Appendix

Standard Integrals

S_dx __ = ~ ln a2 - x2 2a (~) a - x

(forx<a)

---=-tan-S dx 1 _1 (X) a2 + x 2 a a

f dx __ _2_ ln (x - a) x 2 - a2 2a x + a

(for x >a)

f xdx 1 2 2( --- = - 2 ln (a - x ) for x < a) a2 - x2

Length of a Curve and Radius of Curvature

Length of a small segment of a curve

or

ds = [' + (:)')~

ds = [ (: )' + ,, p6

352 BASIC ENGINEERING MECHANICS

Radius of curvature

or

Solutions of Particular Differential Equations

For

the solution is

y =A sin(yKt + cf>)

For

the solution is

APPENDIX 353

Properties of Homogeneous Thin Plates

m is the mass of the body shown.

Plate Area Mass - centre Moment of Inerlio

~ b mb2

~ y0= constant -<o=2 Iyy=12

~ ab y

ma2 fci~y rectangular

a .Yo=2 Ixx=l2

J• b y'l JO=constxx0 2b mb2

xo=-y lyy=18

~] ab 2

Yo • - a ma2 fci xo y· tnangular Yo=3 Ixx=18

j• b y"l y0=constxx02 3b 3mb2 -<o=4 Iyy=ao

~I ab 3 Yo - 3a 1 = 37ma2 fo\ •o y· parabolic ro=w XX 700

j· b •I y0= canst x x0" - (n+1lb L _ mb2(n-t 1) xo= (n+2) YY (n +2)2 (n-t 3)

~;r ob

(n-t1)

YofO x y• (n-t1)a [, = ma2 (n+1)(7n2+4n+1) .Yo=2(2n+1) xx 12(2n+1)2(3n+1)

~ b+c m(b2+c2-bc)

X"o=-3- 1rr= 18

P--+ -¥ x a ab 2

a ma2 ofO x y·b tnangular J6=3 Ixx=18

~ 2r Sin a L = mr2 [1+ s~n2a _!§( Sln2al] X"o=---sa YY 4 2a 9 a2

r 2a x·- Y -x c1ncular

1 = mr2(1_son 2 a) sector xx 4 2a

x-~-x 4r mr2 ( 64 )

X"o= 3,. Iyy=---,;;-- 1- g;2

..,,2

0 X"o 2 mr2

y• sem1c1rcular fxx=4

: Fis;h - 4b mb2 64 •o= 3-rr Iyy=4 (1- 9..,2)

-rrab 4

Yo elhphcal 4o ma2 64 fO~y· quadrant .Yo=3.., J.x=4 (1- 9,;2)

~y rs1na 1 =mr2(l+son2a _ s~n2 a) x=--

x-m-x 0 a YY 2 4a a2

-th1n CirCular

I. =mr2(1- s1n2a) •o--.... annulus I IY' xx 2 4a

354

Properties of Homogeneous Bodies

m is the mass of the body shown.

Body

th1n rod

crrcular cylinder

elliptical cylinder

rectangular parallelepiped

nght c~rcular cone

hem1sphere

BASIC ENGINEERING MECHANICS

volume

=.,. abL

area of curved surface = rr(a+b) L

volume

=~rr/L

area of curved surface =.,. r /(r 2+L2 )

volume

=% 7rr3

area of curved surface = 2.,-r 2

Moment of Inert1a

lxx= lzz

= -&mL2

Ixx= lzz

=tmr2+tzmL2

fxx= lzz

= {omr2 + tomL 2

Ixx =I,, 83 2

= 320mr

Answers to Problems

Chapter 2

2.1 (a) (b) (e) (f) scalar; (c) (d) (g) (h) vector. 2.2 (a) 7.74 L 71.2°; (b) 26.5 L- 10.9°; (c) 213 L 195.9°. 2.3 (a) 2.83 L 28°; (b) 17.3 L 120°; (c) 595 L 35°. 2.4 3.1 L 142° 2.5 (a) ax = 9.64, ay = 11.5; (b) bx =- 17.3, by= 10; (c) Cx =- 8.66,

Cy = 5. 2.6 8.78 L- 52.9°. 2.7 I al = 8.37; (a) 21 o; (b) 45.8°. 2.8 12.8 L- 18.2°. 2.9 a= 14.5 L 39.9°, b = 6.2 L 186.2°.

