Statics and Mechanics of Materials - 2nd Edition - STATICS AND MECHANICS OF MATERIALS 2nd Edition RILEY STURGES AND MORRIS Chapter 1 1-1 Calculate

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Unformatted text preview: STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS Chapter 1 1-1 Calculate the mass m of a body that weighs 600 lb at the surface of the earth. SOLUTION 1-2 Calculate the weight W of a body at the surface of the earth if it has a mass m of 675 kg. SOLUTION W 1-3 600 18.63 slug .................................................... Ans. 32.2 W g m mg 675 9.81 6.62 u 10 3 N 6.62 kN .................................. Ans. If a man weighs 180 lb at sea level, determine the weight W of the man (a) At the top of Mt. McKinley (20,320 ft above sea level). (b) At the top of Mt. Everest (29,028 ft above sea level). SOLUTION Gme m re2 W W0 r02 Therefore: rh (a) r0  h 2.090 u 10 7  2.0320 u 10 4 rh 2.092032 u10 2 7 2 179.7 lb ...................................... Ans. 2.0929028 u 10 7 ft 2.0929028 u 10 2 180 2.090 u 10 7 W0 r02 rh2 7 2 179.5 lb .................................... Ans. Calculate the weight W of a navigation satellite at a distance of 20,200 km above the earth’s surface if the satellite weighs 9750 N at the earth’s surface. SOLUTION W W0 r02 Therefore: r0 where (a) 2.092032 u 10 7 ft r0  h 2.090 u 10 7  2.9028 u 10 4 Wh 1-4 180 2.090 u 10 7 W0 r02 rh2 Wh (b) Gme m 2.090 u 10 7 ft r0 where Wh rh2 rh r0  h Wh Gme m re2 Wh rh2 Gme m 6.370 u106 m 6370  20, 200 W0 r02 rh2 9750 6370 26,570 1 2 26,570 km 2 560 N ........................................... Ans. STATICS AND MECHANICS OF MATERIALS, 2nd Edition 1-5 Compute the gravitational force acting between two spheres that are touching each other if each sphere weighs 1125 lb and has a diameter of 20 in. SOLUTION F Gm1m2 r2 Gms2 d s2 Gms ms rs  rs 2 3.439 u10 1125 32.2 8 2012 1-6 15.11u 106 lb .................................. Ans. 2 Gm1m2 r2 6.673 u10 60 80 11 0.600 0.890 u 10 6 N ........................ Ans. 2 Determine the weight W of a satellite when it is in orbit 8500 mi above the surface of the earth if the satellite weighs 7600 lb at the earth’s surface. SOLUTION W W0 r02 Therefore: r0 where rh ro  h Wh 1-8 2 Two spherical bodies have masses of 60 kg and 80 kg, respectively. Determine the gravitational force of attraction between the spheres if the distance from center to center is 600 mm. SOLUTION F 1-7 RILEY, STURGES AND MORRIS Gm1m2 r2 Wh rh2 Gme mb 2.090 u 10 7 ft 2.090 u 10 7  8500(5280) 6.578 u 10 7 ft Wo ro2 rh2 7600 2.090 u 107 6.578 u107 2 768 lb ..................................... Ans. 2 Determine the weight W of a satellite when it is in orbit 20.2(106) m above the surface of the earth if the satellite weighs 8450 N at the earth’s surface. SOLUTION W W0 r02 Therefore: r0 where rh ro  h Wh Gm1m2 r2 Wh rh2 Gme mb 6.370 u106 m 6.370 u106  20.2 u106 Wo ro2 rh2 8450 6.370 u106 26.570 u10 6 2 2 26.570 u106 m 2 486 N ..................................... Ans. STATICS AND MECHANICS OF MATERIALS, 2nd Edition 1-9 RILEY, STURGES AND MORRIS If a woman weighs 135 lb when standing on the surface of the earth, how much would she weigh when standing on the surface of the moon? SOLUTION Gm1m2 r2 W Therefore on the surface of the earth where 4.095 u 1023 slugs and r me 135 3960 mi G 4.095 u 1023 m 3960 u 5280 2 Gm 1.4413 u 10 7 lb ˜ ft 2 /slug Then on the surface of the moon where mm 5.037 u 1021 slugs and r 1080 mi 1.4413 u10 5.037 u10 7 W 21 1080 u 5280 2 22.33 lb ................................... Ans. 1-10 Determine the weight W of a body that has a mass of 1000 kg (a.) At the surface of the earth. (b.) At the top of Mt. McKinley (6193 m above sea level). (c.) In a satellite at an altitude of 250 km. SOLUTION Gm1m2 r2 W 6.673 u10 5.976 u10 1000 6370 u10 6.673 u10 5.976 u10 1000 6370 u10  6193 6.673 u10 5.976 u10 1000 6370 u10  250 u10 11 (a) W (b) W (c) W 24 3 2 11 9830 N ............................. Ans. 24 2 3 11 9810 N ............................. Ans. 24 3 2 3 9100 N ............................. Ans. 1-11 If a man weighs 210 lb at sea level, determine the weight W of the man (a.) At the top of Mt. Everest (29,028 ft above sea level). (b.) In a satellite at an altitude of 200 mi. SOLUTION W Therefore 210 Gm1m2 r2 Gme m 3960 u 5280 2 Gme m 9.181u1016 lb ˜ ft 2 3 STATICS AND MECHANICS OF MATERIALS, 2nd Edition W (a) W (b) RILEY, STURGES AND MORRIS 9.181u 1016 2 209.4 lb ....................................... Ans. 2 190.3 lb ........................................ Ans. 3960 u 5280  29, 028 9.181u 1016 ª¬ 3960  200 u 5280 º¼ 1-12 A space traveler weighs 800 N on earth. A planet having a mass of 5(1025) kg and a diameter of 30(106) m orbits a distant star. Determine the weight W of the traveler on the surface of this planet. SOLUTION Gm1m2 r2 W Therefore on the surface of the earth where 800 me 5.976 u 1024 kg and r 6370 km G 5.976 u 10 24 m 6370 u10 3 2 Gm 5.432 u 109 N ˜ m 2 /kg Then on the surface of the planet where m 5 u 1025 kg and r 15 u 106 m 5.432 u10 5 u10 15 u10 9 W 25 6 2 1207 N ......................................... Ans. 1-13 The planet Jupiter has a mass of 1.302(1026) slug and a visible diameter (top of the cloud layer) of 88,700 mi. Determine the gravitational acceleration g (a.) At a point 100,000 miles above the top of the clouds. (b.) At the top of the cloud layers. SOLUTION W Gm1m2 r2 3.439 u10 1.302 u10 m 8 W (a) mg 26 ª¬ 44,350  100, 000 u 5280º¼ g 7.71 ft/s 2 ................................................................. Ans. 3.439 u10 1.302 u10 m 8 W (b) mg 2 26 44,350 u 5280 g 2 81.7 ft/s 2 ................................................................ Ans. 1-14 The planet Saturn has a mass of 5.67(1026) kg and a visible diameter (top of the cloud layer) of 120,000 km. The weight W of a planetary probe on earth is 4.50 kN. Determine (a.) The weight of the probe when it is 600,000 km above the top of the clouds. (b.) The weight of the probe as it begins its penetration of the cloud layers. SOLUTION W Gm1m2 r2 4 STATICS AND MECHANICS OF MATERIALS, 2nd Edition Therefore on the surface of the earth where RILEY, STURGES AND MORRIS 5.976 u 1024 kg and r me 6370 km G 5.976 u 1024 m 4500 6370 u10 3 2 Gm 3.055 u 108 N ˜ m 2 /kg 3.055 u10 5.67 u10 8 W (a) 26 ª¬ 60, 000  600, 000 u 103 º¼ 3.055 u10 5.67 u10 60, 000 u10 8 W (b) 2 39.8 N ..................................... Ans. 26 4810 N ...................................... Ans. 3 2 1-15 The first U.S. satellite, Explorer 1, had a mass of approximately 1 slug. Determine the force exerted on the satellite by the earth at the low and high points of its orbit, which were 175 mi and 2200 mi, respectively, above the surface of the earth. SOLUTION F Gm1m2 r2 3.439 u10 4.095 u10 1 8 F 23 ª¬ 3960  175 u 5280º¼ 3.439 u10 4.095 u10 1 8 F 2 29.5 lb ................................... Ans. 23 ª¬ 3960  2200 u 5280 º¼ 2 13.31 lb .................................. Ans. 1-16 A neutron star has a mass of 2(1030) kg and a diameter of 10 km. Determine the gravitational force of attraction on a 10-kg space probe (a.) When it is 1000 km from the center of the star. (b.) At the instant of impact with the surface of the star. SOLUTION F Gm1m2 r2 6.673 u10 2 u10 10 1000 u10 6.673 u10 2 u10 10 5 u10 11 F 30 3 2 11 F 1.335 u 109 N ............................... Ans. 30 3 2 5.34 u1013 N ............................... Ans. 1-17 Determine the weight W, in U.S. customary units, of a 75-kg steel bar under standard conditions (sea level at a latitude of 45 degrees). SOLUTION W mg 75 9.81 735.75 N 0.2248 lb/N 5 165.4 lb ...................... Ans. STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS 1-18 Determine the mass m, in SI units, for a 500-lb steel beam under standard conditions (sea level at a latitude of 45 degrees). SOLUTION W m W g 500 32.2 mg 15.528 slug 14.59 kg/slug 227 kg ........................ Ans. 1-19 An automobile has a 440 cubic inch engine displacement. Determine the engine displacement in liters. SOLUTION 440 in. 16.39 u10 3 V 3 mm3 /in.3 10 1 cm/mm 10 3 L/cm3 3 V 7.21 L ................................................................... Ans. 1-20 How many barrels of oil are contained in 100 kL of oil? One barrel (petroleum) equals 42.0 gal. SOLUTION 100 u10 L 0.2642 gal/L 1 barrel 42 gal 3 V 629 barrel .................. Ans. 1-21* Express the density, in SI units, of a specimen of material that has a specific weight of 0.025 lb/in.3 SOLUTION J J g U Ug · § 1000 mm ·3 in. § 0.025 slug · § 14.59 kg · § ¸¨ ¨ ¸ 3 ¸¨ 3 3 ¸¨ m © 32.2 in. ¹ © slug ¹ ¨© 16.39 u 10 mm ¸¹ © ¹ 3 U 691 kg/m 3 ............................................................... Ans. 1-22 The viscosity of crude oil under conditions of standard temperature and pressure is 7.13(10-3) N-s/m2. Determine the viscosity of crude oil in U.S. Customary units. SOLUTION P 2 § 3 N ˜ s ·§ 0.2248 lb · § 0.0929 m · 3 lb ˜ s ........... Ans. ¸ 0.1489 u 10 ¨ 7.13 u 10 ¸¨ 2 ¸¨ 2 m ¹© N ft ft 2 © ¹© ¹ 1-23 One acre equals 43,560 ft2. One gallon equals 231 in.3 Determine the number of liters of water required to cover 2000 acres to a depth of 1 foot. SOLUTION 3 V § 43,560 ft 2 · § 12 in · § gal · § 3.785 L · 2000 acre ˜ ft ¨ ¸¨ ¸ ¨ ¸ 3 ¸¨ © acre ¹ © ft ¹ © 231 in. ¹ © gal ¹ V 2.47 u 109 L .............................................................. Ans. 1-24 The stress in a steel bar is 150 MPa. Express the stress in appropriate U.S. Customary units (ksi) by using the values listed in Table 1-6 for length and force as defined values. SOLUTION § ft · stress 150 u 10 N/m 0.2248 lb/N 0.0929 m /ft ¨ ¸ © 12 in. ¹ 6 stress 2 2 21.75 u 103 lb/in.2 6 2 2 21.75 ksi ......................................... Ans. STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS 1-25 By definition, 1 hp = 33,000 ft-lb/min and 1 W = 1 N-m/s. Verify the conversion factors listed in Table 1-6 for converting power from U.S. Customary units to SI units by using the values listed for length and force as defined values. SOLUTION ft ˜ lb · § 4.448 N ·§ 0.3048 m ·§ min · N˜m § 1 hp ¨ 33, 000 ............. Ans. ¸¨ ¸¨ ¸¨ ¸ 745.7 min ¹ © lb ¹© ft s © ¹© 60 s ¹ 1-26 The specific heat of air under standard atmospheric pressure, in SI units, is 1003 N-m/kg-qK. Determine the specific heat of air under standard atmospheric pressure in U.S. customary units (ft-lb/slug-qR). SOLUTION lb ·§ 3.281 ft · § 14.59 kg · § 5 qK · ¸¨ ¸¨ ¸¨ ¸ 1003 N ˜ m kgqK §¨© 0.2248 N ¹© m ¹ © slug ¹ © 9 qR ¹ 6000 lb ˜ ft slugqR .......................................................... Ans. 1-27 Newton’s law of gravitation can be expressed in equation form as F G m1m2 r2 If F is a force, m1 and m2 are masses, and r is a distance, determine the dimensions of G. SOLUTION G Fr 2 m1m2 ML T L 2 2 M M § L3 · .......................................... Ans. ¨ 2 ¸ © MT ¹ 1-28 The elongation of a bar of uniform cross section subjected to an axial force is given by the equation G PL EA . What are the dimensions of E if G and L are lengths, P is a force, and A is an area? SOLUTION E PL GA ML T L L L 2 § M · §F· ¨ ¨ 2 ¸ ...................................... Ans. 2 ¸ © LT ¹ © L ¹ 2 1-29 The period of oscillation of a simple pendulum is given by the equation T k L g , where T is in seconds, L is in feet, g is the acceleration due to gravity, and k is a constant. What are the dimensions of k for dimensional homogeneity? SOLUTION 12 k T g L § L T2 · T ¨ ¸ © L ¹ 1 (dimensionless) ............................... Ans. 1-30 An important parameter in fluid flow problems involving thin films is the Weber number (We) which can be expressed in equation form as U v2L V We where U is the density of the fluid, v is a velocity, L is a length, and V is the surface tension of the fluid. If the Weber number is dimensionless, what are the dimensions of the surface tension V? SOLUTION V U v2L We M L L T L 2 3 1 7 §M · §F· ¨ 2 ¸ ¨ ¸ ................................ Ans. ©T ¹ © L ¹ STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS 1-31 In the dimensionally homogeneous equation P Mc  A I V V is a stress, A is an area, M is a moment of a force, and c is a length. Determine the dimensions of P and I. SOLUTION ML2 T 2 L P  I L2 § M · ¨ 2 ¸ © LT ¹ Therefore § M · 2 § ML · P ¨ L ¨ 2 ¸ 2 ¸ © LT ¹ ©T ¹ ML T L L ...................................................... Ans. M LT 2 I F ............................................... Ans. 2 4 2 1-32 In the dimensionally homogeneous equation Pd 1 2 mv 2  12 IZ 2 d is a length, m is a mass, v is a linear velocity, and Z is an angular velocity. Determine the dimensions of P and I. SOLUTION P L 2 L 1 M §¨ ·¸  I §¨ 2 ·¸ ©T ¹ ©T ¹ Therefore § ML2 T 2 · § ML · P ¨ ¸ ¨ ¸ L ¹ © T2 ¹ © I § ML2 · 2 ¨ 2 ¸ T © T ¹ F ................................................ Ans. ML ...................................................... Ans. 2 1-33 In the dimensionally homogeneous equation Tr VQ  J Ib W W is a stress, T is a torque (moment of a force), V is a force, r and b are lengths and I is a second moment of an area. Determine the dimensions of J and Q. SOLUTION § M · ¨ 2 ¸ © LT ¹ Therefore ML T 2 L 2 J 4 ML T L L ..................................................... Ans. M LT 2 J ML T Q L L 2  2 4 2 8 STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS M LT L L ..................................................... Ans. ML T 2 5 3 Q 2 1-34 In the dimensionally homogeneous equation P Tr  A J W W is a stress, A is an area, T is a torque (moment of a force), and r is a length. Determine the dimensions of P and J. SOLUTION ML2 T 2 L P  J L2 § M · ¨ 2 ¸ © LT ¹ Therefore § M · 2 § ML · P ¨ L ¨ 2 ¸ 2 ¸ © LT ¹ ©T ¹ F ............................................... Ans. ML T L L ..................................................... Ans. M LT 2 2 4 J 2 sin at  D is dimensionally homogeneous. If A is a length and t is time, 1-35 The equation x Ae determine the dimensions of x, a, b, and D. SOLUTION t b x L e T b sin ª¬ a T  D º¼ Therefore L 1 1 L ........................................................... Ans. b T ...................................................................... Ans. a 1 T .................................................................... Ans. x D 1 (dimensionless) ........................................................ Ans. 1-36 In the dimensionally homogeneous equation dimensions of a, b, and w? SOLUTION If x w L , then each term has the dimension L3 . x 3  ax 2  bx  a 2 b x , if x is a length, what are the Therefore L ..................................................................... Ans. L L ............................................................... Ans. L L L ............................................................... Ans. L 3 w 3 a 2 3 b 2 Using the last term as a check, 9 STATICS AND MECHANICS OF MATERIALS, 2nd Edition a 2b x L L 2 2 RILEY, STURGES AND MORRIS L 3 L 1-37 Determine the dimensions of a, b, c, and y in the dimensionally homogeneous equation Ae bt cos 1  a 2 bt  c y in which A is a length and t is time. SOLUTION y L eb T cos ª¬ 1  a 2 b T  c º ¼ Therefore y b T 1 L 1 1 L .......................................................... Ans. b 1 T .................................................................... Ans. a 1 (dimensionless) ........................................................ Ans. c 1 (dimensionless) ........................................................ Ans. 1-38 Determine the dimensions of c, Z, k and P in the differential equation m d 2x dx  c  kx 2 dt dt P cosZt in which m is a mass, x is a length, and t is time. SOLUTION M L  c L  k L T 2 T Therefore ML T 2 c §M · ¨ ¸ ......................................................... Ans. ©T ¹ L T ML T 2 k L §M · §F· ¨ 2 ¸ ¨ ¸ .................................................. Ans. ©T ¹ © L ¹ § ML · P ¨ 2 ¸ ©T ¹ Z P cos Z T F ............................................................. Ans. 1 T .................................................................... Ans. 1-39 Round off the following numbers to two significant figures. Find the percent difference between each rounded-off number and the