Low = 25
Average = 73.5
Std Dev = 16.4
20
5
0
20
January 26, 2012
40 60
1.8
Physics 114A - Lecture 11
2.8
80
3.8
100
3/31

Physics 114A - Introduction to Mechanics - Winter-2012
Lecture: Professor John G. Cramer
Textbook: Physics, Vol. 1 (UW Edition), James S. Walker
Week
4
Date L# Lecture Topic Pages Slides Reading HW Due
EXAM 1 - Chapters 1-4
23-Jan-12 E1
24-Jan-12 10 Newton's Laws
26-Jan-12 11 Common Forces
14
11
29
26
5-1 to 5-4
5-5 to 5-7
5
6
27-Jan-12 12 Free Body Diagrams
30-Jan-12 13 Friction
31-Jan-12 14 Strings & Springs
2-Feb-12 15 Circular Motion
3-Feb-12 16 Work & Energy
6-Feb-12 17 Work & Power
7-Feb-12 18 Potential Energy
-
9
12
5
11
7
10
24
27 6-1
-
29 6-2 to 6-4
30 6-5
23 7-1 to 7-2
25 7-3 to 7-4
26 8-1 to 8-2
9-Feb-12 19 Energy Conservation I
10-Feb-12 E2
16 18
EXAM 2 - Chapters 5-8
8-3 to 8-5
HW3
HW4
HW5
January 27, 2012 Physics 114A - Lecture 12
Lab
1-D Dynamics
Newton's Laws
Tension
We are here.
Work-energy
4/23

A horizontal force
F acts on a block that is connected to another block by a string.
Consider the constraints and the forces.
Isolate the string and consider the forces on it:
(F net
) x
= T so
T
A on S
A on S
= T
–T
B on S
B on S
+ m
S
= m
S a

T a
,
B on S
Massless String Approximation:
Assume that m
S

0 so that
T
A on S
= T
B on S
January 27, 2012 Physics 114A - Lecture 12 5/23

Blocks A and B are connected by massless String 2 and pulled across a frictionless surface by massless
String 1. The mass of B is larger than the mass of A.
Is the tension in String 2 smaller, equal, or larger than the tension in String 1?
The blocks must be accelerating to the right, because there is a net force in that direction. We use the massless string approximation to directly relate the string tensions on A and B due to
String 2:
T
A on B
=T
B on A
(F
A net
) x
=T
1
-T
B on A
= T
1
-T
2
= m
A a
Ax so
T
1
= T
2
+ m
A a
Ax
Therefore,
T
1
> T
2
.
January 27, 2012 Physics 114A - Lecture 12 6/23

Massless String Approximation
Strings and ropes often pass over pulleys that change the direction of the tension. In principle, the friction and inertia in the pulley could modify the transmitted tension.
Therefore, it is conventional to assume that such pulleys are
massless and frictionless.
January 27, 2012
Massless and Frictionless
Pulley Approximation
Physics 114A - Lecture 12 7/23

a. T
1
>T
2 b. T
1
=T
2 c. T
1
<T
2
January 27, 2012 Physics 114A - Lecture 12 8/23

January 27, 2012 Physics 114A - Lecture 12 9/23

 Identify all forces acting on the object.
 Draw a coordinate system. Use the axes defined in your pictorial representation. If those axes are tilted, for motion along an incline, then the axes of the free-body diagram should be similarly tilted.
 Represent the object as a dot at the origin of the
coordinate axes. This is the particle model.
 Draw vectors representing each of the identified forces.
Be sure to label each force vector.
 Draw and label the net force vector . Draw this net vector beside the diagram, not on the particle. Or, if appropriate, write 
0 .
 Then check that points in the same direction as the net acceleration vector on your motion diagram.
January 27, 2012 Physics 114A - Lecture 12 10/23

January 27, 2012
F net
 i
N 

1
F i

F
1

F
2
  
F
N superposition of forces
Physics 114A - Lecture 12 11/22

Pull
(contact force)
January 27, 2012
Push
(contact force)
Physics 114A - Lecture 12
Gravity
(long-range force)
12/23

A pair of tug-of-war teams are pulling on the ends of a rope, each team with a force of 1000 N.
If the rope does not move, the tension in the rope is:
A. 2000 N
B. 500 N
C. 1000 N
D. 0 N
E. 2000 kg
January 27, 2012 Physics 114A - Lecture 12 13/23

