Problem Statement
The two forces P and Q act on a bolt A. Determine their resultant.
 
 
 

Solution: R = 97.7 N @ 35.0°

Problem Statement
A barge is pulled by two tugboats. If the resultant of the forces exerted by the tugboats is a 5000-lb force directed along the axis of the barge, determine (a) the tension in each of the ropes knowing that a = 45° , (b) the value of a for which the tension in rope 2 is minimum.
 
 

Solution: (a) T₁ = 3660 lb @ 30.0° , T₂ = 2590 lb @ -45.0° (b) a = 60°

SampleProblem02.02.wm3

Problem Statement
Knowing that a = 20^o, determine the tension (a) in cable AC, (b) in rope BC.
 
 
 

Solution: (a) T_AC = 1244 lb (b) T_BC = 115.4 lb

Comments
The main purpose of the problem is to demonstrate the technique of force balance. The simulation allows control of the angle a and displays the effect of a on the rope tension.

Problem Statement
A 200-kg cylinder is hung by means of two cables AB and AC, which are attached to the top of a vertical wall. A horizontal force P perpendicular to the wall holds the cylinder in the position shown. Determine the magnitude of P and the tension in each cable.
 
 

Solution: P = 235 N,  T_AB = 1244 lb, TA_C = 115.4 lb

Problem Statement
Determine the components of the single couple equivalent to the two couples shown.
 
 
 

Solution: M = - (540 lb-in.) i + (240 lb-in.) j + (180 lb-in.) k

Problem Statement:
A square foundation mat supports the four columns shown. Determine the magnitude and point of application of the resultant of the four loads.
 
 
 

Solution: R = 80 kips (downward) at x = 3.50 ft, z = 3.00 ft

Comments
This simulation shows the effects of changing the magnitude and position of the supporting force on the static equilibrium of the foundation mat.

Problem Statement
It is known that the connecting rod AB exerts on the crank BC a 1.5-kN force directed down and to the left along the center line of AB. Determine the moment of the force about C.
 
 

Solution: M = 42 N-m (CCW)

Problem Statement
Three loads are applied to a beam as shown. The beam is supported by a roller at A and by a pin at B. Neglecting the weight of the beam, determine the reactions at A and B when P = 15 kips.
 
 

Solution: B_x = 0, B_y = 21.0 kips (upward), A = 6.00 kips (upward)

Problem Statement
A 400-lb weight is attached at A to the lever shown. The constant of the spring BC is k = 250 lb/in., and the spring is unstretched when q = 0. Determine the position of equilibrium.
 
 

Solution: q = 0, q = 80.3^o

Problem Statement
A man raises a 10-kg joist, of length 4 m, by pulling on a rope. Find the tension T in the rope and the reaction at A.
 
 
 

Solution: T = 81.9, R = 147.8 N @ 58.6°

Problem Statement
A beam supports a distributed load as shown. (a) Determine the equivalent concentrated load. (b) Determine the reactions at the supports.
 
 
 

Solution: (a) W = 18 kN (downward), X = 3.5 m to the right of A (b) B_x = 0, B_y = 10.5 kN (upward), A = 7.5 kN (upward)

Comments
When the load is distributed in such a way that the resultant force is zero, it is impossible to determine an equivalent concentrated load. For example, in Working Model View, enter 3,000 for the load at point A and -3,000 for the load at point B and see that a special message appears.

Problem Statement
Using the method of joints, determine the force in each member of the truss shown.
 
 
 

Solution:

FAB = 1500 lb (tension)	FAD = 2500 lb (compression)
FDB = 2500 lb (tension)	FDE = 3000 lb (compression)
FBE = 3750 lb (compression)	FBC = 5250 lb (tension)
FEC = 8750 lb (compression)

Problem Statement
A small barrel weighing 60 lb is lifted by a pair of tongs as shown. Knowing that a = 5 in., determine the forces exerted at B and D on tong ABD.
 