Chapter 3

3.1 20 Nat 47°, 30 Nat 29°. 3.2 IPI = INI = 1.13 N. 3.3 37.4°, 6.69 N, 23.8 N. 3.4 (a) 37.8 N L 126.6°; (b) 37.8 N L- 53.4°. 3.5 175400 N. 3.6 IPI=127N,IQ1=58.5N. 3.7 P= 133.6 N L 153.9°, Q = 66.8 N L 63.9°. 3.8 T=16.24N~--+ ____.. ~ 3.9 Note OA = OG + GD + DA and GC = 2GD, where Dis the centre of AB.

Chapter 4

4.1 (a) 16.4 N L 165.4°, 29.3 N m anticlockwise; (b) 16.4 N L 165.4°, 1.79 mL 75.4° fromA.

4.2 P= 10 N, 14.14 N L 135°. 4.3 (a) Yes; (b) R 1 = 425 N, R 2 = 175 N, R 3 = 1000 N. 4.4 (a) Yes; (b) R 1 = 204 N, R2 = 1022 N,R 3 = 162 N. 4.5 (a) No; (b)RA = 375 N,R8 = 41.6 N,Rc = 83.3 N; (c) statically

indeterminate. 4.6 A force 23.2 N L 190.3° plus a couple 2.17 N m anticlockwise. 4.7 F= 115 N;RH = 140 N L 64.3°. 4.8 P= 68 N;RH = 207 N L 109.1°. 4.9 (a) 22.3° (40 kg), 49.5° (20 kg); (b) 11.3°. 4.11 (a) No; (b) 8.53°. 4.12 P= 343 N,Rcx = 98.1 N~,Rcy = 109 Nt,R0 = 60.4 Nt,RH = 263 Nt,

RE = 332 Nt. 4.13 P= 1037 N,R0 = 770 N~,REx = 870 N+-,REy = 273 N~. 4.14 RE = 0.53 kNt, RAx = 2.12 kN~, RAy= 0.18 kNt; ED, 1.06 kN(T);

356 BASIC ENGINEERING MECHANICS

EC, 1.19 kN (S); DC, 1.41 kN (T); DA, 0.35 kN (S); CB, 0.71 kN (T); CA, 1.97 kN (S); BA, 0.71 kN (T).

4.15 BC, 2.73 kN (T); CD, 2.45 kN (S); BD, 1.73 kN (T); DE, 1.73 kN (S); AB, 4.46 kN (T); BE, 2.45 kN (S); AE, 1.73 kN (T). The force in AB would be reduced to 1.73 kN.

4.16 CE, 1.42 kN (T); ED, 2.58 kN (S); CD, 1.33 kN (S); AC, 1.38 kN (T); DF, 3.02 kN (S); AD, 0.72 kN (S).

4.17 Torque= 2.19 Nmclockwise;RA = 13.8 N L 179.4°,R0 = 8.37 N L 126.9°.

4.18 RAy= 0.354 k~t,RAx = 2.6 kN~;Rcy = 0.646 kN.J-,Rcx = 3.63 kN+-; Ray= 0.354 kN.J,, Rsx = 1.9 kN~.

4.19 Torque = 1.53 kN m clockwise; R 0 = 1. 78 kN L - 90°; R E = 14.5 kN L 200.6°;RF = 7.7 kN L 62.6°.

4.20 Torque= 5kNmclockWise;R0 28 kN L - 39f,R0 = 24.7 kN L 243°, RE = 10 kN L 180° (graphical soltltion).

4.21 Torque = 522 N m anticlockwise. 4.22 P= y3 (M + m)g

Chapter 5

5.4 p.>0.5 5.5 (a) No; (b) Nw, 147 to 131, Fw, 0 to 26.3, NF, 490to 464, FF, 147 to

131; (c) No, the friction forces cannot be limiting simultaneously, (d) No. (e) 32.6°.