 Identify “the system” and “the environment.” The system is the object whose motion you wish to study; the environment is everything else.
 Draw a picture of the situation. Show the object—the system—and everything in the environment that touches the system. Ropes, springs, and surfaces are all parts of the environment.
 Draw a closed curve around the system. Only the object is inside the curve; everything else is outside.
 Locate every point on the boundary of this curve where the
environment touches the system. These are the points where the environment exerts contact forces on the object.
 Name and label each contact force acting on the object. There is at least one force at each point of contact; there may be more than one. When necessary, use subscripts to distinguish forces of the same type.
 Name and label each long-range force acting on the object. For now, the only long-range force is weight.
January 27, 2012 Physics 114A - Lecture 12 14/23

January 27, 2012
T w
Physics 114A - Lecture 12 15/23

f k n w
T
January 27, 2012 Physics 114A - Lecture 12 16/23

F thrust
D w
January 27, 2012 Physics 114A - Lecture 12 17/23


F x

F nx

F gx

F x
 ma x
0

0

F x
 ma x a x

F x m

F y

F ny

F gy

F y
 ma y
F n

F g

0
 ma y

0
F n

F g
The vector sum of the forces of a free-body diagram is equal to ma.
18/23 January 27, 2012 Physics 114A - Lecture 12

During your winter break, you enter a dogsled race in which students replace the dogs. Wearing cleats for traction, you begin the race by pulling on a rope attached to the sled with a force of 150 N at 25° above the horizontal. The mass of the sled-passenger-rope system is 80 kg, and there is negligible friction between the sled runners and the ice.
(a) Find the acceleration of the sled
(b) Find the magnitude of the normal force exerted on the surface by the sled.
F n

F g
F ma
F x

F a x

F cos
  ma x
F y

F n
 mg

F sin
  ma y
 m
  
1.7 m/s
2

0
F n
 mg

F sin
January 27, 2012
 
2

Physics 114A - Lecture 12
 
720 N
19/23

You are working for a big delivery company, and you must unload a large, fragile package from your truck, using a 1 m high frictionless delivery ramp.
If the downward component of velocity of the package is greater than 2.50 m/s when it reaches the bottom of the ramp, the package will break.
What is the greatest angle  at which you can safely unload?
F nx

F gx
0 mg sin
  ma x a x
 g sin
 x h / sin
 v x

2 a x
   h
 
2 gh v d
 v x sin

January 27, 2012
 max
 arcsin[ v d
/ v x
]
 

2 arcsin (2.5 m) / 2(9.81 m/s )(1.0 m)

34.4



Physics 114A - Lecture 12 20/23

A picture weighing 8.0 N is supported by two wires with tensions T
2 and T
Find each tension.
2
.
sin 30
  cos 60
  1
2 cos 30
  sin 60
  3
T
1
 
2
F g
 ma

0
2
T
1 cos 30
 
T
2 cos 60 0 0
T
2

T
1
 cos 30 / cos 60

T
1
3
T
1 sin 30
 
T
2 sin 60
 
F g

0
F g

T
1 sin 30
 
T
2 sin 60
 
T
1
(
1
2

3
2
3
)

2 T
1
T
1
 1
2
F g
 1
2
(8.0 N)

4.00 N
T
2

T
1
3

6.93 N
January 27, 2012 Physics 114A - Lecture 12 21/23

As your jet plane speeds along the runway on takeoff, you decide to determine its acceleration, so you take out your yo-yo and note that when you suspend it, the string makes an angle of 22° with the vertical.
(a) What is the acceleration of the airplane?
(b) If the mass of the yo-yo is 40.0 g, what is the tension in the string?
T
 mg
 ma T cos
  mg tan
 
T sin
  ma a
 g tan
  2  
3.96 m/s
2
T
 mg / cos
  2  
0.423 N
January 27, 2012 Physics 114A - Lecture 12 22/23

Suppose that your mass is 80 kg, and you are standing on a scale fastened to the floor of an elevator. The scale measures force and is calibrated in newtons.
(a) What does the scale read when the elevator is rising with acceleration a?
(b) When it is descending with acceleration a’?
(c) When it is rising at 20.0 m/s and its speed is decreasing at 8.0 m/s 2 ?
F n

F g
 ma
F ( a ) na
 
F ( a nb
 
')
F n
 mg
 ma
F nc

(
 a c
)
 
January 27, 2012 Physics 114A - Lecture 12

23/23

 Before Monday, read Walker Chapter 6.1
 Homework Assignment #3 is now due at 11:59
PM on January 27 (tonight). Homework
Assignment #4 is due at 11:59 PM on Thursday,
February 2.
 As of today, 196/205 clickers are registered.
I have sent an E-mail reminder to the nine students without clicker registrations.