original number by using the original number as the reference. (a) 0.015362 (b) 55.33682 (c) 63,746.27 SOLUTION (a) (b) 0.015  0.015362 u 100 2.36 % ............................................. Ans. 0.015362 55  55.33682 u 100 0.609 % ............................................... Ans. 55.33682 10 STATICS AND MECHANICS OF MATERIALS, 2nd Edition 64, 000  63, 746.27 u100 63, 746.27 (c) RILEY, STURGES AND MORRIS 0.398 % ......................................... Ans. 1-40 Round off the following numbers to two significant figures. Find the percent difference between each rounded-off number and the original number by using the original number as the reference. (a) 0.837482 (b) 374.9371 (c) 937,284.9 SOLUTION 0.84  0.837482 u 100 0.301 % ............................................. Ans. 0.837482 370  374.9371 u100 1.317 % ............................................. Ans. 374.9371 940, 000  937, 284.9 u 100 0.290% ......................................... Ans. 937, 284.9 (a) (b) (c) 1-41 Round off the following numbers to three significant figures. Find the percent difference between each rounded-off number and the original number by using the original number as the reference. (a) 0.034739 (b) 26.39473 (c) 55,129.92 SOLUTION 0.0347  0.034739 u 100 0.1123 % ......................................... Ans. 0.034739 26.4  26.39473 u 100 0.01997 % .......................................... Ans. 26.39473 55,100  55,129.92 u 100 0.0543 % ......................................... Ans. 55,129.92 (a) (b) (c) 1-42 Round off the following numbers to three significant figures. Find the percent difference between each rounded-off number and the original number by using the original number as the reference. (a) 0.472916 (b) 826.4836 (c) 339,872.8 SOLUTION 0.473  0.472916 u 100 0.01776 % ......................................... Ans. 0.472916 826  826.4836 u 100 0.0585 % ............................................ Ans. 826.4836 340, 000  339,872.8 u 100 0.0374 % ....................................... Ans. 339,872.8 (a) (b) (c) 1-43 Round off the following numbers to four significant figures. Find the percent difference between each rounded-off number and the original number by using the original number as the reference. (a) 0.056623 (b) 74.82917 (c) 27,382.84 SOLUTION (a) (b) 0.05662  0.056623 u100 5.30 u103 % ..................................... Ans. 0.056623 74.83  74.82917 u 100 1.109 u 103 % ...................................... Ans. 74.82917 11 STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS 27,380  27,382.84 u 100 10.37 u 103 % .................................... Ans. 27,382.84 (c) 1-44 Round off the following numbers to four significant figures. Find the percent difference between each rounded-off number and the original number by using the original number as the reference. (a) 0.664473 (b) 349.3378 (c) 274,918.2 SOLUTION 0.6645  0.664473 u 100 4.06 u 103 % ...................................... Ans. 0.664473 349.3  349.3378 u 100 10.82 u 103 % ...................................... Ans. 349.3378 274,900  274,918.2 u 100 6.62 u 103 % .................................... Ans. 274,918.2 (a) (b) (c) 1-45 The weight of the first Russian satellite, Sputnik I, was 184 lb on the surface of the earth. Determine the force exerted on the satellite by the earth at the low and high points of its orbit which were 149 mi and 597 mi, respectively, above the surface of the earth. SOLUTION Gm1m2 r2 Gme m F 184 Therefore 3960 u 5280 2 Gme m 8.044 u1016 lb ˜ ft 2 F F 8.044 u1016 ª¬ 3960  149 u 5280 º¼ 2 170.9 lb ......................................... Ans. 2 138.9 lb ......................................... Ans. 8.044 u 1016 ª¬ 3960  597 u 5280º¼ 1-46 The planet Jupiter has a mass of 1.90(1027) kg and a radius of 7.14(107) m. Determine the force of attraction between the earth and Jupiter when the minimum distance between the two planets is 6(1011) m. SOLUTION Gm1m2 r2 F 6.673 u10 5.976 u10 1.90 u10...
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