4.2 Moments in Three-Dimensional Force Systems

Transcription

1 4.2 Moments in Three-Dimensional Force Sstems

2 4.2 Moments in Three-Dimensional Force Sstems Procedures and Strategies, page 1 of 2 Procedures and Strategies for Solving Problems Involving Moments in Three-Dimensional Force Sstems 1. To calculate the moment of a force F about a point O, a) Epress F in rectangular component form. possible choice for r F b) Define a position vector r, with tail at point O and head at an point along the line of action of F. If ou have more than one possible choice for r, choose the one that gives the simplest form for r. Epress r in rectangular component form. c) Evaluate the cross product: M O = r F (If ou have a scientific graphing calculator, use its built-in cross-product function). O r r 2. To determine the shortest distance d between a point and a given line, assume that a force of unknown magnitude F acts along the line, and then make use of the fact that two different formulas eist for calculating the magnitude of the moment M about : est choice for r because r has a simple form (onl one component an component) and M = Fd (1) F = Fu M = r F where r is a positive vector with tail at and head on the line. Then a) find a unit vector u parallel to the line, and epress it in rectangular component form; b) epress the force as F = Fu, u d r

3 4.2 Moments in Three-Dimensional Force Sstems Procedures and Strategies, page 2 of 2 c) epress r in rectangular component form, d) compute the cross product r F = r (Fu) = F(r u) Then Eqs. 1 and 2 give Fd = F r u Cancel F. Thus to calculate the shortest distance d from point to a line, just calculate the magnitude of the cross product of r and u.

4 HEVY DUTY 250MM CHIN Moments in Three-Dimensional Force Sstems Problem Statement for Eample 1 1. Use the cross-product definition of the moment of a force to determine the moment of the force about point. lso, compare the sign of the result with that obtained from the scalar definition of positive moment, M = Fd. F = 50 N 200 mm 30

5 HEVY DUTY 250MM CHIN 4.2 Moments in Three-Dimensional Force Sstems Problem Statement for Eample 2 2. force F = 20 N is applied to the end of a string of length L. The other end of the string is tied to the handle of a wrench as shown. Use the cross-product definition of the moment to determine the moment of F about point. Discuss the effect of distance L on our answer. 200 mm L 30 C F = 20 N

6 4.2 Moments in Three-Dimensional Force Sstems Problem Statement for Eample 3 3. shower/bathtub grab bar is being pulled b a force F = 30 lb as shown. Determine the moment of F about the support. lso determine the coordinate direction angles of the moment vector and interpret the result. 5 in. 16 in in. F = 30 lb 60

7 4.2 Moments in Three-Dimensional Force Sstems Problem Statement for Eample 4 4. force F = 15 N acting parallel to the ais is applied to the handle of a socket wrench to turn a bolt at. Determine the moment of the force about the point. lso, state which component of the moment tends to turn the bolt. 80 mm C 100 mm F = 15 N

8 4.2 Moments in Three-Dimensional Force Sstems Problem Statement for Eample 5 5. Pulle is used to drive pulle C. Determine the resultant moment about bearing produced b the belt forces acting on pulle. lso, interpret our result. Q = 55 N 30 F C Radius = 70 mm D elt forces E P = 30 N 40 mm

9 4.2 Moments in Three-Dimensional Force Sstems Problem Statement for Eample 6 6. child on a biccle collides with a mailbo and eerts the force F shown. If the base of the pole at O will fail if the magnitude of the moment there eceeds 60 N m, determine if the mailbo will fall over. F = {80i + 12j 10k} N 900 mm O 250 mm 75 mm

10 4.2 Moments in Three-Dimensional Force Sstems Problem Statement for Eample 7 7. Copper tubing emerges from the wall at and is subjected to a force F at its free end. The tubing will fail if the magnitude of the moment at eceeds 3 N m. Determine the largest value of the force F that can be applied to the free end of the tubing. 200 mm mm mm 40 F

11 4.2 Moments in Three-Dimensional Force Sstems Problem Statement for Eample 8 8. Two forces, P = 60 N and Q = 80 N act on the vertices of a cube as shown. Determine the moment of each force about point O, if the length of each edge of the cube is 2 m. lso, determine the shortest distance from O to the line F. Each edge is 2 m long. E F P = 60 N Q = 80 N C D O G

12 4.2 Moments in Three-Dimensional Force Sstems Problem Statement for Eample 9 9. Determine the moment about the screw at of the force F = 2 N applied to the sheet-metal bracket shown. lso, determine the shortest distance from to the line connecting and C. 50 mm 60 mm 30 mm C 80 mm 70 mm F = 2 N mm 60 mm

13 4.2 Moments in Three-Dimensional Force Sstems Problem Statement for Eample If the tension in the cable C is T = 80 lb, determine the moment about point of the cable force acting on the frame at point. lso, determine the shortest distance from to the line through and C. C 12 in. T = 80 lb 18 in. 10 in. 32 in.

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