Problems (Force Systems)

Transcription

1 1. Problems (orce Sstems)

2 Problems (orce Sstems). Determine the - components of the tension T which is applied to point A of the bar OA. Neglect the effects of the small pulle at B. Assume that r and are known. Also determine the n-t components of the tension T for T=100 N and =35 o.

3 - coordinates AB r - r sin r r cos r 3 cos - sin r - rsin b T b r r cos 1 cos cos b r 3 cos - sin 3 cos - sin r - r sin 1- sin sin b r 3 cos - sin 3 cos - sin T T T cos b T -T sin b T 1 cos 3 cos - sin sin -1 3 cos - sin r + rcos n-t coordinates (for =35 o and T=100 N) 1- b arctan 1 T T n t T cos T sin sin cos b 100cos b 100 sin N o N b T

4 Problems (orce Sstems) 3. In the design of the robot to insert the small clindrical part into a close-fitting circular hole, the robot arm must eert a 90 N force P on the part parallel to the ais of the hole as shown. Determine the components of the force which the part eerts on the robot along aes (a) parallel and perpendicular to the arm AB, and (b) parallel and perpendicular to the arm BC.

5 the robot arm must eert a 90 N force P on the part force which the part eerts on the robot vertical P=90 N (a) parallel and perpendicular to the arm AB P=90 N AB horiontal P AB 15 o vertical 45 o 30 o P //AB 60 o //AB P //AB = P AB =90 cos45=63.64 N (b) parallel and perpendicular to the arm BC. vertical //BC horiontal P=90 N P //BC 30 o 45 o 15 o 45 o 45 o P //BC =90 cos30=77.94 N P BC =90 sin30=45 N P BC BC

6 Problems (orce Sstems) 4. The unstretched length of the spring is r. When pin P is in an arbitrar position, determine the - and -components of the force which the spring eerts on the pin. Evaluate our answer for r=400 mm, k=1.4 kn/m and =40.

7 Problems (orce Sstems) 5. Three forces act on the bracket. Determine the magnitude and direction of so that the resultant force is directed along the positive u ais and has a magnitude of 50 N. 3 =5 N 1 =80 N

8 if R=50 N =? =? Resultant R R cos 5i - Rsin 5 j R 50cos 5i - 50sin 5 j R i j 1 80i cos 5 i - sin 5 j 5 5 i 13 R i j 0i 3 cos sin cos 5 - sin j 48 j =5 N 1 =80 N R sin cos tan N

9 Problems (orce Sstems) 6. The turnbuckle T is tightened until the tension in cable OA is 5 kn. Epress the force Ԧ acting on point O as a vector. Determine the projection of Ԧ onto the -ais and onto line OB. Note that OB and OC lies in the - plane.

10 O plane Ԧ acting on point O 65 o C =5sin50=3.83 kn =5cos50=3.14 kn = sin65=3.14sin65=.91 kn = cos65=3.14cos65=1.358 kn 1.358i.91j 3.83k Projection onto ais j. 91 kn OB OB OB Projection onto line OB O Unit vector of line OB n OB 30 o n OB cos30i cos60 j B n 1.358i.91j 3.83k 0.866i 0.5 j OB kn

11 7. The cable BC carries a tension of 750 N. Write this tension as a force T acting on point B in terms of the unit vectors Ԧi, Ԧj and k. The elbow at A forms a right angle. T T acting on point B T Tn BC T r r C / B C / B The coordinates of points B and C are B (1.6; 0.8sin30; 0.8cos30) B (1.6; 0.4; 0.693), C (0; 0.7; 1.) The position vector BC is r B -1.6i 1.1 j 0.507k rc / B BC i 1.1 j 0.507k The unit vector of T r BC is nc / B nbc nt i j 0.53k Tension T acting on point B in vector form T Tn BC C / i j 0.53k i 411j k 750 m

12 8. In opening a door which is euipped with a heav-dut return mechanism, a person eerts a force P of magnitude 40 N as shown. orce P and the normal n to the face of the door lie in a vertical plane. Epress P as a vector and determine the angles, and which the line of action of P makes with the positive -, - and -aes. P =40sin30=0 N P =40cos30=34.64 N plane 0 o // P = P cos0=34.64cos0=3.55 N // n P P = P sin0=34.64cos0= N P 3.55i j 0k angles, and which the line of action of P makes with the positive -, - and -aes a cos a cos a cos 40 60

13 Problems (orce Sstems) 9. The spring of constant k =.6 kn/m is attached to the disk at point A and to the end fitting at point B as shown. The spring is unstretched when A and B are both ero. If the disk is rotated 15 clockwise and the end fitting is rotated 30 counterclockwise, determine a vector epression for the force which the spring eerts at point A.

