# Equilibrium & Elasticity

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1 PHYS 101 Previous Exam Problems CHAPTER 12 Equilibrium & Elasticity Static equilibrium Elasticity 1. A uniform steel bar of length 3.0 m and weight 20 N rests on two supports (A and B) at its ends. A block of weight W = 30 N is placed at a distance 1.0 m from A (see figure 1). What are the magnitudes of the forces on the supports A and B, respectively? (Ans: 30 N and 20 N) 2. Two scales are 2.0 m apart. A uniform 40 kg beam of the same length is placed on top of them (see figure 2). A 10-kg block is placed on the beam after which the right scale reads 22 kg and the left scale reads 28 kg. How far from the right scale is the center of gravity of the block located? (Ans: 1.6 m) 3. A uniform meter stick has mass M = 1.25 kg. As shown in figure 3, this meter stick is supported by two vertical strings, one at each end, in such a manner that it makes an angle of 20 with the horizontal. Find the magnitude of the tension in each string. (Ans: T 1 = 6.1 N, T 2 = 6.1 N) 4. Figure 4 shows three boxes of masses m1, m 2 and m 3 hanging from a ceiling. The crossbars are horizontal and have negligible mass and same length L. If m 3 = 1.0 kg, then what is m 1? (Ans: 12 kg) 5. A uniform rod AB is 1.2 m long and weighs 16 N. It is suspended by strings AC and BD, as shown in figure 5. A block P, weighing 96 N, is attached at point E, 0.30 m from A. What is the magnitude of the tension in the string BD? (Ans: 32 N) 6. A 5.0-m long uniform ladder (with mass m = 12.0 kg) leans against a wall at a point 4.0 m above a horizontal floor as shown in figure 6. Assuming the wall is frictionless (but the floor is not), determine the magnitude of the normal force exerted on the ladder by the wall. (Ans: 44 N) 7. A man weighing 720 N stands halfway up a 5.0 m ladder of negligible weight. The base of the ladder is 3.0 m from the wall, as shown in figure 7. Assume that the wall-ladder contact is frictionless. With what force does the wall push against the ladder? (Ans: 270 N) 8. Figure 8 shows a stationary 50-N uniform rod (AB), 1.2 m long, held against a wall by a rope (AC) and friction between the rod and the wall. Find the magnitude of the force (T) exerted on the rod by the rope. (Ans: 50 N) 9. A 240-N weight is hung from two ropes AB and BC, as shown in figure 9. What is magnitude of the tension in the horizontal rope AB? (Ans: 416 N) 10. A thin right angled rod is made of a uniform material. The shorter end is half the length of the longer end. It is hanging by a string attached at point O (see figure 10). At equilibrium, what is the angle α between the shorter rod and the vertical? (Ans: 76 ) Dr. M. F. Al-Kuhaili PHYS 101 Chapter 12 Page 1

2 11. Figure 11 shows a uniform beam, with a weight of 60.0 N and a length of 3.20 m, that is hinged at its lower end. A horizontal force F of magnitude 50.0 N acts at its upper end. The beam is held vertical by a cable that makes an angle θ = 30.0 with the ground and is attached to the beam at a height h = 1.60 m. What is the tension (T) in the cable? (Ans: 115 N) 12. A 5.0-m weightless rod (AC), hinged to a wall at A, is used to support an 800-N block, as shown in figure 12. What are the horizontal and vertical components (F H, F V ) of the force of the hinge on the rod? (Ans: FH = 600 N, F V = 800N) 13. A uniform beam, having a mass of 60 kg and a length of 2.8 m is held in place at its lower end by a pin (P), as shown in figure 13. Its upper end leans against a vertical frictionless wall. What is the magnitude of the force by the pin on the beam? (Ans: 706 N) 14. The system in figure 14 is in equilibrium. A mass (M) of 5.0 kg hangs from the end of the uniform beam of mass = 10.0 kg. What is the magnitude of the tension in the cable? (Ans: 190 N) 15. Figure 15 shows a uniform ball of 600 N weight suspended by a string AB and rests against a frictionless vertical wall. The string makes an angle of 30.0 with the wall. What is the magnitude of the tension in the string? (Ans: 693 N) 16. A horizontal steel rod of length 81 cm and radius 9.5 mm is fixed at one end. It stretches by 0.90 mm when a horizontal force of magnitude F is applied to its free end. Find the magnitude of F. (Young modulus of steel is N/m 2 ) (Ans: 63 kn) 17. A 500-kg mass is hung from the ceiling with a steel wire. The wire has length = 45.0 cm, radius = 4.00 mm and has negligible mass. Calculate the change in the length of the wire. (Young s modulus of steel = N/m 2 ) (Ans: 0.22 mm) 18. Figure 16 shows a uniform block of mass m = 100 kg, which is held in a horizontal position by two vertical steel rods at its ends. Each of the rods has length = 1.0 m, cross sectional area = m 2 and Young s modulus = N/m 2. What is the increase (ΔL) in the length of any one of the rods? (Ans: m) 19. A horizontal aluminum rod (shear modulus = N/m 2 ) projects L = 5.0 cm from the wall, as shown in figure 17. The cross sectional area of the rod is m 2. A shearing force of 500 N is applied at the end of the rod. Find the vertical deflection x of the end of the rod. (Ans: m) 20. A wire stretches 1.0 cm when a force F is applied to it. The same force is applied to a wire of the same material but with twice the diameter and twice the length. How much does the second wire stretch? (Ans: 0.50 cm) 21. A cube of volume 8.0 cm 3 is made of a material with a bulk modulus of N/m 2. When it is subjected to a pressure of Pa, what is the change in its volume ( V)? (Ans: cm 3 ) 22. A solid copper sphere has a diameter of 85.5 cm. How much stress must be applied to the sphere to reduce its diameter to 85.0 cm? The bulk modulus of copper is N/m 2. (Ans: N/m 2 ) 23. Two masses m1 = 2.5 kg and m 2 = 5.0 kg are hanging as shown in figure 18. Find the tension T in the horizontal string between the two masses. Assume the strings are massless. (Ans: 42 N) 24. A 90-kg mountain climber is tied to one end of an elastic rope of unstretched length 15 m and diameter 9.6 mm. The climber falls, and the rope is stretched by 3 cm. Find Young s modulus of the rope. (Ans: N/m 2 ) Dr. M. F. Al-Kuhaili PHYS 101 Chapter 12 Page 2