 

Solution: B = 94.9 lb @ 18.4^o (left-downward), D = 94.9 lb @ 18.4^o (right-downward)

Problem Statement
In the frame shown, determine the internal forces (a) in member ACF at point J, (b) in member BCD at point K. This frame has been previously considered in Sample Prob. 6.5.
 
 

Solution: (a) M = 2160 N-m (CCW), F = 1344 N (right-downward), V = 1197 N (left-downward)
               (b) M = 1800 N-m (CW), F = 0, V = 1200 N (upward)

Problem Statement
A 100-lb force acts as shown on a 300-lb block placed on an inclined plane. The coefficients of friction between the block and the plane are m_s = 0.25 andm_k = 0.20. Determine whether the block is in equilibrium, and find the value of the friction force.
 
 

Solution: Block not in equilibrium, F_actual = 48 lb (right-upward)

Problem Statement
The movable bracket shown may be placed at any height on the 3-in.-diameter pipe. If the coefficient of static friction between the pipe and bracket is 0.25, determine the minimum distance x at which the load W can be supported. Neglect the weight of the bracket.
 
 

Solution: x = 12 in.

Comments
Students can control the location at which the load W is applied by adjusting a slider. Depending on where they put the load, the bracket either stays stationary or moves downward. Also, students can change the coefficient of static friction between the pipe and the bracket. In doing so, they gain the intuition that with a larger coefficient of static friction, they can place the load closer to the pipe without the bracket sliding down.

Problem Statement
Assuming a uniform acceleration of 11 ft/s² and knowing that the speed of a car as it passes A is 30 mi/h, determine (a) the time required for the car to reach B, (b) the speed of the car as it passes B.
 
 

Solution: (a) 2.72 s (b) 50.4 mi/h

Comments
Graphs and meters allow the students to look up the final position of the car (160 feet) and find the corresponding time and velocity at that point by scrolling along the time line.

HomeworkProblem11.036.wm3

Problem Statement
In a boat race, boat A is leading boat B by 120 ft and both boats are traveling a constant speed of 105 mi/h. At t = 0, the boats accelerate at constant rates. Knowing that when B passes A, t = 8 s and v_A = 135 mi/h, determine (a) the acceleration of A, (b) the acceleration of B.
 
 

Solution: (a) 5.50 ft/s² (b) 9.25 ft/s²

Comments
By trial and error and by looking at the meters at the bottom of the simulation, students can find the correct acceleration for each boat such that the conditions in the problem statement are satisfied. The simulation is intended to be a realistic boat race that is exciting and motivational.

Problem Statement
A projectile is fired with an initial velocity of 800 ft/s² at a target B located 2000 ft above the gun A and at a horizontal distance of 12,000 ft. Neglecting air resistance, determine the value of the firing angle a .
 
 

Solution: a = 29.5^o, a = 70.0^o

Comments
The answer to this problem can be determined by trial-and-error interaction with the Working Model file. Alternately, closed form solutions from hand calculations can be verified via simulation. With different combinations of initial projectile velocity and trajectory angle, students can experiment to see when the target would be hit.

Problem Statement
The two blocks shown start from rest. The horizontal plane and the pulley are frictionless, and the pulley is assumed to be of negligible mass. Determine the acceleration of each block and the tension in each cord.
 
 

Solution: a_A = 8.40 m/s², a_B = 4.20 m/s², T₁ = 840 N, T₂ = 1680 N

Problem Statement (altered from text)
A 12-lb block B starts from rest and slides down a 30-lb wedge A, which is supported by a horizontal surface. Neglecting friction, determine (a) the acceleration of the wedge, (b) the acceleration of the block relative to the wedge.
 
 

Solution: (a) 5.07 ft/s² (b) 20.5 ft/s² @ 30^o

Comments
In the Working Model file, the mass of the block can be varied. By keeping the mass of the wedge constant while changing that of the block, students can gain an intuitive feel for the relationship between mass and acceleration of the block and wedge.