5.7 0.52 m. 5.10 (a) 334 N; (b) 864 N; (c) (i) 56.6°, (ii) 33.4°. 5.11 (a) 51.4°; (b) 68.2 N. 5.12 (a) 16.7°; (b) Yes; (c) (i) 232 N, (ii) 65.7 N. 5.13 3.34 m. 5.14 Belt first slips at B when TtfT2 = 1.702; 132 N m; Yes. 5.15 52.5°. 5.16 515 N. 5.17 (a) 27 N m; (b) 52.7 N m in direction of motion. 5.18 (a) 11.7 N m; (b) 3.45 N m; 0.08 m. 5.19 (a) 327 N m; (b) Yes; (c) 0.205 m. 5.20 125 N m. 5.21 r 1 = 0.126 m, r2 = 0.378 m, Mr = 437 N m; P max = 100 kN/m2 ,

Mr=403 Nm. 5.22 (a) 503 N; (b) 208 N. 5.23 IJ.a > Ro IJ.o/ra· 5.24 (a) At the axle; (b) 8 = 9.4°, P= 650 N.

Chapter 6

6.1 P= y3 (M + m)g

ANSWERS TO PROBLEMS

Chapter 7

7.1 x = 1.67m,y = 0.833 m. 7.2 2.77 m. 7.3 (a) 0.266 m; (b) 0.197 m. 7.4 27.3°. 7.5 x = 1.54 m, y = 0.937 m. 7.6 x =2m, y = z = 0.85 m. 7.7 L =Ry'(2/3). 7.8 0.163 m. 7.9 (a) 0.35p g; (b) 1.96 m. 7.10 (a) 6280 kg; (b) 0.975 m. 7.11 A= 0.141 m2 , V= 0.00471 m3 •

7.14 Area= KR 3 /3; x = 3R/4; volume of revolution= rrKR4 /2; 14 rad/s.

Chapter 8

8.1 (a) 23.1 s; (b) 46.7 s. 8.2 v = 30 m/s, a= 5/6 m/s2 , 540 m. 8.3 (a) 15 m/s; (b) 400/3 s, 200 s, 800/3 s. 8.4 (a) No; (b) No, it has a limiting value ofO.OS m/s2 ; (c) 240.5 s;

(d) 1360 m; (e) 1/45 m/s2 , 1/25 m/s2 •

8.5 (a) 10 m/s; (b) 1.35 s, 8.47 m; (c) It falls until it reaches a steady value of 10 m/s.

357

8.6 (a) 29.3 s, 170.7 s; (b) 100 s; (c)- 1.414 m/s2 , + 1.414 m/s2 ; (d)- SO m/s; (e)+ 690 m,- 4023 m; (f) 4713 m; (g) 63.4 s, 236.6 s; (h) 9426 m.

8.7 v = 20.6 m/s L 61.7°;a = 167.7 m/s2 L 262f; 1.69 m L 57°. 8.8 (a) 4.47 m L 26.6°; (b) 2.0 s; (c) At this time the particle is at its nearest

distance fort> 2 s; (d) v = 2.24 m/s L 116.6°, a= 3.16 m/s2 L 161.6°. 8.9 7.65 m L 87.7°, v = 5 m/s L 144.5°,a = 3.55 m/s2 L 234.5°. 8.11 (a) Vm = 31.4 m/s, am= 1974 m/s2 ; (b) v = + 25.1 m/s, a=- 1184 m/s2 ;

(c) 0.0356 s. 8.12 (a) Urn= 6.58 rad/s; (b)~ = 82.7 rad/s2 ; (c) n = 6.2 rad/s, a=- 27.6

rad/s2 •

8.13 T= 12 s, A= 0.3 m. 8.14 x = 6.4 sin (4t + 38.7°). 8.15 (a) 0.603 m L 12.7°; (b) v = 17.8 m/s L 204.8°; (c) a= 262 m/s2

L 153.7°. 8.16 23.1 m/s, 46.2 m. 8.17 74.5 min, 6 km, 54 min. 8.18 (a) 18.75 km/h; (b) 13.4 min.

Chapter 9

9.1 29.62 N; beam balance will indicate 19.62 N. 9.2 v = y'[2gs (sin 0 - 11 cos 0)]. 9.3 a= g [m2 - m1 (p. cos 0 +sin 0)] /(m 1 + m2 ).