3 25. A uniform ladder of length L and mass M rests against a smooth, vertical wall. The coefficient of static friction between the ladder and the ground is Find the angle θ between the ladder and the ground, at which the ladder is about to slide. (Ans: 51.3 o ) 26. A uniform horizontal beam of length 6.00 m and mass M = 100 kg is attached to the vertical wall by a rope making an angle θ = 60.0 º with the wall; the beam can rotate about the fixed pivot O, (see figure 19). A mass m = 80.0 kg hangs from P at a distance of 4.00 m from O. Find the tension in the rope attached to the wall. (Ans: N) 27. What increase in pressure is necessary to decrease the volume of a solid aluminum sphere by %? (Bulk Modulus of Aluminum = N/m 2 ) (Ans: Pa) 28. A certain wire stretches 0.90 cm when outward forces with magnitude F are applied to each of its ends. The same forces are applied to a second wire of the same material but with two times the diameter and three times the length. How much does the second wire stretch? (Ans: 0.68 cm) 29. A non-uniform meter stick of mass 200 g is pivoted at the 0 cm mark, and balances horizontally in a vertical plane. In addition to the reaction at the pivot and weight, two vertical forces are acting on it, as shown in figure 20. Find the distance of the center of mass of the rod from the pivot. (Ans: 76 cm) 30. A uniform beam is 5.00 m long and has a mass of 53.0 kg. The beam is supported in a horizontal position by a hinge and a cable, as shown in figure 22. Find the x and y components, respectively, of the force of the hinge on the beam. (Ans: 150 N, +260 N) 31. In figure 23, the applied force of magnitude F = 19.6 N keeps the 6.00-kg block and the pulleys in equilibrium. The pulleys have negligible mass and friction. Find the tension T in the upper cable. (Ans: 78.4 N) 32. A man holds horizontally a non-uniform rod AB of length = 6.0 m and weight = 30 N in equilibrium, by exerting a downward force F 1, with one hand, at the end A and an upward force F 2, with the other hand, at a point 1.0 m from A as shown in figure 25. If the magnitude of F 1 is half that of F 2, find the location of the center of mass of the rod from the point A. (Ans: 2.0 m) 33. As shown in figure 26, a uniform beam of length 4.20 m is suspended by a cable from its center point O. A 65.0-kg man stands at one end of the beam. Where should a 190-kg block be placed on the beam so that the beam is in static equilibrium? (Distances are measured from the center point O of the beam)? (Ans: m) 34. Three parallel forces of magnitudes 10.0 N, 20.0 N, and 20.0 N, respectively, act on a body, as shown in figure 27. The perpendicular distances from a given point O to their lines of action are shown. What single force can replace these forces? (Ans: 50.0 N, 10.0 cm to the right of point O) 35. In Figure 28, PQ is a horizontal uniform beam weighing 155 N. It is supported by a string and a hinge at point P. A 245 N block is hanging from point Q at the end of the beam. Find the horizontal component of net force on the beam from the hinge. (Ans: 580 N) 36. A meter stick balances horizontally on a knife-edge at the 50.0 cm mark. With two 5.00 g coins stacked over the 10.0 cm mark, the stick is found to balance at the 45.5 cm mark. What is the mass of the meter stick? (Ans: kg) 37. When water freezes, it expands by about 9.0 %. What would be the pressure increase inside the radiator of your car if the water in it freezes? The bulk modulus of ice is N/m 2. (Ans: Pa) 38. A uniform and symmetric sign board is supported by two strings as shown in figure 30. Find the ratio of the tension in string 1 to the weight of the sign board. (Ans: 0.33) Dr. M. F. Al-Kuhaili PHYS 101 Chapter 12 Page 3

