Moment Forces Moment The moment of a force

Moment Forces

Moment The moment of a force is a measure of the tendency of the force to rotate the body upon which it acts.

Terminology =F lever arm pivot distance =d The distance must be perpendicular to the force.

Moments Formula M=dx. F pivot =F distance =d

Units for Moments Force Distance Moment English Customary Pound force (lbf) Foot (ft) lb-ft SI Newton (N) Meter (m) N-m

Rotation Direction In order to add moments, it is important to know if the direction is clockwise (CW) or counterclockwise (CCW). CCW is positive CW is negative

Right-Hand Rule Curl your fingers to match the direction of rotation. Thumb is pointing. . Up = Moment is positive Down = Negative Toward You = Positive Away from You = Negative +

Right-Hand Rule POSITIVE MOMENT B M U TH TS N POI ARD TOW U YO

Right-Hand Rule NEGATIVE MOMENT THU M POIN B TS AWA Y FR YOU OM

Moment Calculations Wrench F = 20. lb M=dx. F ¯ Use the right-hand rule to determine positive and negative. d = 9. 0 in. =. 75 ft M = - (20. lb x. 75 ft) d = 9. 0 in. M = - 15 lb-ft (15 lb-ft clockwise)

Moment Calculations Longer Wrench F = 20. lb M=dx. F ¯ M = - (20. lb x 1. 0 ft) M = - 20. lb-ft d = 1. 0 ft

Moment Calculations L - Shaped Wrench F = 20. lb d = 3 in. =. 25 ft 3 in. M=dx. F M = - (20. lb x. 25 ft) ¯ M = - 5 lb-ft d = 1. 0 ft

Moment Calculations Z - Shaped Wrench F = 20. lb 9 in. d = 8 in. + 10 in. = 1. 5 ft M=dx. F M = - (20. lb x 1. 5 ft) ¯ M = - 30. lb-ft 8 in. 10. in.

Moment Calculations Wheel and Axle d = r = 50. cm = 0. 50 m r = 50. cm M=dx. F Use the right-hand rule to determine positive and negative. + F = 100 N M = 100 N x 0. 50 m M = 50 N-m

Moment Calculations Wheel and Axle r = 50. cm Fy = Fsin 50. ° = (100. N)(. 766) Fy = 76. 6 N d = r = 50. cm = 0. 50 m M = d x Fy M = 76. 6 N x 0. 50 m M = 38 N-m 40. o 50. o F = 100. N Fy

What is Rotational Equilibrium? The sum of all moments about any point or axis is zero. . • This occurs in two cases: 1. Object is not rotating 2. Object is spinning at a constant speed • In either case rotation forces are balanced: ΣM = 0 M 1 + M 2 + M 3. . . = 0

Moment Calculations See-Saw

Moment Calculations ΣM = 0 See-Saw or a cantilevered beam problem M 1 + M 2 = 0 Use the right-hand rule to determine positive and negative. M 1 = -M 2 F 2 = 40. lb F 1 = 25 lb d 1 x F 1 = -d 2 x F 2 25 lb x -4. 0 ft = -40. lb x d 2 -100 lb-ft = -40. lb x d 2 -40. lb + d 1 = 4. 0 ft 2. 5 ft = d 2 = ? ft -40. lb

Moment Calculations Select A as the pivot location. Solve for RBy Simply-supported beam problem Step 1: ΣM = 0 MB + MC = 0 MB = -MC d. AB x RBy = -(d. AC x FC) d. AB = 10. 00 ft d. AC= 3. 00 ft 10. 00 ft x RBy = -(3. 00 ft x 35. 0 lb) 10. 0 ft x RBy = 10. 00 ft C A B -105 lb-ft 10. 00 ft RBy = -10. 5 lb Step 2: ΣF = 0 FC = 35. 0 lb RAy RBy 35. 0 lb + RAy + RBy = 0 RAy = 10. 5 lb - 35. 0 lb = -24. 5 lb

Moment Calculations Question: Truss calculation What are the reaction forces at A and D? FB = 500. lb 12 ft B A 24 ft C 8 ft d. AC = 24 ft d. CD = 8 ft d. CB = 12 ft d. AD = 32 ft Fc = 600. lb D

Moment Calculations Truss calculation Step 1: FB = 500. lb 12 ft B RAx A 24 ft C 8 ft D d. AC = 24 ft d. CD = 8 ft RAy d. CB = 12 ft d. AD = 32 ft Fc = 600. lb RDy Replace the pinned and roller supports with reaction forces.

Moment Calculations Step 2: Truss Select A as the axis of rotation. Solve for RDY ΣM = 0 12 ft RAx A 24 ft C MD + MB + MC = 0 (d. AD x RDy) +(d. CB x FB) +(d. AC x FC) = 0 (32 ft x RDy) + (12 ft x 500. lb) + (24 ft x 600. lb) = 0 12 ft B FB = 500. lb (RDy x 32 ft) + 6000 lb-ft + 14400 lb-ft =0 8 ft D (RDy x 32 ft) = -20400 lb-ft 32 ft RDy = -640 lb RAy Fc = 600. lb RDy = -640 lb 32 ft

Moment Calculations Truss Step 3: Solve for the other vertical support, RAy ΣFy = 0 12 ft B RAy + FC + RDy = 0 RAy + 600 lbs + (-640 lbs) = 0 12 ft RAy = 640 lbs - 600 lbs = 40 lbs RAy = 40 lbs RAx FB = 500. lb RAy = +40 lbs Surprised? ? 24 ft C 8 ft A Fc = 600. lb D RDy = -640 lb

Moment Calculations Step 4: Truss Solve for the other horizontal reaction, RAx 12 ft B FBx = 500. lb RAx = -500 lbs 12 ft 24 ft C FBx + RAx = 0 500 lbs + RAx = 0 RDY = -640 lb RAy = 40 lb ΣFX = 0 8 ft A D RAx = -500 lbs FCy = 600. lb RDy = -640 lb