358 BASIC ENGINEERING MECHANICS

9.6 64.3°. 9.7 (a)gy(4 sin2 8 + cos2 8); (b) y(2gR sin 8); (c)3mg sin 8. 9.8 (a) 40.19 m/s; (b) 6.84 s; (c) 175m. 9.9 TAB= 4g/3 N, T8c = 2g/3y3 N. 9.10 (a)x =- na sin 8;x=- il2a cos 8; (b)N= m(g- il2a cos 8); (c)

il=y(gfa). 9.11 (a)j =- 4.16 m/s; (b).X =- 4.13 m/s2 ,y =- 2.37 m/s2 •

9.13 8=44.4°. 9.14 8.68 m/s. 9.15 t = 0.055 s, s = 1.39 m. 9.16 T= 2rry(R/g). 9.17 (a) 1 m/s; (b) 0.2 m; (c) 0.943 s. 9.18 v = 10 m/s L 38.7°;a = 18.87 L 96.7°;P= 1 N;N= 1.6 N;P does not

vary. 9.19 a1 =-21g/51,a2 =15g/51,T1 =72mg/51.

Chapter 10

10.1 6 m/s to the left. 10.2 Body comes instantaneously to rest after 0.64 s, velocity after 5 s is

46.8 m/s to the right. 10.3 (a) t2 = 2.94 s (it comes to rest at this time in its downward motion and

10.4 10.5 10.6 10.7 10.8

10.9 10.10 10.11

10.16 10.17 10.18

instantaneously reverses); (b) t3 = 3.88 s; (c) 24 m/s. v = 6.61 m/s, 17850 N. 2m, 300 Nm. my(5gR). 0.9R; it carries on in the same direction with velocity 2y(gR.f5). Taking the direction of the velocity of A before the impact as positive (a) vA =- 6.57 m/s, v8 = + 1.99 m/s; (b) 12.8 N s; (c) 13.7 N m. (a) 125 m/s; (b) 41.7 m/s; (c) 55.2 m/s. 904 m/s, 58.1 N s. (a) 10 m/s; (b) 3 m/s; (c) 3.75 m/s, 0.968 m; (d) vA = 7.5 m/s, v8 = - 2.5 m/s; (e) v A·· 0 to 7.5 m/s, v8 , 10 to- 2.5 m/s. (mv2 tan2 cp)/2 + mgR (1 -cos cp). 3. Taking the velocity of A as positive (a) v A= (L 1 + L 2 }\/[Km2 /m 1

(m2 + md], Va =- vAmdm2;(b)sA =sa= (m2L2- m1Ld/ (m1 + m2); (c) sA= [m2(L + L2)- m1Ld /(m2 + m!), sa= [m2L2 -m1 (L + Ld] /(m2 + m!).

10.21 10 w.

Chapter 11

11.1 EVB = 4.35 m/s L 36.6°, E0B = 10.14 m/s2 L 249.5°. 11.2 n = 2.38 rad/s direction indeterminate; o: = 1.66 rad/s2 clockwise;

28 m/s2 L- 30.4°. 11.3 EvH = 4.07 m/s L 0°; ilsoF = 5.7 rad/s clockwise.

ANSWERS TO PROBLEMS

11.4 11.5

(a) Eve= 0.68 m/s L 147°; (b) nHJ = 1.86 rad/s clockwise. (i) (a)- 5 rad/s, (b)+ 7 rad/s; (ii) (a)- 35 rad/s, (b) 1 rad/s; (iii) (a)- 20 rad/s, (b)+ 4 rad/s. nAc = 1.73 rad/s anticlockwise, Evn = 2.75 m/s L 23.7°.