Problem Statement
A 60-kg wrecking ball B is attached to a 15-m-long steel cable AB and swings in the vertical arc shown. Determine the tension in the cable (a) at the top C of the swing, (b) at the bottom D of the swing, where the speed of B is 4.2 m/s.
 
 

Solution: (a) 553 N (b) 659 N

Comments
While the main purpose of the problem is to learn about rope tension, centripetal force, and centripetal acceleration, we have added an additional element to the Working Model file. By using a slider to vary the mass of the wrecking-ball one can see the effect on the distance that the windows are thrown when the ball hits them.

HomeworkProblem12.045.wm3

Problem Statement
A 20-lb collar slides without friction along a vertical rod as shown. The spring attached to the collar has an undeformed length of 4 in. and a constant of 3 lb/in. If the collar is released from rest in position 1, determine its velocity after it has moved 6 in. to position 2.
 
 

Solution: v₂ = 4.91 ft/s (downward)

Problem Statement
A 30-kg block is dropped from a height of 2 m onto the 10-kg pan of a spring scale. Assuming the impact to be perfectly plastic, determine the maximum deflection of the pan. The constant of the spring is k = 20 kN/m.
 
 

Solution: h = 225 mm

Problem Statement
A 700-kg sphere A moving with a velocity v₀ parallel to the ground strikes the inclined face of a 2.1-kg wedge B which can roll freely on the ground and is initially at rest. After impact the sphere is observed from the ground to be moving straight up. Knowing that the coefficient of restitution between the sphere and the wedge is e = 0.6, determine (a) the angle q that the inclined face of the wedge makes with the horizontal, (b) the energy lost due to the impact.

Solution: (a) 62.7^o (b) 3.50 N-m

Problem Statement
A 3-lb sphere A strikes the frictionless inclined surface of a 9-lb wedge B at a 90^o angle with a velocity of magnitude 12 ft/s. the wedge can roll freely on the ground and is initially at rest. Knowing that the coefficient of restitution between the wedge and the sphere is 0.50 and that the inclined surface of the wedge forms an angle q = 40^o with the horizontal, determine (a) the velocities of the sphere and of the wedge immediately after impact, (b) the energy lost due to the impact.

Solution: (a) v_A = 3.82 ft/s (left-upward @ 50^o), v_b = 3.39 ft/s (to the right)  (b) 4.42 ft-lb

Problem Statement
Ball B, of mass m_B, is suspended from a cord of length l attached to cart A, of mass m_A, which can roll freely on a frictionless horizontal tract. If the ball is given an initial horizontal velocity v_o while the cart is at rest, determine (a) the velocity of B as it reaches its maximum elevation, (b) the maximum vertical distance h through which B will rise. (It is assumed that v_o² < 2gl.)

Solution:

(a)     (vB)2 = (vA)2 =  mb  ma + mb vo

(b)        h =mA vo2 / 2g (mA+mB)

Comments

In the horizontal direction, there is an initial momentum of the ball, but no forces that retard the motion. Hence, a horizontal motion of the system results.

Because the sample problem is analytical and the final answers are symbolic, we made the length of the rope 5 m and treated the other variables as slider-controlled inputs. Students are encouraged to verify the equations found in the solution of the problem by running the Working Model simuation with different speed and mass ratios and checking the validity of the equations.

Problem Statement
In the engine system shown, the crank AB has a constant clockwise angular velocity of 2000 rpm. For the crank position indicated, determine (a) the angular velocity of the connecting rod BD, (b) the velocity of the piston P.
 
 

Solution: (a) v_P = v_D = 43.6 ft/s (to the left) (b) w_BD = 62.0 rad/s (CCW).

Problem Statement
Crank AB of the engine system of Sample Prob. 15.3 has a constant clockwise angular velocity of 2000 rpm. For the crank position shown, determine the angular acceleration of the connecting rod BD and the acceleration of point D.
 
