Post on 09-Jul-2020
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
Lecture 23
Chapter 12
Static Equilibrium
Physics I
My favorite subject
… Static Equilibrium
Course website:https://sites.uml.edu/andriy-danylov/teaching/physics-i/
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
Today we are going to discuss:
Chapter 12:
Static Equilibrium: Section 12.8
IN THIS CHAPTER, you will discuss static equilibrium of an object
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
Angular Momentum Conservation helps to solve many problems
I11 I22
IL For a rigid body21 LL
Launch: leg and hands are out to make I large
Flight: leg and hands are in to make large
Landing: leg and hands are out to “dump” large
1212 )( II
large1 I mallI s2
Example Figure Skater’s Jump
Angular Momentum stays constant throughout the whole jump
1 2
small-1 large-2
ConcepTest Figure Skater
A) thesame
B) largerbecauseshe’srotatingfaster
C)smallerbecauseherrotationalinertiaissmaller
Afigureskaterspinswithherarmsextended.Whenshepullsinherarms,shereducesherrotationalinertiaandspinsfastersothatherangularmomentumisconserved.Comparedtoherinitialrotationalkineticenergy,herrotationalkineticenergyaftershepullsinherarmsmustbe:
KErot =I2 =(I) =L (usedL=I ).BecauseL isconserved,largermeanslargerKErot.
12
12
12
12
The“extra”energycomesfromtheworkshedoesonherarms.
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
Bicycle wheel/turntable as a demo of Angular Momentum Conservation
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
Bicycle wheel/Ang. Mom. Conservation
xL
wzL
pzL
Initial situation Final situation
finz
inz LL
Angular Momentum Conservation (z comp):
pz
wz
finz LLL 0 w
zpz LL wwpp II
wp
wp I
I )( counterclockwiseclockwise
0inzL w - wheel
p - person
ConcepTest Spinning Bicycle Wheel
You are holding a spinning (CCW) bicyclewheel while standing on a stationaryturntable. If you suddenly flip the wheelover so that it is spinning in the oppositedirection, the turntable will:
ThetotalangularmomentumofthesystemisL upward,anditisconserved.Soifthewheelhas−L downward,youandthetablemusthave+2L upward.
A)remainstationary
B)starttospininthesamedirectionasbeforeflipping(CCW)
C)starttospininthesamedirectionasafterflipping(CW)
finz
inz LL
tablemeLLL zLL tableme 2
L
L2L?
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
Static Equilibrium
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
The 1st Condition for Equilibriumprevents translational motion
0F
0 xF 0 yF 0 zF
amF
N. 2nd law describes translational motion
He doesn’t want to have any sliding of a ladder, i.e. 0a
Force equilibrium
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
The 2nd Condition for Equilibriumprevents rotational motion
0
I
Rotational N. 2nd law describes rotational motion:v
He doesn’t want to have any rotation of a ladder, i.e. 0
There must be no net torque around any axis (the choice of axis is arbitrary).
0 x 0 y 0 z
Torque equilibrium
Six conditions!
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
Reduce#ofEquilibriumEquations
For simplicity, we will restrict the applications to situations in which all the forces lie in the xy plane.
1st condition:
2nd condition:
0 xF 0 yF 0 zF
0 x 0 y 0 z
0)1 xF 0)2 yF 0)3 z
There are three resulting equations, which we will use
Fr
Fr ,
Simplification!
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
Axisofrotationforthe3rd equation
Does it matter which axis you choose for calculating torques?
0 z
0F
Anyaxisofrotationworks.Itisuptoyouwhichonetochoose
If an object is in a force equilibrium and the net torqueis zero about one axis, then the net torque must be zero about any other axis
We should be smart to choose a rotation axis to simplify problems
So.Thechoiceofanaxisisarbitrary
EH
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
Anyaxisofrotationworksforthe3rd equation(proof)(Read only if you want)
The choice of an axis is arbitrary 0 z
End of Class
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
Concurrent/Nonconcurrent forces
1F 2F
3F
Concurrentforces:whenthelinesofactionoftheforcesintersectatacommonpoint,therewillbenorotation.So
1F 2F
3F
0 xF 0 yF0 z 0 xF 0 yF0 z
Nonconcurrent forces:whenthelinesofactionoftheforcesdonotintersectatacommonpoint,therewillberotation.So
Automaticallysatisfied
What’s a difference?
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
Recallhowtocalculatetorque(shortcut)
Fr
F
r
• Draw a line of force • Find the perpend. distance (r┴) from the axis of rotation
to that line.• Magnitude of torque is r┴F• Torque is positive if it tries to rotate an object CCW
sinrF
Line of force (action)
r Fr
Lecture 20
ConcepTest Static equilibriumConsideralightrodsubjecttothetwoforcesofequalmagnitudeasshowninfigure.Choosethecorrectstatementwithregardtothissituation:
(A) The object is in force equilibrium but not torque equilibrium.(B) The object is in torque equilibrium but not force equilibrium(C) The object is in both force equilibrium and torque equilibrium(D) The object is in neither force equilibrium nor torque equilibrium
F
extforce equilibrium
torque equilibrium
Here,the1stconditionissatisfiedbutthe2nd isn’t,sotherewillberotation.So,tohavestaticsituation,bothconditionsmustbesatisfied.
origin
Fr
1
2
0
FF
0 X21
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
Example
0 z
0 xF
A 3.0‐m‐long ladder leans against africtionless wall. The coefficient ofstatic friction between the ladder andthe floor is 0.40.
What is the minimum angle the laddercan make with the floor withoutslipping?
Ladder stability (12.58)
0 yF
The forces are nonconcurrent, so we need all equilibrium
conditions
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
Find the tension in the two wires supporting the traffic light
Example Traffic light
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
1. Choose one object at a time, and make a free-body diagram by showing all the forces on it and where they act.
2. Choose a coordinate system and resolve forces into components.
3. Write equilibrium equations for the forces.
4. Choose any axis perpendicular to the plane of the forces and write the torqueequilibrium equation. A clever choice here can simplify the problem enormously.
5. Solve.
Solving Statics Problems
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture23A.Danylov
That’s all, Folks!!