# ME 221 Statics ME 221 Statics LECTURE #4 Sections: 3.1 - 3.6 ME 221 Lecture 5 1 Announcements HW #2 due Friday 5/28 Ch 2: 23, 29, 32, 37, 47, 50, 61, 82, 105, 113 Ch 3: 1, 8, 11, 25, 35 Quiz #3 on Friday, 5/28 Exam #1 on Wednesday, June 2 ME 221 Lecture 5 2

Chapter 3 Rigid Bodies; Moments Consider rigid bodies rather than particles Necessary to properly model problems Moment of a force about a point Moment of a force about an axis Moment of a couple Equivalent force couple systems Problems ME 221 Lecture 5 3 Rigid Bodies

The point of application of a force is very important in how the object responds F F We must represent true geometry in a FBD and apply forces where they act. ME 221 Lecture 5 4 Transmissibility A force can be replaced by an equal magnitude force provided it has the same line of action and does not disturb equilibrium B A ME 221 Lecture 5

5 Moment A force acting at a distance is a moment M M A O d F d is the perpendicular distance from Fs line of action to O Defn. of moment: M = F d Transmissibility tells us the moment is the same about O or A ME 221

Lecture 5 6 Vector Product; Moment of Force Define vector cross product trig definition component definition cross product of base vectors Moment in terms of cross product ME 221 Lecture 5 7 Cross Product The cross product of two vectors results in a vector perpendicular to both. AxB

A B A B sin n B A B The right-hand rule decides the direction of the vector. n^ = ME 221 A AxB=-BxA BxA AxB AxB

Lecture 5 8 Base Vector Cross Product Base vector cross products give us a means for evaluating the cross product in components. i i 0 ; j i k ; k i j i j k ; j j 0 ; k j i i k j ; j k i ; k k 0 Here is how to remember all of this: ME 221 j j i + k i - k Lecture 5

9 General Component Cross Product Consider the cross product of two vectors A i A j A k B i B j B k x y z x y z Ax By k Ax Bz j Ay Bx k Ay Bz i Az Bx j AzBy i Or, matrix determinate gives a convenient calculation ME 221

i j k A B Ax Bx Ay By Az Bz Lecture 5 10 i j

A B Ax Bx Ay By i j k i j k Az A B Ax Bx Bz Ay By

Az Bz A+ B Ax Bx Ay By Az Bz k = (AyBz-AzBy) i - (AxBz-AzBx) j + (AxBy-AyBx)k ME 221 Lecture 5 11

Example Problems If: A = 5i + 3j & B = 3i + 6j Determine: AB The angle between A and B AxB BxA ME 221 Lecture 5 12 ME 221 Lecture 5 13 Vector Moment Definition The moment about point O of a force acting at point A is: F O

A MO = rA/O x F rA/O Compute the cross product with whichever method you prefer. ME 221 Lecture 5 14 200 N Example Method # 1 O.4 0.2 tan 60=0.2m/x x=0.115m

sin 60=d/0.285m 60 o A 0.285 60 o x d d = 0.247 m MA =200N *0.247m= 49.4 Nm ME 221 Lecture 5 15 200 sin 60

200 N Method # 2 O.4 60 o 200 cos60 0.2 A + M =200N (sin 60)(0.4m)- 200N (cos 60)(0.2m) = 49.4 Nm Note: Right-hand rule applies to moments ME 221 Lecture 5 16 200 N

Method # 3 O.4 0.2 60 o r A F=200N cos 60 i + 200N sin 60 j r =0.4 i + 0.2 j ^i j^ 0.2 MA= 0.4 200cos60 200sin60 ME 221 ^k 0 =200 (sin 60)(0.4) - 200 (cos 60)(0.2) 0 = 49.4 Nm Lecture 5

17 200 N Method # 4 O.4 60 o 0.2 r =0.285 i A F=200N cos 60 i + 200N sin 60 j r =0.285 i i j k 0 0 = 49.4 Nm MA= 0.285 200cos60 200sin60 0

ME 221 Lecture 5 18 Moment of a Force about an Axis y ^ |Mn| =MAn A ^ =n(r B/A x F ) Same as the projection of MA along n n^

rAB=rB/A F B O x z nx ny r r B/Ax B/A y |Mn|= Fx Fy ME 221 nz r B/Az

Fz Lecture 5 19 Resolve the vector MA into MA two vectors one parallel and y Mp one perpendicular to n. A ^ Mn=|Mn|n Mn ^ n F

rAB=rB/A B O Mp = MA - Mn x z ^ ^ =n x [(r B/A x F) x n] ME 221 Lecture 5 20 Moment of a Couple Let F1 = -F2 B

y Mo=rA x F2+ rB x F1 =(rB - rA ) x F1 =rAB x F1= C |C|=|F1| d ME 221 rB O F1 rAB=rB/A F2 d A rA x The Moment of two equal and opposite forces is called

a couple z Lecture 5 21 Moment of a Couple (continued) The two equal and opposite forces form a couple (no net force, pure moment) The moment depends only on the relative positions of the two forces and not on their position with respect to the origin of coordinates ME 221 Lecture 5 22 Moment of a Couple (continued) Since the moment is independent of the origin, it can be treated as a free vector, meaning that it is the

same at any point in space The two parallel forces define a plane, and the moment of the couple is perpendicular to that plane ME 221 Lecture 5 23 Example ME 221 Lecture 5 24