 

Solution: a _BD = 9940 rad/s² (CCW), a_D = 9290 ft/s² (to the left)

Problem Statement
The linkage ABDE moves in the vertical plane. Knowing that in the position shown crank AB has a constant angular velocity w₁ of 20 rad/s counterclockwise, determine the angular velocities and angular accelerations of the connecting rod BD and of the crankDE.
 
 

Solution: w _BD = - (29.33 rad/s) k, w_DE = (11.29 rad/s) k
                a _BD = - (645 rad/s²) k, a_DE = (809 rad/s²) k

Problem Statement
The Geneva mechanism shown is used in many counting instruments and in other applications where an intermittent rotary motion is required. Disk D rotates with a constant counterclockwise angular velocity w_Dof 10 rad/s. A pin P is attached to disk Dand slides along one of several slots cut in disk S. It is desirable that the angular velocity of disk Sbe zero as the pin enters and leaves each slot; in the case of four slots, this will occur if the distance between the centers of the disks is l = Ö 2 R. At the instant whenf = 150^o, determine (a) the angular velocity of diskS, (b) the velocity of pin P relative to diskS.
Solution: (a) v_P/S = v_P/S =477 mm/s @ 42.4^o (b) w _S = w _S = 4.08 rad/s (CW)

Comments
The visualization of the Geneva mechanism shown in the Sample Problem 15.9 AVI and the associated Working Model file makes it very easy to understand how the Geneva mechanism works. With only sketches or pictures, mechanisms such as the Geneva mechanism are difficult to visualize.

Problem Statement
In the Geneva mechanism of Sample Prob. 15.9, disk D rotates with a constant counterclockwise angular velocity w_Dof 10 rad/s. At the instant when f = 150^o, determine the angular acceleration of disk S.
 
 

Solution: a _S = a_S = 233 rad/s² (CW)

Comments
The visualization of the Geneva mechanism shown in the Sample Problem 15.10 AVI and the associated Working Model file makes it very easy to understand how the Geneva mechanism works. With only sketches or pictures, mechanisms such as the Geneva mechanism are difficult to visualize.

Problem Statement
The rod AB, of length 7 in., is attached to the disk by a ball-and-socket connection and to the collar B by a clevis. The disk rotates in the yz plane at a constant rate w ₁ = 12 rad/s, while the collar is free to slide along the horizontal rod CD. For the position q = 0, determine (a) the velocity of the collar, (b) the angular velocity of the rod.
 

Solution: (a) v_B = - (12 in./s)i (b) w = (3.69 rad/s)i + (1.846 rad/s)j + (2.77 rad/s)k

Comments
The 3-D motion shown in the Sample Problem 15.12 AVI and the associated Working Model file is the most important contribution to this exercise. The motion of the collar-clevis-disk mechanism is difficult to show on paper or to picture mentally. The moving parts in this simulation convince users that this mechanism does indeed work.

Problem Statement
The thin plate ABCD of mass 8 kg is held in the position shown by the wire BH and two links AE and DF. Neglecting the mass of the links, determine immediately after wire BH has been cut (a) the acceleration of the plate, (b) the force in each link.
 
 

Solution: (a) a = 8.50 m/s² @ 60^o (b) F_AE = 47.9 N in tension, F_DF = 8.70 N in compression

Comments
In Sample Problem 16.2 Working Model file, the mass of the thin plate can be varied with a slider. In doing so, one realizes that the acceleration of the thin plate is not affected its mass. However, the mass does affect the force in the links.

Problem Statement
The extremities of a 4-ft rod weighing 50 lb can move freely and with no friction along two straight tracks as shown. If the rod is released with no velocity from the position shown, determine (a) the angular acceleration of the rod, (b) the reactions at A and B.
 
 

Solution: (a) a = 2.30 rad/s² (CCW) (b) R_B = 22.5 lb @ 45° , R_A = 27.9 lb (upward)

Comments
Working Model adds visually appealing animation to an otherwise mundane problem.

