1. MOMENTS Created by The North Carolina School of Science and Math.The North Carolina School of Science and Math Copyright 2012. North Carolina Department of Public Instruction.North Carolina Department of Public Instruction Created by The North Carolina School of Science and Math.The North Carolina School of Science and Math Copyright 2012. North Carolina Department of Public Instruction.North Carolina Department of Public Instruction

2. If our parrot isn’t sitting in the center, how do I calculate T1 and T2? W T2T1 We need another equation! Creative commons image from: http://upload.wikimedia.org/wikipedia/commons/thumb/0/02/Amazona_albifrons_-in_tree2-3c.jpg/444px-Amazona_albifrons_-in_tree2-3c.jpghttp://upload.wikimedia.org/wikipedia/commons/thumb/0/02/Amazona_albifrons_-in_tree2-3c.jpg/444px-Amazona_albifrons_-in_tree2-3c.jpg

3. Static Equilibrium If a system is in static equilibrium –Σ F x = 0 –Σ F y = 0 and … –Σ M = 0 Just like with forces, moments have direction. Creative commons image from: http://upload.wikimedia.org/wikipedia/commons/thumb/0/02/Amazona_albifrons_-in_tree2-3c.jpg/444px-Amazona_albifrons_-in_tree2-3c.jpghttp://upload.wikimedia.org/wikipedia/commons/thumb/0/02/Amazona_albifrons_-in_tree2-3c.jpg/444px-Amazona_albifrons_-in_tree2-3c.jpg

4. Moments A moment of a force is a measure of its tendency to cause a body to rotate about a point or axis. A moment (M) is calculated using the formula: Moment = Force * Distance M = F * D Always use the perpendicular distance between the force and the point!

5. Moment = Force * Perpendicular Distance M = F * D What distance would you use here? L F = 20 lb L = 2 ft Find the moment about point O: M = F * D M = (20 lb) * (2 ft) M = 40 ft-lb clockwise Moments

6. Moment = Force * Perpendicular Distance M = F * D Can you visualize the result of the force acting on the beam? The beam has a tendency to rotate clockwise about point O. Moments

7. Moment = Force * Distance M = F * D w L F = 20 lbf L = 0.5 ft Find the moment about point O: M = F * D M = (20 lb) * (0.5 ft) M = 10 ft-lb clockwise Moments

8. Moment = Force * Distance M = F * D Can you visualize the result of the force acting on the beam? The beam has a tendency to rotate clockwise about point O. Moments

9. Moment = Force * Perpendicular Distance M = F * D What distance would you use here? L Moments – Direct Method d'd' M = F * d’ Use trigonometry to find d’ θ Where is θ in the triangle ? 12 Imagine the force rotating so that θ gets larger and smaller. Which angle (1 or 2) has the same behavior? L d'd',θ,θ 1 2 θ,θ,θ θ is at location 1. Therefore: d‘ = L cos(θ) M = F * L cos(θ)

10. Moment = Force * Perpendicular Distance M = F * D What distance would you use here? L Moments – Direct Method d'd' M = F * d’ Use trigonometry to find d’ θ 12 θ d‘ = L cos(θ) M = F * L cos(θ)

11. Moment = Force * Perpendicular Distance M = F * D Alternative Method … L Moments – Component Method M = F x * D x + F y * D y θ Let’s find the components Fx = F sin(θ)Fy = F cos(θ) Dx = 0Dy = L M = F x * D x + F y * D y M = F sin(θ) * 0 + F cos(θ) * L M = F * L cos(θ) (same as before) FxFx FyFy

12. Advantages of Component Method Use static equilibrium equations directly to find reaction forces. Find reaction forces for the beam in the prior slide. –Σ F x = 0: F px – F x = 0 -> F px = F x = F sin( θ ) –Σ F y = 0: F py + F y = 0 -> F py = - F cos( θ ) –Σ M = 0: M p + F y L + F x 0 = 0 -> M p = - F L cos( θ ) L θ FxFx FyFy

13. H OW TO SOLVE A MOMENT PROBLEM Draw a free body diagram that shows all forces. List the knowns and unknowns Use the following three formulas to determine unknowns (system is in static equilibrium) –Σ F x = 0 –Σ F y = 0 –Σ M = 0

14. Moment Problem, example Use static equilibrium equations directly to find reaction forces. Find reaction forces for the beam in the prior slide. –Σ F x = 0: F px – F x = 0 -> F px = F x = F sin( θ ) –Σ F y = 0: F py – F y = 0 -> F py = - F cos( θ ) –Σ M = 0: M p – F y L + F x 0 = 0 -> M p = - F L cos( θ ) (CCW) L θ FxFx FyFy

15. If our parrot is not sitting in the center, how do I calculate T1 and T2? W T2T1 Parrot on the Perch: Revisited 41 Σ M = 0 (about where?) : Σ F y = 0

16. The sum of moments is zero around any point! Choose any location, but the same for the whole problem. Some places are easier than others. W T2T1 Where to take the moment 41 Σ M = 0 (this time about W) Σ F y = 0 Solve:

17. Supports are translated into forces and moments in a free body diagrams. The following are three common supports and the forces and moments used to replace them. Roller: Pin Connection: Fixed Support: FyFy FyFy FxFx FxFx FyFy MoMo Review: Types of Supports

18. 8kN9kN 5kN compressed spring ΣF X = 0, therefore R Ax = 0 ΣF y = 0 = R Ay + R B - 8kN + 5kN - 9 kN R Ay + R B = 12kN -> R Ay = 12kN - R B ΣM A = 0 = -8kN·2m + 5kN·4m + R B ·5m - 9kN·7m (16 - 20 + 63) kN-m = R B ·5m R B = 11.8 kN Put into ΣF y result: R Ay = 22kN-11.8 kN = 10.2 kN R Ax R Ay RBRB 8kN 5kN 9kN 2m 1m 2m 1m 2m

19. C A B 30 ° 3m 2m 3kN The rod AC is hinged at point A and is suspended by a cable (BD) at Point B. A weight of 3kN hangs off the rod at Point C. Calculate the reaction forces at Point A and the tension in the cable BD. D

20. C A B 30 ° 3m 2m 3kN R Ay R Ax F By F Bx 5 sin(30)=2.5m 3 sin(30) = 1.5m 3 cos(30)=2.6m 5 cos(30) = 4.3m D Free Body Diagram

21. 3kN R Ay R Ax F By F Bx 5 sin(30)=2.5m 3 sin(30) = 1.5m 3 cos(30)=2.6m 5 cos(30) = 4.3m F By = F b sin(30 °) = 0.5 F B F Bx = F b cos(30 °) = 0.87 F B ΣM A = 0 = -3kN·4.3m + F By ·2.6m + F Bx ·1.5m 12.9kN-m = 0.5 F B ·2.6m + 0.87 F B ·1.5m F B = 4.95 kN

22. 3kN R Ay R Ax F By F Bx 5 sin(30)=2.5m 3 sin(30) = 1.5m 3 cos(30)=2.6m 5 cos(30) = 4.3m F B = 4.95 kN F By = F b sin(30 °) = 0.5 F B = 0.50*4.95 kN = 2.5 kN F Bx = F b cos(30 °) = 0.87F B = 0.87*4.95 kN = 4.3 kN ΣF x = 0 = R Ax – F Bx  R ax = F Bx = 4.3 kN ΣF y = 0 = R Ay + F By - 3kN  R ay = F Bx = 3kN – F by = 3kN – 2.5kN = 0.5 kN