Problems (Equilibrium of Particles)

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2 1. he -kg block rests on the rough surface. Length of the spring is 18 mm in the position shown. Unstretched length of the spring is mm. Determine the coefficient of friction required for the equilibrium. ( m =? ) D of lock =mg o or spring: l=18 mm, l o = mm (spring is compressed)..18 spring kl 5 5 Equations of equilibrium 9.81cos 368. : mg cos sin 18. : spring friction mg sin friction 5 1 friction friction m m.5

3 . -kg sphere rests on the smooth parabolic surface. Determine the reaction of the surface on the sphere and the mass m required to hold it in the equilibrium position. 6 o m =.5. m

4 3. Determine the stretch in springs C and for equilibrium of the kg block. he springs are shown in the equilibrium position. SOLUIO: D of C 5 o o rom equilibrium of : =mg=(9.81) 1 : cos36.87 cos 5 C 9.81 : sin sin C C C m m

5 . continuous cable of total length m is wrapped around the small pulles at,, C and D. If each spring is stretched 3 mm, determine the mass m of each block for equilibrium. eglect the weight of the pulles and cords. he springs are unstretched when d= m. D of Pulle C spring q q C q spring kl sinq C' q 9 d m.7 C q arcsin..7 m 1 m 1 ' D of Pulle mg q q 75 sinq cosq m g m kg

6 5. he pail and its contents have a mass of 1 kg. If the length of cable C is 7 m, determine the horiontal distance for equilibrium. lso find the tension in the cable. he mass of the small pulle at is small and can be neglected. m 1 m C

7 1 m L=7 m, =? (for equilibrium). =? q q' C 1 D of rom equilibrium of : q rom geometr: cosq q' =mg a 7 1 cosq 7 a 7 m : cosq cosq : sinq (*) a q a a a tanq m a 7 C q q 1. (*) sin

8 6. he clinder of mass 1 kg having a ver small diameter is held against a semi-clinder with a much larger diameter b two identical springs, which are fied to points C and C on h r C,C the ground. he springs are unstretched when at point. Knowing that the small clinder is in C equilibrium at point, what is the spring constant? r= mm h=1 mm C

9 7. wo bodies weighing 15 and, respectivel, rest on a clinder and are connected b a rope as shown. ind the reactions of the clinder on the bodies, the tension in the rope and the angle q. ssume all surfaces to be smooth. rope 1 = =15 9q q

10 SOLUIO: D of Particle 1 1 sin q cosq 1 cosq sin q 9q q 1 q q 1 = 1 cosq sin q cosq cosq sin q sin q cos q sin q sin q sin q

11

12 8. force of =1 holds the - crate in equilibrium. Determine the coordinates (,, ) of point if the tension in cords C and is 7 each. m 5 m 5 m SOLUIO: C =7 D of =7 =1 = Coordinates:,, 5,, C 5,, C 5i j k 7 5 5i j k j k

13 5 5 7 k j i C k j i j 1 k ' 5 1 3' ' 3' 1 ' m m

14 9. he shear leg derrick is used to haul the kg net of fish onto the dock. Determine the compressive force along each of the legs and C and the tension in the winch cable D. ssume the force in each leg acts along its ais. SOLUIO: D of D C =(9.81) Coordinates:,,,, C,, D, 5.6, C D i j k 6 i j k C j k D k

15 6 k j i k k j i C C k j D D D C Equations of Equilibrium here are three unknowns C D C D C 3 k k k D C

16 1. small peg P rests on a spring that is contained inside the smooth pipe. he spring eerts an upward force of 8 on the peg. Determine the point of attachment (,, ) of cord P so that the tension in cords P and PC equals 13 and 8, respectivel. P.6 m. m. m.3 m C

17 P spring = 8, = 13, C = 8 ; determine coordinates of point (,, ). D of peg P Coordinates C C. m.6 m. m C.3 m i j.6k.6 a. j.6k j 18.33k.3i. j.6k C i j 7k spring 8k spring C,,,., C.3,., P,,.6 rom equilibrium of peg P a a a a a a 3 1 a 36.8 m a a m a =

18 11. Crate weighing 58 rests on the inclined surface b the cable and force P which is parallel to the -ais. Determine the tension in the cable and force P for equilibrium. Since the crate is mounted on casters, the force eerted b the incline on the crate is perpendicular to the incline. 1.5 m 3 m. m C 3 m E P O m D

