Chapter 1

1.2 The radius and length of a steel cylinder are 60mm and 120 mm, respectively. If the mass density of steel is 7850 kg/m3, determine the weight of the cylinder in pounds.

1.4 The mass moment of inertia of a certain body is I = 20 kg · m2. Express in terms of the base units of the U.S. Customary system.

1.6 In a certain application, the acceleration a and the position coordinate x ofa particle are related bya = gkxW where g is the gravitational acceleration, k is a constant, and W is the weight ofthe particle. Show that this equation is dimensionally consistent if the dimensionof k is [F/L].

1.8 In some applications dealing with very high speeds, the velocity is measuredin mm/μs. Convert 25 mm/μs into (a) m/s; and (b) mi/h.

1.10 The mass moment of inertia I of a homogeneous sphere about its diameteris I =(2/5)mR2, where m and R are its mass and radius, respectively. Find thedimension of I in terms of the base dimensions of (a) a gravitational [FLT] systemand (b) an absolute [MLT] system.

1.12 In a certain vibration problem the differential equation describing themotion of a particle of mass m iswhere x is the displacement of the particle and t is time. What are the dimensionsof the constants c, k, P0, and ω in terms of the base dimensions of a gravitational[FLT] system?

1.14 The typical power output of a compact car engine is 120 hp. What is theequivalent power in (a) lb · ft/s; and (b) kW?

1.16 Two identical spheres of radius 8 in. and weighing 2 lb on the surface ofthe earth are placed in contact. Find the gravitational attraction between them.Use the following data for Problems 1.17–1.21: mass of earth = 5.9742×1024 kg,radius of earth = 6378 km, mass of moon = 0.073 483 × 1024 kg, radius ofmoon = 1737 km.

1.18 Use Eq. (1.4) to show that the weight of an object on the moon isapproximately 1/6 its weight on earth

1.20 Find the elevation h (km) where the weight of an object is one-tenth itsweight on the surface of the earth.

1.22 The magnitudes of the two velocity vectors are v1 = 3 m/s and v2 = 2 m/s.Determine their resultant v = v1 + v2.

1.24 The total aerodynamic force F acting on the airplane has a magnitude of6250 lb. Resolve this force into vertical and horizontal components (called the liftand the drag, respectively).

1.26 The velocity vector of the boat has two components: v1 is the velocity ofthe water, and v2 is the velocity of the boat relative to the water. If v1 = 3 mi/hand v2 = 5 mi/h, determine the velocity vector of the boat.

1.28 The 500-N weight is supported by two cables, the cable forces being F1 F F2 1

and F2. Knowing that the resultant of F1 and F2 is a force of magnitude 500Nacting in the y-direction, determine F1 and F2.

1.30 Resolve the position vector A of the car (measured from fixed point O)into components parallel to OB and OC.

1.32 The supporting cables AB and AC are oriented so that the components ofthe 360-lb force along AB and AC are 185 lb and 200 lb, respectively. Determinethe angles α and β.

1.34 The resultant of the two forces has a magnitude of 650 lb. Determine thedirection of the resultant and the magnitude of P.

1.36 A surveyor sights a target at C from points A and B, recording the angles. Determine the magnitudes of the position vectors a and b.

1.40 Obtain the rectangular representation of the force P, given that its magnitudeis 30 lb.

1.42 (a) Compute the angle θz between the force vector F and the z-axis. (b) Determine the rectangular representation of F given that F = 240 N.

1.44 The slider travels along the guide rod AB with the velocity v = 6 m/s.Determine the rectangular representations of (a) the unit vector directed from Atoward B; and (b) the velocity vector v.

1.46 The magnitude of the force F is 120 lb. Find its rectangular representation.

1.48 Find the angles between the force F = 1200i + 800j – 1500k N and thex-, y-, and z-axes. Show your results on a sketch of the coordinate system.

1.50 Determine the resultant of the two forces shown.

1.52 Given that P =120 lb and Q =130 lb, find the rectangular representation

1.54 If R is the resultant of the forces P and Q, find P and Q.

1.56 The vertical post is secured by three cables. The cables are pre-tensioned so that the resultant of the cable forces F, Q, and P is directed along the z-axis.If F = 120 lb, find P and Q.

1.58 Compute the cross product C=A × B for each of the cases given inProb. 1.57. Identify the units of each product.

1.60 Compute A × B and C × B for the position vectors shown.

1.62 Use the dot product to find the angle between the position vectors A and B.

1.64 Determine which of the following position vectors B is perpendicular toA=3i − 5j + 2k m:(a) B=5i + 3j − 2k m(b) B=2i + 3j + 4k m(c) B=i + j + k m(d) B=3i + j − 2k m

1.66 The three points A (0, −2, 2), B (−1, 4, 1), and C (3, 0, 0) define a plane.The coordinates are in inches. Find a unit vector that is perpendicular to this plane.

