LEC23 - Lecture notes 23 PDF

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PHYS 172: Modern Mechanics Spring 2021

Lecture 23 Thursday, April 15th

MID TERM EXAM 3 • SEE EMAIL/ANNOUCEMENT IN Brightspace. • Exam will be administered via Examity and LON CAPA – Be sure to complete Examity Process Practice (see email)

• WHEN: 24 Hr. Window to access 2 Hr. Exam – Starting 9:00AM ET, Thur. 4/22, Closes 8:59AM ET, Fri. 4/23. – You must start the exam in Examity 2 hours prior to closing

• The exam will consist of up to 30 MC questions • No books, equations sheets, notes, human assistance, internet searches, chat rooms, etc. • Accommodations as per DRC letters (if you have not provided one, please do so ASAP!)

How to Prepare • Practice Exam posted in Brightspace – Solutions will be posted 12 Noon Thurs. (April 15th)

• ‘What You Should be Able To Do’ – Review this sheet (formula sheet neither allowed nor provided on exam)

• Review : Problem Quiz 5, HW (19-24), Recitation (11-13), Lecture Quiz problems • Read relevant sections of text, analyze example problems

EXAM 3 • FORMAT SIMILAR TO EXAM 1 : Multiple Choice • COVERS: ALL MATERIAL SINCE EXAM 2. • PREPARATION: – RECITATION PROBLEMS – LECTURE : LECTURE Qs AND PROBLEMS – LAB, QUIZZES, HW, EXAMPLES IN TEXTBOOK, END CH PROBLEMS

• PRACTICE EXAM 3 POSTED: YOUR MAIN RESOURCE! – TAKE IT LIKE A REAL EXAM (TIMED, WITH CALCULATOR, NO FORMULA SHEET, NO HELP FROM OTHERS) – CHECK YOUR ANSWERS & SOLUTIONS (POSTED 12 NOON Thurs. 4/15 ) : FIGURE OUT WHY YOU GOT QUESTIONS WRONG – MAKE SURE YOU UNDERSTAND HOW TO SOLVE THE PROBLEMS & HOW THEY CAN BE TWEAKED 4

Exam 3 Review • Collisions: elastic & inelastic – 1D and 2D collisions

• Angular Momentum: translational & rotational • Angular Momentum Principle – Change rate of angular momentum and torque – Right hand rules

• • • •

Angular Momentum Conservation (zero net torque) Equilibrium Problems Angular Acceleration Problems Combination Problems (apply angular momentum principle, energy principle, and/or momentum principle)

5

THREE FUNDAMENTAL PRINCIPLES Momentum  dP  = Fnet , ext dt

Angular Momentum

 dLA  = t net , ext , A dt

Energy

DE = W + Q

If Next External Force: If Net External Torque: If Work is done: Momentum changes. Angular momentum Energy of system changes changes If No Net External Force: Momentum of system is constant

If No Net External If No Work is Torque: Angular done: Energy of momentum of system system is constant about any point A is constant 6

EQUILIBRIUM For a system to be in Equilibrium Momentum Principle

Equilibrium so no change in momentum of system

Angular Mom Principle

Equilibrium so no change in angular momentum of system

7

Equilibrium Problem A rigid beam of mass , length , has two supports. One support is located at the beam’s left end, while the other is a distance from the left edge. A block of mass is situated such that its center of mass is located a distance from the beam’s left end as shown in the figure. What is the ratio of the normal force exerted by support 1 to the normal force exerted by support 2 on the beam? 



 

 

 

 

 

 

 

 





8

PROBLEM

C

You plan on supporting a 50 kg sign with a rigid beam B and a cable C.

45o

1m

(a) What is the tension in the cable?

B 1m

(b) What is the force exerted by the beam? 9

PROBLEM : Solution Step 1: Identify system, surroundings, and reference directions • System: Sign • Surroundings: – Beam B, – Cable C, – Earth

+y

• Reference Frame: +x +z out of page 10

PROBLEM : Solution Step 2: Draw Free Body diagram showing…

45°

• Each Force, , and • Point of application of each force 

+y +x +z out of page 11

PROBLEM : Solution Step 3: Set up 45°

X direction: Y direction: 

+y +x +z out of page

12

PROBLEM : Solution Step 4: Identify… • Point A, about which to calculate Torque, due to each force, • Position of point of application of each force relative to point A.

45°

A 

+y +x +z out of page

13

PROBLEM : Solution Step 5: Set up…







45°

• Find torque about A due to each Force. A

  

 

• Net torque about A = 0





+y +x +z out of page

14

Torque due to Gravity Q1. What is the torque, about point A exerted on the sign by the force of gravity, ? A. B.







