1. Forging new generations of engineers

2. Moments A lesson on understanding and evaluating moments of forces

3. In This Lesson What is a moment? How are moments calculated? How are moments evaluated in dynamics problems? How are moments evaluated in static equilibrium problems? How do moments affect unconstrained objects?

4. What Is a Moment? The moment or torque of a force is a measure of the tendency of the force to rotate the body upon which it acts about an axis.

5. Terminology = F pivot distance = D lever arm

6. Formula for Moment = F pivot Moment M M = F x D distance = D

7. Units for Moments ForceDistanceMoment English Customary Pound (lb)Foot (ft)lb-ft SINewton (N)Meter (m)N-m

8. The Right Hand Rule +

9. Right Hand Rule THUMB POINTS TOWARD YOU POSITIVE

10. Right Hand Rule THUMB POINTS AWAY FROM YOU NEGATIVE

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

12. Moment Calculations Longer Wrench F = 20 lb D = 1 ft M = -(F x D) M = -(20 lb x 1 ft) M = -20 lb-ft ¯

13. Moment Calculations L-shaped Wrench F = 20 lb D = 3 in. =.25 ft M = -(F x D) M = -(20 lb x.25 ft) M = -5 lb-ft ¯ 3 in.

14. Moment Calculations Offset Wrench F = 20 lb D = 8 in. + 10 in. = 1.5 ft M = -(F x D) M = -(20 lb x 1.5 ft) M = -30 lb-ft ¯ 8 in. 10 in.

15. D = r = 50 cm =.5 m M = F x D ***Use the right hand rule to determine positive and negative. M = 100 N x.5 m M = 50 N-m Moment Calculations Wheel and Axle F = 100 N r = 50 cm +

16. 50 o F y = Fsin50° = (100 N)(.766) F y = 76.6 N D = r = 50 cm =.5 m M = F y x D M = 76.6 N x.5 m M = 38.3 N-m Moment Calculations Wheel and Axle F = 100 N r = 50 cm FyFy FxFx

17. What Is Equilibrium? The state of a body or physical system at rest or in unaccelerated motion in which the resultant of all forces acting on it is zero. The sum of all moments about any point or axis is zero. ΣM = 0 M 1 + M 2 + M 3... = 0

18. Moment Calculations See-Saw

19. ΣM = 0 M 1 + ( – M 2 ) = 0 *** Use the right hand rule to determine positive and negative. M 1 = M 2 F 1 x D 1 = F 2 x D 2 25 lb x 4 ft = 40 lb x D 2 100 lb-ft = 40 lb x D 2 Moment Calculations See-Saw F 1 = 25 lb F 2 = 40 lb D 1 = 4 ft D 2 = ? ft 40 lb 40 lb 2.5 ft = D 2 + ¯

20. ΣM = 0 M B + (–M C ) = 0 M B = M C R B x D AB = F C x D AC R B x 10 ft = 35 lb x 3 ft R B x 10 ft = 105 lb-ft Moment Calculations Loaded Beam D AB = 10 ft D AC = 3 ft A C B RARA F C = 35 lb RBRB 10 ft 10 ft R B = 10.5 lb R A + R B = 35 lb R A = 35 lb – 10.5 lb = 24.5 lb Select the pivot location A. Solve for R B.

21. A B C D F c = 600 lb Moment Calculations Truss 24 ft 8 ft 12 ft F B = 500 lb Replace the pinned and rolling supports with reaction forces. R AY R AX R DY D AC = 24 ft D CD = 8 ft D CB = 12 ft D AD = 32 ft

22. A B C D F c = 600 lb Moment Calculations Truss D AC = 24 ft D CD = 8 ft D CB = 12 ft D AD = 32 ft 24 ft 8 ft 12 ft F B = 500 lb R AY R AX R DY Select the pivot at A. Solve for R DY. 12 ft ΣM = 0 M D – M B – M C = 0 M D = M B + M C R DY x D AD = (F B x D CB ) + (F C x D AC ) R DY x 32 ft = (500 lb x 12 ft) + (600 lb x 24 ft) R DY x 32 ft = 6000 lb-ft + 14400 lb-ft R DY x 32 ft = 20400 lb-ft 32 ft 32 ft R DY = 637.5 lb

23. Moments on An Airplane AILERON Roll ELEVATORS Pitch RUDDER Yaw

24. PATH WITH MOMENTS Translation and Rotation PATH WITHOUT MOMENTS Pure Translation Moments on an Airplane

25. References Halpern, A.M. (1988). Schaum’s 3000 solved problems in physics. New York, NY: McGraw-Hill. NASA. (n.d.). The beginners' guide to aeronautics. Retrieved June 11, 2008, from http://www.grc.nasa.gov/WWW/K- 12/airplane/ Nave, C.R. (2005). HyperPhysics. Retrieved June 12, 2008, from http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html National Institute of Standards and Technology. (2000). The NIST reference on constants, units and uncertainty. Retrieved June 11, 2008, from http://physics.nist.gov/cuu/Units/

26. Writer: Wendy DeMane Content Editor: Wes Terrell Production Work: CJ Amarosa Credits