1.68 Compute the orthogonal component of F=6i + 20j − 12k lb in thedirection of the vector A=2i − 3j + 5k ft.

1.70 Resolve A = 3i + 5j − 4k in. into two vector components—one parallelto and the other perpendicular to B = 6i + 2k in. Express each of your answersas a magnitude multiplied by a unit vector.

1.72 Determine the value of the scalar a if the following three vectors are to liein the same plane: A=2i−j+2k m, B=6i+3j+ak m, and C=16i+46j+7k m.

1.74 It can be shown that a plane area may be represented by a vector A= Aλ,where A is the area and λ represents a unit vector normal to the plane of the area.Show that the area vector of the parallelogram formed by the vectors a and bshown in the figure is A=a × b.

1.76 Show that |a × b · c| equals the volume of a parallelepiped that has a, b,and c as its edges. (Hint: See Prob. 1.74.)

Chapter 2

2.2 Two men are trying to roll the boulder by applying the forces shown. Determine the magnitude and direction of the force that is equivalent to the two appliedforces.

2.4 Determine P and θ so that the three forces shown are equivalent to thesingle force R = 85i + 20j kN.

2.6 The forces P1 = 110 lb, P2 = 200 lb, and P3 = 150 lb are equivalent to asingle force R. Determine (a) the magnitude of R; and (b) the coordinates of thepoint where the line of action of R crosses the yz-plane.

2.8 The magnitudes of the three forces acting on the plate are T1 z =100 kN, T2 = 80 kN and T3 = 50 kN. Replace these forces with a singleequivalent force R. Also, find the coordinates of the point where R intersectsthe plate.

2.10 The force R is the resultant of the forces P1, P2, and P3 acting on therectangular plate. Find P1 and P2 if R =40 kN and P3 =20 kN.

2.12 Knowing that the forces P and Q are equivalent to a single force R that passes through point A, determine P and R. the rectangular components of this force and the point of intersection of its line ofaction with the plate.

2.14 Find the forces Q1, Q2, and Q3 so that the two force systems areequivalent.

2.16 The three forces acting on the beam can be replaced with a singleequivalent force R. Determine the angle θ and R.

2.18 Replace the three forces acting on the guy wires by a single, equivalent

2.20 The three forces, each of magnitude F, are applied to the crate. DetermineF so that the three forces are equivalent to a single 600-lb force.

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2.21 Determine the resultant force R that is equivalent to the forces exertedby the three tugboats as they maneuver the barge. Specify the coordinate of thepoint on the x-axis through which R passes. (Hint: First determine the resultantforce for the two forces at point A, and then combine this result with the force atpoint B.)

2.22 Determine the magnitude and sense of the moment of the 800-N force

2.23 Find the magnitude and sense of the moment of the 60-lb force aboutpoints A and B.

2.24 The two forces can be replaced by an equivalent force R acting at point Bon the beam. Determine the distance b that locates B. (Hint: The combinedmoment of the two forces about any point is equal to the moment of R aboutthe same point.)

2.26 A force P in the xy-plane acts on the triangular plate. The moments of Pabout points O, A, and B are MO =80 N · m counterclockwise, MA =200 N · mclockwise, and MB =0. Determine P.

2.27 Determine the moment of the force F=9i + 18j lb about point O bythe following methods: (a) vector method using r × F; (b) scalar method using

rectangular components of F; and (c) scalar method using components of F thatare parallel and perpendicular to the line OA.

2.28 Given that T =28.3 kN and W =25 kN, determine the magnitude andsense of the moments about point B of the following: (a) the force T; (b) theforce W; and (c) forces T and W combined.

2.30 Knowing that the forces P and Q are equivalent to a single force R thatpasses through point A, determine P. (Hint: The combined moment of P and Qabout A is zero.)

2.32 The tow truck’s front wheels will be lifted off the ground if the moment ofthe load W about the rear axle exceeds the moment of the 6200-lb weight of thetruck. Determine the largest W that may be safely applied.

2.34 Compute the moment of the force P about point A.

2.36 The magnitude of the force Q is 250 N. Determine the moments of Qabout (a) point O; and (b) point C.Determine the magnitude of P.

2.38 The magnitude of the force P is 50 kN. Determine the moment of P about(a) point A; and (b) point B.

2.40 Find the combined moment of the forces P and Q about point O. Themagnitudes of the forces are P = 80 lb and Q = 60 lb.

2.42 The magnitudes of the two forces shown are P =16 lb and Q =22 lb. Determine the magnitude of the combined moment of P and Q about point O andthe direction cosines of this moment vector.

