4.4 Moment of a Force About a Line

Transcription

1 4.4 Moment of a orce bot a Line

2 4.4 Moment of a orce bot a Line Eample 1, page 1 of 3 1. orce is applied to the end of gearshift lever DE. Determine the moment of abot shaft. State which wa the lever will rotate abot. 6 in. 40 D E {i j + 0.5k} lb E {i j + 0.5k} lb 5 in. D 5 in. 1 The moment M L of a force abot a line L is k M L r where is a nit vector along L, and r is a position vector from an point on L to an point on the line of action of. Here, the nit vector,, is jst the base vector, k.

3 4.4 Moment of a orce bot a Line Eample 1, page of 3 6 in. {i j + 0.5k} lb 40 D E 4 The best wa to evalate the scalar triple prodct is to se a calclator that has bilt-in fnctions for the dot and cross prodcts. If sch a calclator is not available, then evalate the triple prodct b sing a determinant in which the sccessive rows are, r, and. 5 in. M r E k r k r E hoose position vector r as r E : r r E (6 in.)(sin 40 )i + [5 in. + (6 in.)(cos 40 )]j {3.857i j} in (1)[3.857( 1) 9.596()] 3.0 lb in.

4 4.4 Moment of a orce bot a Line Eample 1, page 3 of 3 E 5 The negative sign in M 3.0 lb in. indicates that M has a sense opposite to. This moment will case the lever DE to rotate clockwise abot, when viewed from end. ns. D 3.0 lb in. 6 We cold have saved a little work b noting that M r E k r E M (moment of abot ) k component of M So we onl need to calclate M and look at its k component; we didn't need to evalate the determinant for the scalar triple prodct, r E.

5 4.4 Moment of a orce bot a Line Eample, page 1 of 3. Thread is plled off a spool as shown. Determine the moment of the force abot the ais of the spool. 30 mm 0.5 N 60 1 The moment M L of a force abot a line L is mm M L r where is a nit vector along L and r is a position vector from an point on L to an point on the line of action of.

6 4.4 Moment of a orce bot a Line Eample, page of 3 3 The best choice (simplest form) for r is r {30i} mm. The nit vector is jst the base vector, j. 30 mm N 4 (0.5 N) cos N 5 (0.5 N) sin N 6 (0.165 N) sin N 7 (0.165 N) cos N 8 In vector form, {0.1658i j 0.139k} N

7 4.4 Moment of a orce bot a Line Eample, page 1 of 3 9 M ais of bobbin r [30( 0.139) 0(0.1658)] N mm ns.

8 4.4 Moment of a orce bot a Line Eample 3, page 1 of 3. The lg wrench is sed to trn the lg nt holding the wheel on the hb. Determine the moment of the force abot the ais of the bolt to which the nt is fastened. {3i + 10j 5k } N 1 The moment M L of a force abot a line L is M L r 0.3 m 0.15 m 0.5 m where is a nit vector along L and r is a position vector from an point on L to an point on the line of action of.

9 4.4 Moment of a orce bot a Line Eample 3, page of Since we want the moment abot the bolt ais, which is the ais, the nit vector is the base vector: i {3i + 10j 5k} N 3 hoose r as the position vector from to : r r {0.3i j + 0.5k} m 4 M bolt r 0.15 m 0.3 m 0.5 m i {0.3i j + 0.5k} {3i + 10j 5k} [0.15(5) 0.5(10)] N m ns.

10 4.4 Moment of a orce bot a Line Eample 4, page 1 of 5 4. orces P 1, P, P 3, each of magnitde P, act on the edges of a cbe of side "a" as shown. Determine the moment of each force abot diagonal. lso epress each moment in vector form. orce P lies on the ais. P P G P 1 P H 1 The moment M L of a force abot a line L is a P 3 P M L r where is a nit vector along L and r is a position vector from an point on L to an point on the line of action of.

