2.001 - MECHANICS AND MATERIALS I Lecture #1 9/6/2006 Prof. Carol Livermore A ﬁrst course in mechanics for understanding and designing complicated systems.PLAN FOR THE DAY: 1. Syllabus 2. Review vectors, forces, and moments 3. Equilibrium 4. Recitation sectionsBASIC INGREDIENTS OF 2.001 1. Forces, Moments, and Equilibrium → Statics (No acceleration) 2. Displacements, Deformations, Compatibility (How displacements ﬁt to- gether) 3. Constitutive Laws: Relationships between forces and deformations Use 2.001 to predict this problem.HOW TO SOLVE A PROBLEMRULES: 1. SI inits only. 2. Good sketches. • Labeled coordinate system.
• Labeled dimensions and forces. 3. Show your thought process. 4. Follow all conventions. 5. Solve everything symbolically until the end. 6. Check answer ⇒ Does it make sense? • Check trends. • Check units. 7. Convert to SI, plug in to get answers.VECTORS, FORCES, AND MOMENTSForces are vectors (Magnitude and Direction) i j ˆ F = Fxˆ + Fy ˆ + Fz kOR ∗ ∗ ˆ F = Fx∗ iˆ + Fy∗ jˆ + Fz∗ k ∗Where do forces come from? • Contact with another object • Gravity • Attractive of repulsive forcesMoments are vectors. (Moment is another word for torque.) • Forces at a distance from a point • Couple
Moments are vectors. M = My ˆ j. EX:2D: FB = FBxˆ + FBy ˆ i j rAB = rABxˆ + rABy ˆ i j MA = rAB × FB = = (rABxˆ + rABy ˆ × (FBxˆ + FBy ˆ i j) i j) = (rAB FB − rAB FB )k ˆ x y y x OR MA = |rAB | FB sin φ using Right Hand Rule 3D: i j ˆ rAB = rABxˆ + rABy ˆ + rABz k i j ˆ FB = FBxˆ + FBy ˆ + FBz k ˆ MA = rAB ×FB = (rABx FBy −rABy FBx )k +(rABz FBx −rABx FBz )ˆ j+(rABy FBz −rABz FBy )ˆ i 3
EX: Wrench rAB = dˆ j FB = FBˆ i j i ˆ MA = rAB × FB = dˆ × FBˆ = −dFB kEquations of Static Equilibrium ΣFx = 0 ΣF = 0 ΣFy = 0 ΣFz = 0 ΣM0 x = 0 ΣM0 = 0 ΣM0 y = 0 ΣM0 z = 0
4 F =0 i=1 F1 + F2 + F3 + F4 = 0 F1x + F2x + F3x + F4x = 0 F1y + F2y + F3y + F4y = 0 F1z + F2z + F3z + F4z = 0 Mo = 0 ⇒ r01 × F1 + r02 × F2 + r03 × F3 + r04 × F4 = 0 i, j, ˆa. Expand in ˆ ˆ k .b. Group.c. Get 3 equations (x,y,z).Try a new point? Try A. rA1 = rA0 + r01 ... MA = 0 rA1 × F1 + rA2 × F2 + rA3 × F3 + rA4 × F4 = 0(rA0 + r01 ) × F1 + (rA0 + r02 ) × F2 + (rA0 + r03 ) × F3 + (rA0 + r04 ) × F4 = 0rA0 × (F1 + F2 + F3 + F4 ) + (r0 1 × F1 + r0 2 × F2 + r0 3 × F3 + r0 4 × F4 ) = 0 rA0 × (F1 + F2 + F3 + F4 ) = 0 F =0 (r0 1 × F1 + r0 2 × F2 + r0 3 × F3 + r0 4 × F4 ) = 0 M0 = 0So taking moment about a new point gives no additional info. 5