2.44 Determine the magnitude of the moment of the 150-N force about point Oand find the direction cosines of the moment vector.

2.46 The force F = −20i + 4j + 6k lb acts at point A. Determine the coordinatesof point B where the line of action of F intersects the xy-plane. (Hint: Themoment of F about B is zero.)

2.48 Determine the moment of the 40-kN force about each of the followingaxes: (a) AB; (b) CD; (c) CG; (d) CH; and (e) EG.

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2.50 The magnitude of the force F is 75 lb. Calculate the moment of F about the x-axis using (a) the scalar method; and (b) the vector method.

2.52 The moment of the force F about the x-axis is 1080 N · m. Determine themoment of F about the axis AB.

2.54 To lift the table without tilting, the combined moment of the four parallelforces must be zero about the x-axis and the y-axis (O is the center of the table).Determine the magnitude of the force F and the distance d.

2.56 The trap door is held open by the rope AB. If the tension in the rope isT = 40 lb, determine its moment about the y-axis.

2.58 The magnitude of the force P is 480 N. Determine the moment of P aboutthe axis CD. Express the result in vector form.

2.60 Determine the magnitude of the force F given that its moment about theaxis BC is 150 lb · ft.perpendicular to the plane ABC and passes through point O. Express your answerin vector form.

2.62 Calculate the moment of the force P about the axis AD using (a) point Aas the moment center; and (b) point D as the moment center.

2.64 The force F= F(0.6i + 0.8j) kN is applied to the frame at thepoint D (0, 0, zD). If the moment of F about the axis BC is zero, determine thecoordinate zD.

2.65 Determine the combined moment of the four forces acting on the pulleysabout the axis AB (points A and B are the centers of the pulleys).0.5 m 0.8 m

2.66 The flexible shaft AB of the wrench is bent into a horizontal arc with aradius of 24 in. The two 20-lb forces, which are parallel to the z-axis, are appliedto the handle CD, as shown. Determine the combined moment of the two 20-lbforces about the x-axis (the axis of the socket at point B).

2.68Which of the systems are equivalent to the couple in (a)?

2.70 Replace the two couples shown by a single equivalent couple.

2.72 Determine the magnitude of the single couple that is equivalent to the twocouples shown.

∗2.74 Determine the couple-vector that is equivalent to the three couples actingon the gear box, given that C1 =200 lb · in., C2 =140 lb · in., and C3 =220 lb · in.corresponding couple-vector.

2.76 The couple acts on the handles of a steering mechanism. In the positionshown, the moment applied by the couple about the z-axis is zero. Determine thedistance b. Use F = 200i − 110j − 80k kN.

2.77 The force-couple system shown can be replaced by a single equivalentcouple CR. Determine CR.

2.78 A couple of magnitude 360 lb · ft is applied about portion AB of the driveshaft (the drive shaft is connected by universal joints at points B and C). Computethe moment of the applied couple about the portion CD when the drive shaft is inthe position shown.

2.80 The figure shows one-half of a universal coupling known as the Hooke’sjoint. The coupling is acted on by the three couples shown: (a) the input coupleconsisting of forces of magnitude P, (b) the output couple C0, and (c) the coupleformed by bearing reactions of magnitude R. If the resultant of these couples iszero, compute R and C0 for P =600 lb.

2.82 Which of the systems are equivalent to the force-couple system in (a)?

2.84 The bracket, which is fastened to a wall by anchor bolts at A and B, isloaded by the force P =120 N and the couple C =140 N·m. Replace P and C with(a) an equivalent force-couple system, the force of which acts at A; and (b) twovertical forces, one acting at A and the other at B.

2.85 The three forces shown are equivalent to a 50-kN upward force at A and a P Q

170-kN ·m counterclockwise couple. Determine P and Q.

2.86 Replace the two forces shown by a force-couple system with the forceacting at O. bile (the torsion bar appears in cross section at A). If the three forces and thecouple C = 900 lb · ft are equivalent to a upward vertical force R = 1200 lbacting at D, determine H and V.

2.88 The table can be lifted without tilting by applying the 100-N force atpoint O, the center of the table. Determine the force-couple system with the forceacting at corner A that will produce the same result.

2.89 The magnitude of the force F acting at point A on the plate is 160 kN.Determine the equivalent force-couple system with the force acting at point O.

2.90 Replace the force-couple system acting on the pipe with an equivalentforce-couple system with the force acting at point O.

2.92 Determine the force-couple system, with the force acting at point O, thatis equivalent to the force and couple acting on the arm CD of the industrial robot.Note that the arm ABCD lies in a vertical plane that is inclined at 40◦ to theyz-plane; the arm CD makes an angle of 30◦ with the vertical.