11 4.4 Moment of a orce bot a Line Eample 4, page of 5 To determine the nit vector, first define the position vector r. r ai aj Ths r r ai i aj a + ( a ) j a r P P G P 1 P H P 3 P

12 4.4 Moment of a orce bot a Line Eample 4, page 3 of 5 3 To calclate the moment of P 1 abot line, the best choice (simplest form) for r is r G (Note that G lies on the line of action of P 1 ). r r G ( a)k a r G M G 4 P 1 P Vector form of the force P 1 Pi H 6 5 or force P 1, M r G P [0(0) a) P)] a 1 0 a ap P a P 0 ns P 0 The vector representation of the moment abot is jst the prodct of the magnitde of the moment, M, times the nit vector, since is parallel to. Ths ap M ap i j ap ( i + j) ns.

13 4.4 Moment of a orce bot a Line Eample 4, page 4 of 5 7 To calclate the moment of P abot line, a good choice for r is r (Note that is on the line of action of P ). 9 or force P, M r P r r aj a 0 G H 0 0 P 1 a P P a a r r wold be an eqall good choice r M P P Vector form of the force P Pk 10 1 [ a) P) 0(0)] ap ns. Vector form M ap ap i j ap ( i + j ) ns.

14 4.4 Moment of a orce bot a Line Eample 4, page 5 of 5 11 To determine the moment of P 3 abot line, note that the line of action of P 3 passes throgh point. Ths G the force does not tend to rotate the cbe abot line : H M 0 and in vector form ns. M 0 ns. a P 3 P In mathematical terms, we cold choose a position vector of ero length (from to ), r 0. This wold give M r 0 0

15 4.4 Moment of a orce bot a Line Eample 5, page 1 of 4 5. Determine the moment abot the line throgh points and of the force shown. Epress the reslt in vector form. 10 m 1 The moment M L of a force abot a line L is 0 m 18 m D 15 m 6 kn 1 m M L r where is a nit vector along L and r is a position vector from an point on L to an point on the line of action of.

16 4.4 Moment of a orce bot a Line Eample 5, page of 4 10 m 3 The best choice (simplest form) for r is r r r [10 m ( 0 m)]j 18 m r 15 m 6 kn {30j} m Note that point lies on the line of action of the force. 0 m r D 1 m To determine the nit vector along, first introdce position vector r. r (0 0 m)j + (0 18 m)k { 0j 18k} m Then, r r 0j 18k ( 0) + ( ) j k

17 4.4 Moment of a orce bot a Line Eample 5, page 3 of 4 10 m r D 4 To epress the force in rectanglar components, first introdce a position vector from D to. r D { 15i + 10j 1k} m 18 m 0 m D 15 m 6 kn 1 m Then the force is, (6 kn) (6 kn) r D r D 15i + 10j 1k ( 15) + (10) + ( 1) { i j 3.347k} kn

18 4.4 Moment of a orce bot a Line Eample 5, page 4 of 4 5 M r ( 0. 33) ) [0( 3.347) 0( )] [0(.7705) 30( )] kn.m ns. 6 Vector form M M M is the moment in the direction of line, which is defined b the nit vector,. (.41){ j k} {6.0j k} kn m ns.

19 4.4 Moment of a orce bot a Line Eample 6, page 1 of 3 6. n electrical condit is held in place b brackets, D and E. The effect of the bracket at on the condit can be represented b the force as shown. Determine the reslting moment that prodces that tends to twist the portion G of the condit abot its ais. 3 m D E G 1 The moment M L of a force abot a line L is.5 m M L r { 0i +1j + 16k} N m 3.5 m where is a nit vector along L and r is a position vector from an point on L to an point on the line of action of.

20 4.4 Moment of a orce bot a Line Eample 6, page of 3 D 3 m r G E G To determine the nit vector, first introdce a position vector from to G. r G {3i +.5j} m Then the nit vector along G is, { 0i +1j + 16k} N r m.5 m 3.5 m r G r G 3i +.5j (3) + (.5) {0.768i j} 3 hoose the position vector r as r r { 3.5j + k} m

21 4.4 Moment of a orce bot a Line Eample 6, page 3 of 3 4 M G r [ 3.5(16) (1)] (16) ()( 0) 87.1 N m ns.

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