45°

+y

C. +x +z out of page

D.

A

E.  



 15

Torque due to Tension in Cable Q2. What is the torque, about point A exerted on the sign by the force of tension in cable, ? A.    B. 45°

+y

C. D.

+x +z out of page

E.

A  



 16

Torque due to Force of Beam Q3. What is the torque, about point A exerted on the sign by the force of tension in cable, ? A.    B. 45° +y C. D. +x +z out of page E.

A  



 17

PROBLEM : Solution Step 5:







45°

A  





+y +x +z out of page

18

PROBLEM : Solution Step 6: Solve for Unknowns







45° X direction:  

A  

Y direction:  





+y +x19 +z out of page

PROBLEM : Climbing the Ladder What is the maximum distance that a person, mass can climb a ladder of length and negligible mass, that is resting against a wall, making angle with the floor that has a coefficient of static friction with the bottom of the ladder? 0

• System: ladder & climber • Surr. : Earth, wall floor Momentum Principle:

PROBLEM : Solution ,





  ,

Resolving into components: •

,

+y

• +x 21 +z out of page

PROBLEM : Solution

Results So Far: •

,

• Ang. Mom Principle:



 Substituting from above:  

 ,

, ,

But: Substituting :

+y

+x 22 +z out of page

PROBLEM : Tweaking It If the mass of the ladder, , in the situation we’ve analyzed was not negligible compared to the climber’s mas… • There would be an additional contribution to the torque on the ladder-climber system about point A. What is it? • Would there be any other additional contribution to ?

,

 ,

,

+y

+x 23 +z out of page

PROBLEM : Tilting Book A book is standing vertically on a bookshelf as shown. The thickness of the book is , and height is . Assuming the book has a uniform density throughout, what is the maximum angle with respect to the vertical that the book can be tilted, such that it goes back to its standing position when let go, rather than tipping over?

𝒎𝒂𝒙

24

PROBLEM : Tilting Book - Solution The book will go back to its original standing position, if the net torque on the book about point A is counter clockwise. i.e. is CCW • Net torque • The weight,

The book will tip over, if the net torque on the book about point A is clockwise. i.e is CW A

A

about A is due to weight,

of book.

acts through the CM. 25

PROBLEM : Tilting Book - Solution We need to find the point of transition from the… Figure on the left ( is CCW)

A

Transition point occurs when

Figure on the right ( is CW)

A

A

:

passes vertically thru A. 26

PROBLEM : Tilting Book - Solution At the transition point… Using trigonometry …

A 27

SYSTEMS WITH NON-ZERO NET TORQUE NON Equilibrium Systems Momentum Principle

Angular Mom Principle

• Find Net Force,

• Find Net Torque,

• Direction of Net Force gives direction of change in momentum i.e. which direction CM will translate

• Direction of Net Torque gives direction of change in angular momentum i.e. which direction it will rotate 28

Movement of Spool Q4. A spool with axle radius and outer radius is initially at rest on the floor. You begin to pull gently upward with constant force, on the light string wrapped around the axle as shown. In which direction does the spool begin to move? • Torque due to causes it to A. Left rotate counter clockwise. B. Right • Friction is to left, it opposes C. Neither rotation at point of contact Friction • Friction causes translation toward left.

Force

29

PROBLEM: Acceleration of Spool A spool with axle radius and outer radius is initially at rest on the floor. You begin to pull gently to the right with constant force, on the light string wrapped around the axle as shown, so that the spool rolls without slipping. What is the acceleration of the CM?

30

PROBLEM: Acceleration of Spool +y +x +z out of page

Momentum Principle 

,

Angular Mom Principle ,

,,

Rolling without Slipping : Negative sign in this case, b/c spool rotates counterclockwise & translates to right. The point of contact with the ground can only be at rest if we have this negative sign. This is opposite to usual case e.g., in a bicycle wheel where it rotates clockwise & translates to the right.

Be Sure To… • Complete – Lecture Quiz 22 by 11:59 PM Thursday, April 15th. – HW 23 due 11:59 PM Friday, April 16th. – PROBLEM QUIZ 5 in 24-hour window 9:00AM, Friday, April 16th – 8:59 AM, Saturday, April 17th – Recitation 13 by 12:00 Noon Saturday, April 17th. – LAB 11 by 11:59 PM Saturday, April 17th. – HW 24 due 11:59 PM Wednesday, April 21st.

• Read relevant sections in text as per syllabus, before next lecture. SEE YOU NEXT LECTURE! 32...