4 39. Figure 31 shows the stress-strain curve for a material. What is Young s modulus of this material? (Ans: N/m 2 ) 40. A block with a weight of 10.0`kN is supported by a massless rope attached to a 4.00`m massless beam that can pivot at the base, as shown in figure 32. Find the magnitude of the tension (T) in the rope. (Ans: 5.08 kn) 41. A horizontal uniform meter stick is supported at the 50-cm mark. A mass of 0.50 kg is hanging from it at the 20-cm mark and a 0.30 kg mass is hanging from it at the 60-cm mark (see figure 33). Find the position on the meter stick at which one would hang a third mass of 0.60 kg to keep the meter stick balanced. (Ans: 70 cm) 42. A uniform rigid rod having a mass of 50 kg and a length of 2.0 m rests on two supports A and B as shown in figure 35. When a block of mass 60 kg is kept at point C at a distance of x from the center, the rod is about to be lifted off A (the normal force on the rod at A is zero). What is the value of x? (Ans: 0.92 m) 43. Figure 36 shows a uniform rod (mass M = 5.0 kg, length L = 1.2 m) dangling over a frictionless edge at point P, but secured by a horizontal rope with tension T = 5.0 N. If the angle θ= 30 o, then what is the distance d needed to keep the rod in equilibrium? (Ans: 3.3 cm) 44. The uniform sphere in figure 37 has a mass m = 2.0 kg and is held in place by a massless rope of length L = 20 cm, touching a rough wall. Find the force of friction between the sphere and the wall if the tension in the rope is 10 N, and the angle θ = 30 o. (Ans: 11 N) A B C D E Conceptual Problems 1. Three identical uniform rods are each acted on by the forces shown in figure 21, all perpendicular to the rods and all equal in magnitude. Which of the rods could be in static equilibrium if an additional force is applied at the center of mass of the rod? A. Only 3 B. Only 2 C. Only 1 D. Only 1 and 2 E. All three 2. Three forces, F 1 = 5.0 N, F 2 = 10 N, and F 3 = 15 N are acting on a body as shown in figure 24. Choose the correct answer (COM stands for center of mass): A. The body is in both translational and rotational equilibrium. B. The body is only in rotational equilibrium. C. The body is only in translational equilibrium. D. The body is neither in rotational equilibrium nor in translational equilibrium. E. The body is moving horizontally with a uniform acceleration. 3. A picture is to be hung from the ceiling by means of two wires as shown in figure 29. Order the following arrangements of the wires according to the tension in wire B, from least to greatest. A. II, I, III B. I, II, III C. III, I, II D. I, III, II E. III, II, I Dr. M. F. Al-Kuhaili PHYS 101 Chapter 12 Page 4

5 4. Figure 32 shows three situations in which a horizontal rod is supported by a hinge on a wall at one end and a cord at the other end. Rank the situations according to the magnitude of the force on the rod from the cord, greatest first. A) (1 and 3 tie) then 2 B) 2 then (1 and 3 tie) C) All tie D) 1, 2, 3 E) 3, 2, 1 5. The diagrams in figure 34 show forces applied to a wheel of weight W = 20 N. Which diagram is the wheel in equilibrium? A. diagram (3) B. diagram (2) C. diagram (1) D. diagram (4) E. none of them Dr. M. F. Al-Kuhaili PHYS 101 Chapter 12 Page 5

6 Dr. M. F. Al-Kuhaili PHYS 101 Chapter 12 Page 6

7 Figure 26 Figure 27 Figure 28 Figure 29 Dr. M. F. Al-Kuhaili PHYS 101 Chapter 12 Page 7

8 Figure 30 Figure 31 Figure 32 Figure 32 Figure 33 Figure 34 Figure 35 Figure 36 Figure 37 Dr. M. F. Al-Kuhaili PHYS 101 Chapter 12 Page 8

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