359

11.6 11.7 (a) nac = 5 rad/s anticlockwise, aac = 43.3 rad/s2 clockwise; (b) EaG =

86.6 m/s2 L 210°. 11.8

11.9

11.10

11.11

11.12 11.13 11.14

11.15 11.16

11.17

11.18

(a) EvP = 5.7 m/s L 30°; (b) EaP = 18 m/s2 L 210°; (c) EaG = 96 m/s2

L 260°; (d) ncP = 2.54 rad/s clockwise; (e) 28 rad/s2 clockwise; (f) retarding. (a) EvF = 3.9 m/s L 174°, EaF = 42.5 m2 L - 8°; (b) nAB = 4.58 rad/s anticlockwise, aAB = 28.3 rad/s2 anticlockwise. (a) EvF = 0.96 m/s L 30°; (b) nAB= 0.98 rad/s antic1ockwise; (c) a 80 =

24.3 rad/s2 anticlockwise. EaD = 10.6 m/s2 L 0°; sliding acceleration of C relative to the slot is 13.5 m/s2 L 117.3°. EVA= 2.04 m/s L 0°; EaA = 152.4 m/s2 L 0°. EaFG = 10.7 m/s2 L- 90°; aac = 22.8 rad/s2 anticlockwise. nAB= 2.886 rad/s clockwise; aAB = 5.966 rad/s2 anticlockwise; EVD = 3.82 m/s L 10.9°; EaD = 5.31 m/s2 L 253.6°. Eva= 2.89 m/s L 0°; Eaa = 53.9 m/s2 L 180°. nAB = 0.88 rad/s clockwise; aAB = 0.052 rad/s2 anticlockwise; dr/dt = + 3.09m/s; d2 r/dt2 = + 7.63 m/s2 •

(a) nPQ = 17.6 r~d/s clockwise, G:p9 = 2590 r~d/s2 clockwise; (b) EVz =

9.93 m{s L- 45 , Eaz = 1004 m/s L- 65.3 ; (c) EvQ = 12.5 m/s L- 82.5°, EaQ = 1680 m/s2 L 253.7°. nAB= 2.886 rad/s clockwise, G:AB = 4.81 rad/s2 anticlockwise; EaG = 4.81 m/s2 L- 90°.

Chapter 12

12.3 (!xx)G = ptR4 (1T/8- 8/91f), (!zz)G = ptR4 (1T/4- 8/91f). 12.4 lyy = 13000 kg m2 •

12.5 lxx = 4050 kg m2 •

12.6 lzz = 16611 kg m2 •

12.7 lzt zt = lz3 z3 = 1624p kg m2 , lz2 z2 = 595p kg m2 •

12.8 lxx = 47.57 kg m2 •

12.9 ~Y = 44.25p kg m2 , lxx = 56.025p kg m2 , lytyt = 57.75p kg m2 •

12.10 ~lxx)G = 10.687p kg m2 , (/zz)G = 17.062p kg m2 •

12.11 (a) I axis = 8.247 kg m2 ; (b) /dia = 6.1 kg m2 .

12.12 9.675 kg m2 , 85.2%.

Chapter 13

13.1 Body accelerates to the right with a= 100 m/s2 L 0° and a= 41.7 rad/s2

clockwise. 13.2 P= mg, a= g. 13.3 P = 1.9mg.

360 BASIC ENGINEERING MECHANICS

13.4 (a) a= 0.000768 rad/s2 clockwise; (b) n = 0.0577 rad/s (direction indeterminate); (c) a X = 2.04 m/s2 L 0°' a;. = 2.33 m/s2 L - 90°' Ts = 238000 N, Ta = 496000 N L - 70.1 °.

13.6 p. = 0.2. 13.7 (a) a=- (2gcos 8)/L; (b) n =y[4g(sin{3- sin 8)/L]; (c)ax =gcos 8

(2 sin {3- 3 sin 8), ay = g(2 sin28- cos28- 2 sin {3 sin 8); (d) y[(4g sin {3)/L].

13.8 a= 12.1 m/s2 ,P= 1490 N,NE = 85 N,N0 = 510 N. 13.9 F=9.26N L0°. 13.10 (a) 595 N m clockwise; (b) 187.5 N L 120°. 13.11 a= (3g cos 8)/4, n = y'[(3g sin 8)/2], 174 N L 129.3°. 13.13 (a) 1.93 m/s2 ; (b) 1080 N m; (i) 1022 N in the direction of motion;