14 Problems (orce Sstems) 10. An overhead crane is used to reposition the bocar within a railroad car-repair shop. If the bocar begins to move along the rails when the -component of the cable tension reaches 3 kn, calculate the necessar tension T in the cable. Determine the angle between the cable and the vertical - plane.

15 T =3 kn, calculate tension T, the angle between the cable and the vertical - plane. Unit vector of T 5i 4 j k nt n 0.77i j 0.154k T T in vector form T T 0.77i j 0.154k -component of T 0.77T 3 T and -components of T T T kn (magnitude of T) kn T kn i.4 j 0.6k T T T kn T 3.84 cos T

16 Problems (orce Sstems) 11. The rectangular plate is supported b hinges along its side BC and b the cable AE. If the cable tension is 300 N, determine the projection onto line BC of the force eerted on the plate b the cable. Note that E is the midpoint of the horiontal upper edge of the structural support.

17 If T=300 N, determine the projection onto line BC of the force eerted on the plate b the cable. Coordinates of points A, B, C and E with respect to coordinate sstem A 400, 0, 0 B (0, 0, 0) C 0, 100sin5, 100cos5 C (0, , ) E 0, 100sin5, 600cos5 E (0, , ) - 400i j - T TnT T -14.1i j k Projection of T onto line BC T T n BC BC k Unit vector of line BC -cos 5k sin 5 j 0.43 j k n BC i j k 0.43 j k T BC N n BC 5 o 5 o

18 Problems (orce Sstems) 1. The and scalar components of a force are 100 N and 00 N, respectivel. If the direction cosine l=cos of the line of action of the force is -0.5, write Ԧ as a vector. 100 N 00 N l -0.5 l m n 1 m n m n N cos m cos n N cos N -19.1i 100 j 00k

19 13. Determine the parallel and normal components of force Ԧ in vector form with respect to a line passing through points A and B. Problems (orce Sstems) Cartesian components of kn 5 3 A B (6, 4, 8) m cos kn sin kn kn (3,, 5) m =75 kn 41.34i j 38.59k

20 41.34i j 38.59k Unit vector of line AB 3i j 3k n AB 0.639i 0.46 j 0.639k 3 3 Parallel component of to line AB (its scalar value) : // nab 41.34i j 38.59k 0.639i 0.46 j 0.639k // // kn Parallel component of to line AB (in vector form): n i 0.46 j 0.639k 46.14i j 46.14k // // AB Normal component of to line AB (in vector form): i j // k B n (6, 4, 8) m AB A (3,, 5) m =75 kn

21 14. Determine the magnitude and Problems (orce Sstems) direction angles of the resultant force acting on the bracket. Resultant R 1-450cos 45sin 30i i 450cos 45cos30 j j k 450sin 45k

22 Direction angles for Direction cosines cos cos l m n cos cos 45 cos 60 cos 1 cos 0.5 cos cos cos 45i 600cos60 j 600 cos10k 44.6i 300 j 300k -

23 -159.1i j k Resultant R 1 R 44.6i 300 j i j k R 65.16i j 18.k 300k Magnitude of Resultant orce R R Direction Cosines of Resultant orce cos cos R R cos R R Direction angles for R arccos( 0.418) 65.3 cos cos R R arccos(0.907) 4.9 N cos arccos(0.09) 88.3

24 Problems (orce Sstems) 15. Epress the force Ԧ as a vector in terms of unit vectors Ԧi, Ԧj and k. Determine the direction angles, and which Ԧ makes with the positive -, -, and -aes.

25 Position vector k j i AB r k j i AB r A B A B / / Unit vector k j i n k j i r r n A B A B / / - - k j i k j i n n m l n m l Direction cosines Direction angles cos cos cos n m l