Problem Statement
Each of the 3-kg bars AB and BC is of length L = 500 mm. A horizontal force P of magnitude 20 N is applied to bar BC as shown. Knowing that b = L (P is applied at C), determine the angular acceleration of each bar.
 
 

Solution: (a) a _AB = 11.43 rad/s² (CW) (b) a_BC = 57.1 rad/s² (CCW).

Problem Statement
Each of the 3-kg bars AB and BC is of length L = 500 mm. A horizontal force P of magnitude 20 N is applied to bar BC. For the position shown, determine (a) the distance b for which the bars move as if they formed a single rigid body, (b) the corresponding angular acceleration of the bars.
 

Solution: (a) 227 mm (b) 7.27 rad/s² (CCW)

Comments
The Working Model file for Homework Problem 16.150 allows the user to position force P anywhere along bar BC. By trial and error, one may determine the point where force P should be applied in order for the bars to move initially as if they were a single rigid body.

Problem Statement
Each of the two slender rods shown is 0.75 m long and has a mass of 6 kg. If the system is released from rest with b = 60^o, determine (a) the angular velocity of rod AB when b = 20^o, (b) the velocity of point Dat the same instant.
 
 

Solution: (a) w _AB = 3.90 rad/s (CW), (b) v_D = 2.00 m/s (to the right)

Problem Statement
Rod AB has a mass of 3 kg and is attached to a 5-kg cart C. Knowing that the system is released from rest in the position shown and neglecting friction, determine (a) the velocity of point B as rod AB passes through a vertical position, (b) the corresponding velocity of the cart C.
 
 

Solution: (a) 6.975 m/s (left) (b) 1.610 m/s (right)

Comments
By viewing the Homework Problem 17.92 AVI, one may find the motion of this relatively simple system to be surprisingly non-intuitive.

Problem Statement
A 2-kg sphere moving horizontally to the right with an initial velocity of 5 m/s strikes the lower end of an 8-kg rigid rod AB. The rod is suspended from a hinge at A and is initially at rest. Knowing that the coefficient of restitution between the rod and the sphere is 0.80, determine the angular velocity of the rod and the velocity of the sphere immediately after the impact.
 

Solution: w ’ = 3.21 rad/s (CCW), v_s’ = 0.143 m/s (to the left).

Problem Statement
A slender rod AB of length L = 8 ft and weight W =40 lb is pinned at A to a vertical axle DE which rotates with a constant angular velocity w of 15 rad/s. The rod is maintained in position by means of a horizontal wire BC attached to the axle and to the endBof the rod. Determine the tension in the wire and the reaction at A.
 

Solution: T = 384 lb, A = - (175 lb) I + (40 lb) J

Problem Statement
A 50-kg block moves between vertical guides as shown. The block is pulled 40 mm down from its equilibrium position and released. For each spring arrangement, determine the period of the vibration, the maximum velocity of the block, and the maximum acceleration of the block.
 
 

Solution: (a) springs attached in parallel:
               t_n = 0.444 s, v_m = 0.566 m/s (vertically), a_m= 8.00 m/s² (vertically)
               (b) springs attached in series:
               t_n = 0.907 s, v_m = 0.277 m/s (vertically), a_m= 1.920 m/s² (vertically)

Problem Statement
A motor weighing 350 lb is supported by four springs, each having a constant of 750 lb/in. The unbalance of the rotor is equivalent to a weight of 1 oz located 6 in. from the axis of rotation. Knowing that the motor is constrained to move vertically, determine (a) the speed in rpm at which resonance will occur, (b) the amplitude of the vibration of the motor at a speed of 1200 rpm.
 

Solution: (a) Resonance speed = 549 rpm (b) x_m = 0.001352 in. (out of phase)

Comments
The Working Model file for Sample Problem 19.05 allows users to vary the motor speed. It is easy to read the graphs which display the amplification of the vertical displacement of the motor. Note: If the weight of the offset particle is increased to 10 oz., the effect is much more visually apparent.

SampleProblem19.05.wm3