(ii) 57.9 N in the opposite direction to motion. 13.15 f= 0.249 Hz, 2.49 rev/s. 13.16 (a) a= 1.504 rad/s2 clockwise; (b) 7.06 Nat wall, 11 Nat floor. 13.17 2673 N, 16.04 kW. 13.18 24.25 N m anticlockwise; power in= 50 W, power out= 70 W. 13.19 10.65 N m anticlockwise, 30.74 W out. 13.20 n = 10.93 rad/s, v = 9.87 m/s. 13.21 QA = + 53.6 rad/s; rJ.a =- 134 rad/s. 13.22 rJ.A = rJ.a, rJ.A = rJ.c + 200, rJ.c =- 21.05 reV/min, rJ.A = 179 rev/min. 13.23 n = 31/Sma. 13.24 V= 6.3 m/s, a=- 2.01 rad/s2 (that is, opposite to its angular velocity). 13.25 (kG 2 = 0.0428 m2 ) (a) 5.23 rad/s; (b) 0.284 m/s; (c) 28370 N; (d)

11890 N. 13.26 DAB= rJ.Bc = 6u/1L. 13.27 6.85 rad/s. 13.29 (a) 250 N m; (b) 39.8 N m, 18.03 rad/s. 13.30 (a) 140503 N m; (b) 197050 N m. 13.31 v= 11.083 m/s, 10908 Nm. 13.34 a = 3p.g, a = p.g. 13.35 1.19 m. 13.36 Slipping does not take place at P; a A = 0.477g, aa = 0.147g, a= 0.11g

rad/s2 •

13.38 Energy dissipated= mgs p. cos {3 (sin {3- 3p. cos {3)/(sin {3- p. cos {3). 13.39 (a) nt = y(2Mtgr)/(IG + Mtr2 ); (b) v2 = r1.tL(IG + Mtr2 )/(IG + Mtr2 +

M2 L 2 );(c)v3 =2v2;(d)s=4M1L2 r(IG +M1r2)/p.(IG +M1 r 2 +M2L2i. 13.40 (b) 0.0122 m.

Index

acceleration 1 25, 164 absolute 232 angular 143 Coriolis 156, 244 diagram 241 gravitational 1 7 image 241 relative 148, 231 sliding 245 summation 150

acceleration components, intrinsic 139

polar 140 rectangular 139

acceleration diagram 242, 244 active force diagram 95 amplitude 129 analytical method 248 angular impulse, particle 295

rigid body 296

bearing and axle friction 81 - 3 belt and cable friction 75 -77

centre of gravity 25, 104- 12 circular wire 108 composite bodies 119 sector plate 109 triangular plate 107

centre of mass 112 centre of oscillation 290 centre of parallel forces 24 centre of percussion 288, 302 centre of suspension 290 centroid 113- 18 clutch friction 78-81 coefficient of friction 58 coefficient of restitution 195, 205 components 8 - 1 0

rectangular 13 - 15 composite problems 332-8 concept 1 configuration 41, 98 connected bodies 33 -7, 94 connected particles 1 7 5 constrained link 239-42

constraint 93 relaxation of 96 types of 33 workless 93

Coriolis 156 component 156

couple, definition 23 effective 286 inertia 290 work of 92

definition 1 deformation 193 diagram, acceleration 241

active force 95 free-body 5, 11 velocity 240

dimension 1, 2 derived 1 primary 2

displaceJUent 124 virtual 93

elastic constant 96 energy, ldnetic 203, 312

potential 213, 215 total mechanical 215

entity 1 derived 1

equations of motion, particle 164 -70 particle system 170- 84 rigid body 283-95

equilibrium 3 conditions for 15, 30, 94 definition 15 dynamic 167, 291

fixed-axis rotation 228, 288 force 1, 13

active 95 applied 31, 93, 167 body 33,171 conservative 213 constraint 31, 9 3 contact 171 effective 166, 283

362

force (cont.) external 21 , 1 71 , 286 extraneous 216 field of 167, 214 friction 57 gravitational 16, 31, 167 impulsive 192 inertia 21, 166,290 internal 171 moment of 25 reaction 31 resultant 13, 164 surface 32 unit 165 work of 91

force- couple set 27- 8 equivalent 286

frame, reference 165 inertial 165

frames, simple 40 virtual work 94

free body 5, 33, 41 diagram 5, 11

freedom, degrees of 94, 98, 175 frequency 129

circular 129 friction 57-90, 325

angle of 59- 60 applications 67 - 83 characteristics of 57-9 circle 82 coefficient 58 - 9 dry or Coulomb 57 in rigid body motion 325- 31 law of 58 limiting value 58