19 SOLUIO: D of Crate P (.; 1.; 1.5) 3 m (3cos37; 3 3sin37; 1.5) (; 5.; ) a 3 m m Reaction (normal force) eerted b incline on crate 3 m a m (Reaction force (normal force) must be perpendicular to the incline.) a a 37 o a orces acting on the crate 58 j P Pk. 9. i j 1. 5k. 9 cos37 j sin37i. 6i. 8j. i j 1. 5 k

20 orces acting on the crate 58 j P Pk. i j 1. 5k. 9. 6i. 8j P Equations of Equilibrium P P. 9 3 rom 1, and 3 ; =395, =3, P=1

21 1. Collars and are connected b wire and can slide freel on frictionless rods. If a force P=31 is applied to collar, determine (a) the tension in the wire, (b) the magnitude of the force Q required to maintain the equilibrium of the sstem and reactions on collar b the vertical rod. SOLUIO: Coordinates:,155,,,6 155 mm mm Problems (Equilibrium of Particles) D of collar P orces acting on collar i 155 j 6k 55 P 351j 1 j 6 mm Reactions eerted b the vertical bar on collar : i k Equations of Equilibrium P

22 1 i k P 351j i 155 j 6k 55 1 j P D of collar Q (correct sense) Reactions eerted b the horiontal on collar : i 155 j 6k j 6 Q Q i Q Qk j

23 13. If = =1, determine the force P, and the reactions between the collars and bars.

24 1. he 1-kg collar rests on the smooth straight fied bar CD b the cable. Determine the tension in the cable and the reaction between the collar and bar CD. C

25 15. he 1-kg collar rests on the smooth straight fied bar CD b the cable. Determine the tension in the cable and the reaction between the collar and bar CD. D of collar D C C Length of fied bar CD: CD 7 9 m 6 3 Coordinates of point :. 67 m m (.67;.33;.67) 9 9 (; 7; ) orces acting on the collar j 981j.67i.67 j 1.33k.8i.8 j.k 5.5 i j k (Reaction eerted b bar on the collar)

26 orces acting on the collar.8i.8 j.k 981j i j k (here are four unknowns:,,, ) (ut, there are there equations of equilibrium.) C (; ; ) D (; 7; ) SOLUIO MEHOD: Reaction force between the collar and bar CD is perpendicular to the ais of the bar. ( CD or n CD ). So, scalar product of these vectors must be equal to ero. dditionall, since n CD n CD n n n n CD CD CD CD C n CD i 7 j k 9 i 7 j k. 8i. 8 j. k 981j i j. 8k 88. 3i j. 8k

27 16. Collar weighing 17, which can slide freel.5 m on the quarter circle, is held in equilibrium b cable. a) Determine the tension in the cable. b) Determine the magnitude and the components of the contact force acting on the collar from the circle. D of collar 3. m 15 r =3. m 8 Coordinates of points and : ; 3. ; (3; 1.6; ) (; 3.;.5) 17 17

28 plane plane.5 m orces acting on the collar 3i 1. 8 j. 5k i j k Equation of Equilibrium 17 j (Reaction eerted b bar on the collar) 3. m rom 1, and 3 ; =85, =17, =5 15 r =3. m 8

29 17. Smooth collars, and C, each weighing 36, are connected b the wires and C and ma slide freel on the smooth rod having the shape shown. Determine the magnitude of the horiontal force P which must be applied to the collar to maintain equilibrium. DEG portion of the rod is parallel to -plane.

30 D of collar our unknowns P D of collar D of collar C C C our unknowns C hree unknowns C

31 D of collar C C C C (;9;3) (;5;7) C (6;1;) 1.5 m.5 m m 1 6 E C C C i j 7k C 9 36 j C i C k 81 9 C C C 36

32 D of collar C E q o sinq.6.5 m q q 6 m cosq.6 E C C C 36 i 36 j 63 k i j k 6 36 j.6 i.8 j k

33 D of collar P 36 j i P Pk 36 i 7 j j 7 k P 7 P P 7

34 18. Smooth collars and are connected b the spring. Spring has a constant of 1 /cm and its unstretched length is 3 cm. Determine the magnitude of the force P which must be applied to the collar to maintain equilibrium and the reaction between the collar and bar. eglect the weight of the collars. ake (;;) and (;;8). Q P cm cm cm 8 cm 8 cm 3 cm

35 D of collar P P spring spring spring P i j k 1(6 3) 6 i 1 j k Pi Pk i P 5 5 spring 1 q 3 P 5 5 P q 3 cm k j P 3 cm i j k cm cm cm Q 8 cm ( ) i ( ) j (8 ) k cm 1 Correct sense 6 8 cm cm

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