Physics 114A - MechanicsLecture 12 (Walker: Ch.5)

Free Body DiagramsJanuary 27, 2012

Physics 114A - MechanicsLecture 12 (Walker: Ch.5)

Free Body DiagramsJanuary 27, 2012

John G. CramerProfessor Emeritus, Department of Physics

Office: B451 PABjcramer@uw.edu

AnnouncementsAnnouncements 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 2/23Physics 114A - Lecture 12

Out[12]=

2 0 4 0 6 0 8 0 1 0 00

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1 5

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January 26, 2012 Physics 114A - Lecture 11 3/31

Exam 1 Statistics (Revised)

Exam 1 Statistics (Revised)

Average = 73.5Std Dev = 16.4

1.8 2.8 3.8

High = 100Median = 76Low = 25

Physics 114A - Introduction to Mechanics - Winter-2012

Lecture: Professor John G. Cramer

Textbook: Physics, Vol. 1 (UW Edition), James S. Walker

Week Date L# Lecture Topic Pages Slides Reading HW Due Lab

423-Jan-12 E1 EXAM 1 - Chapters 1-4  

1-D Dynamics24-Jan-12 10 Newton's Laws 14 29 5-1 to 5-4  

26-Jan-12 11 Common Forces 11 26 5-5 to 5-7

27-Jan-12 12 Free Body Diagrams - 24 -  HW3

530-Jan-12 13 Friction 9 27 6-1  

Newton's Laws Tension

31-Jan-12 14 Strings & Springs 12 29 6-2 to 6-4  

2-Feb-12 15 Circular Motion 5 30 6-5 HW4

3-Feb-12 16 Work & Energy 11 23 7-1 to 7-2  

66-Feb-12 17 Work & Power 7 25 7-3 to 7-4  

Work-energy7-Feb-12 18 Potential Energy 10 26 8-1 to 8-2  

9-Feb-12 19 Energy Conservation I 16 18 8-3 to 8-5 HW5

10-Feb-12 E2 EXAM 2 - Chapters 5-8  

Lecture Schedule (Part 2)

Lecture Schedule (Part 2)

We are here.

January 27, 2012 4/23Physics 114A - Lecture 12

A horizontal force F acts on a block that is connected to another block by a string. Consider the constraints and the forces.

The Massless String Approximation

The Massless String Approximation

Isolate the string and consider the forces on it:(Fnet)x = TA on S–TB on S = mSa,so TA on S = TB on S + mSa TB on S

Massless String Approximation:Assume that mS0 so that TA on S = TB on S

January 27, 2012 5/23Physics 114A - Lecture 12

Example: Comparing Tensions

Example: Comparing Tensions

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: TA on B=TB on A

(FA net)x=T1-TB on A

= T1-T2 = mAaAx

so T1 = T2 + mAaAx

Therefore, T1 > T2.

January 27, 2012 6/23Physics 114A - Lecture 12

PulleysPulleys

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.

Massless String Approximation

Massless and Frictionless Pulley Approximation

January 27, 2012 7/23Physics 114A - Lecture 12

Clicker Question 1Clicker Question 1

a. T1>T2 b. T1=T2 c. T1<T2

January 27, 2012 8/23Physics 114A - Lecture 12

Free-Body DiagramsFree-Body Diagrams

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

Drawing a Free-Body Diagram

Drawing a Free-Body Diagram

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 vector beside the diagram, not on the particle. Or, if appropriate, write . Then check that points in the same direction as the acceleration vector on your motion diagram.

net 0F

netF

anetF

January 27, 2012 10/23Physics 114A - Lecture 12

Two Forces Applied to a Box

Two Forces Applied to a Box

1 21

N

net i Ni

F F F F F

superposition of forces

January 27, 2012 11/22Physics 114A - Lecture 12

Examples of Force Vectors

Examples of Force Vectors

Pull(contact force)

Push(contact force)

Gravity(long-range force)

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

Clicker Question 2Clicker Question 2

A. 2000 N

B. 500 N

C. 1000 N

D. 0 N

E. 2000 kg

A pair oftug-of-war teamsare pulling on theends of a rope,each team with aforce of 1000 N. If the rope does not move, the tension in the rope is:

January 27, 2012 13/23Physics 114A - Lecture 12

Identifying ForcesIdentifying Forces 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 14/23Physics 114A - Lecture 12

The Forces ona Bungee Jumper

The Forces ona Bungee Jumper

T

w

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

nfk

The Forces on a SkierThe Forces on a Skier

T

w

January 27, 2012 16/23Physics 114A - Lecture 12

The Forces on a RocketThe Forces on a Rocket

Fthrust

wD

January 27, 2012 17/23Physics 114A - Lecture 12

Using Free-Body Diagrams

Using Free-Body Diagrams

x nx gx x xF F F F ma 0 0 x xF ma

xx

Fa

m

y ny gy y yF F F F ma 0 0n g yF F ma

n gF F The vector sum of the forces of a free-body diagram is equal to ma. January 27, 2012 18/23Physics 114A - Lecture 12

Example: A Dogsled RaceExample: A Dogsled Race 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. n gF F F ma

cosx xF F ma sin 0y n yF F mg F ma 2cos / (150 N)(cos 25 ) /(80 kg) 1.7 m/sxa F m

2sin (80 kg)(9.81 m/s ) (150 N)(sin 25 ) 720 NnF mg F January 27, 2012 19/23Physics 114A - Lecture 12

Example: Unloading A Truck

Example: Unloading A Truck

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? 0 sinnx gx xF F mg ma sinxa g / sinx h

2 2( sin )( / sin ) 2x xv a x g h gh

sind xv v max

2

arcsin[ / ]

arcsin (2.5 m) / 2(9.81 m/s )(1.0 m)

34.4

d xv v

x

January 27, 2012 20/23Physics 114A - Lecture 12

Example: Picture Hanging

Example: Picture Hanging

A picture weighing 8.0 N is supported by two wires with tensions T2 and T2.

Find each tension.

1 2 0gT T F ma

1 2cos30 cos 60 0 0T T

1 2sin 30 sin 60 0gT T F

2 1 1cos30 / cos 60 3T T T

1 31 2 1 12 2sin 30 sin 60 ( 3 ) 2gF T T T T

1 11 2 2

(8.0 N) 4.00 NgT F

2 1 3 6.93 NT T

1

2sin 30 cos 60

3

2cos30 sin 60

January 27, 2012 21/23Physics 114A - Lecture 12

Example:An Accelerating Jet Plane

Example:An Accelerating Jet Plane

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

cosT mg sinT ma

tan /a g

2 2tan (9.81 m/s )(tan 22 ) 3.96 m/sa g

2/ cos (0.040 kg)(9.81 m/s ) /(cos 22 ) 0.423 NT mg January 27, 2012 22/23Physics 114A - Lecture 12

Example:Weighing Yourself in an

Elevator

Example:Weighing Yourself in an

ElevatorSuppose 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/s2?

n gF F ma

nF mg ma

( )naF m g a

( ')nbF m g a

2 2( ) (80.0 kg)[(9.81 m/s ) (8.00 m/s )] 144.8 Nnc cF m g a January 27, 2012 23/23Physics 114A - Lecture 12

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.

End of Lecture 12End of Lecture 12