gravitation, law of 210 gravity 16

acceleration due to 211 centre of 25, 104

hodograph 137

impact 192 central 193 direct 193 eccentric 193 elastic 193 oblique 193

oblique central 199 plastic 196 rigid bodies 305 - 11

impulse 186, 295 angular 295 linear 186

INDEX

impulse- momentum equation, particle 187

rigid body 296 inertia 3

couple 290 force 290

inertial frame 165 instantaneous centre 232, 235

jamming 68, 71

kinematics, particle 123 - 63 rigid bodies 226-61

kinetics, particle 123 equations of motion 165 - 85 integrated forms 185 - 225

rigid body 283- 350 kinetic energy, particle 203

rigid body 312

law 1 length 1 linear impulse, particle 186

tigid body 299 linear momentum, particle 186

rigid body 298 machine 41 mass 3

inertial 165 unit 165

mass-centre 112, 190 definition 173 motion of 1 72

mechanical energy 215 mechanism 41

virtual work 96 moment, of components 26

of couple 26 -7 of force 25 - 6

moment-centre 25 moment of area 113 moment of inertia 262- 81

circular cylinder 272 circular plate 268 complex bodies 274

INDEX

cylinder theorem 273 elliptical plate 269 parallel axis theorem 265 perpendicular axis theorem 262 rectangular plate 268 rectangular prism 271 sphere 274 triangular plate 269

momentum, angular 295 linear 186

conservation of 190 motion, angular 143

constrained 236 curvilinear 134-42 equations of 164- 84, 283- 95 fixed-axis rotation 228 general plane 228 impending 61 oscillatory 134- 42 plane 226 rectilinear 122-34 relative 148-58 rigid body 226-32 second law of 164 simple harmonic 128, 145 translation 226

Newton 1, 164 first law 15 law of gravitation 210 second law 164 third law 21

Pappus and Guldinus 114 theorems 114

parallelogram law 7 particle, definition 5

equations of motion 164 -70 particle system 189, 207

equations of motion 170- 84 pendulum, compound 289

simple 289 periodic time 128 phase angle 146 polygon law 13 position 123

components 135 relative 148

postulate 1 potential energy, particle 213

rigid body 21 5

power 205-6 of torque 314

principle 2 of motion of mass-centre 174 of transmissibility 22 of virtual work 94

radius of gyration 262

363

relative motion, curvilinear 149-58 rectilinear 148

restitution 193 coefficient of 195

resultant, coplanar forces 28- 9 definition 13 parallel forces 22-4

retardation 125 rigid body, definition 5

equations of motion 283-95 rolling disc 236-8 rules of thumb 248

scalar, definition 6 screw friction 72 - 5 self-locking 68-71 simple harmonic motion 128, 145 solution of problems 11, 31, 67, 174 speed 125 spring force, work of 212 strain energy 31 5 strut 38 surface of revolution 114 surroundings 5

work of 209 system, boundary 5

definition 5 of particles 207

three-force member 34 thrust bearing 7 8 -81 tie 38 time 1 torque 27

work of 92 translation 150, 227

curvilinear 227 rectilinear 227

triangle law 7 tribology 57 truss, simple 37-40

virtual work 94 two-force member 34

364

unit 22 British Engineering 1 0 coherent 165 force 165 length 165 mass 165 SI 10, 165 time 165

unit vector 8

Varignon 26 theorem 26

vector 6 addition 7 component 8 definition 7 subtraction 8

velocity, absolute 232 angular 143 definition 124 diagram 240 image 240 notation 239 relative 148, 231 sliding 245 summation 150

INDEX

velocity components, intrinsic 137 polar 136 rectangular 136

velocity diagram 240 simple mechanisms 242

velocity- time graph 125 virtual work 91 - 103

elastic member 96 friction 96 principle of 94 simple frame 94 simple truss 94

volume of revolution 115

wedge friction 67 - 8 weight 17, 168

intensity 1 06 work 91, 203

definition 91 in general plane motion 313 of couple 92 of gravity force 210- 12 of spring force 212 virtual 91-103

work -energy equation 216 rigid body 315

work- kinetic-energy equation 204