# 3.7: Chapter 3 Homework Problems

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## Exercise $$\PageIndex{1}$$

An 18-inch shelf is supported by a pin joint at point A, and a cable at point B. The shelf itself has a weight of 60 lbs. If we want the net moment about point A to be zero, what should the tension in the cable be? Figure $$\PageIndex{1}$$: problem diagram for Exercise $$\PageIndex{1}$$; a horizontal shelf is attached to a wall with a pin joint at one end and a cable at the other end.
Solution

$$T = 52.30 \, lbs$$

## Exercise $$\PageIndex{2}$$

What is the moment the force shown below exerts about Point A? About Point B?

Hint: use Varginon's Theorem. Figure $$\PageIndex{2}$$: problem diagram for Exercies $$\PageIndex{2}$$; a rectangular slab is bolted to a wall at two points, A and B, with a force is exerted on one of the rectangle's unattached corners.
Solution

$$M_A = 10.88 \, kNm$$

$$M_B = 21.02 \, kNm$$

## Exercise $$\PageIndex{3}$$

You are attempting to rotate a heavy table about the base of one leg at point O and are going to exert a 100-lb force at the opposite end. Person A recommends pulling straight up, while person B recommends pulling up at 30° from vertical. What would be the moment about point O, in inch-pounds, in either case? Figure $$\PageIndex{3}$$: problem diagram for Exercise $$\PageIndex{3}$$; a table is rotated about the base of one leg, with one of the two proposed forces applied at the tabletop at the opposite end.
Solution

$$M_{AO} = 6000 \, \text{in-lbs}$$

$$M_{BO} = 6996.15 \, \text{in-lbs}$$

## Exercise $$\PageIndex{4}$$

Determine the moment vectors that each of the three tension forces in the diagram below exerts about the origin point O. Provide the answers in vector form with units. Figure $$\PageIndex{4}$$: problem diagram for Exercise $$\PageIndex{4}$$; two straight beams lying in the $$xy$$-plane, joined perpendicularly to each other, experience 3 upward tension forces (in the +$$z$$ direction).
Solution

$$M_{AO} = [0, \, -400, \, 0] \, Nm$$

$$M_{BO} = [-300, \, 0, \, 0] \, Nm$$

$$M_{CO} = [300, \, 0, \, 0] Nm$$

## Exercise $$\PageIndex{5}$$

You exert a 50-lb force on the side of a fridge as shown below. Assuming the fridge is sitting on a rough surface and not moving, what is the magnitude of the moment exerted by the couple consisting of the pushing force and the friction force? Figure $$\PageIndex{5}$$: problem diagram for Exercise $$\PageIndex{5}$$; a force is exerted on a refrigerator that sits on a flat surface, without causing it to move.
Solution

$$M = -100 \text{ft-lbs}$$

## Exercise $$\PageIndex{6}$$

A space station consists of a large ring that spins in order to provide an artificial gravity for the astronauts in the station. To start the station spinning, a pair of thrusters is attached to the outside of the ring, each pointing in opposite directions as shown below.

a) If we want to exert a 10 kN-m moment with the thrusters, and the ring has a diameter of 45 meters, what thrust force should each thruster produce?

b) If we were to use the same thrusters on a 60-meter diameter ring, what moment would they exert? Figure $$\PageIndex{6}$$: problem diagram for Exercise $$\PageIndex{6}$$. An example of the spinning ring-shaped space station as described in the problem (left); a diagram of the locations and directions of the thruster forces on the ring (right).
Solution

a) $$F_{thruster} = 222.22 \, N$$

b) $$M_{thruster} = 13.33 \, kNm$$

## Exercise $$\PageIndex{7}$$

A 60-N force is applied in the $$yz$$ plane, 40° from the $$y$$ direction, on an L-shaped bar as shown below.

a) What is the moment vector this force exerts about point O?

b) What is the overall magnitude of the moment about point O? Figure $$\PageIndex{7}$$: problem diagram for Exercise $$\PageIndex{7}$$; an L-shaped bar lies in the $$xz$$ plane with one end located at the origin, point O, and the other experiencing a force in the $$yz$$ direction.
Solution

$$M_O = [-11.49, \, -11.57, \, 13.79] \, Nm$$

$$|M| = 21.36 \, Nm$$

## Exercise $$\PageIndex{8}$$

What is the moment that the force shown in the diagram exerts about point O? About the axis of the cylindrical shaft (the $$y$$-axis)? Figure $$\PageIndex{8}$$: problem diagram for Exercise $$\PageIndex{8}$$; an L-shaped part that extends in the $$y$$ and $$z$$ directions, with one end located at the origin O, experiences a force in the $$xz$$ direction applied at the opposite end.
Solution

$$M_O = [16484, \, 17046, \, -20979] \, in-lbs$$

$$M_y = 17046 \, in-lbs$$

## Exercise $$\PageIndex{9}$$

The diving board shown below is supported by a pin joint at A and frictionless support at B. A 150-lb diver is standing at the end of the board. Determine the reaction forces acting on the diving board at points A and B. Figure $$\PageIndex{9}$$: problem diagram for Exercise $$\PageIndex{9}$$; a diving board is supported by a pin joint at its leftmost end (point A) and a frictionless support two feet to the right (point B), with a 150-lb diver standing 6 feet to the right of B.
Solution

$$F_{AX} = 0$$

$$F_{AY} = -450 \, lbs$$

$$F_{BY} = 600 \, lbs$$

## Exercise $$\PageIndex{10}$$

A simplified crane is shown lifting a 400-kg load. The crane is supported by a pin joint at A, and a cable at B. Assuming the crane arm is in equilibrium, what are the reaction forces at A and the tension at B? Figure $$\PageIndex{10}$$: problem diagram for Exercise $$\PageIndex{10}$$; a crane arm is attached to a vertical support at one end (A), is connected to a horizontal cable at its midpoint (B), and lifts at a 400-kg load at the other end.
Solution

$$F_{AX} = 9352.9 \, N$$

$$F_{AY} = 3924 \, N$$

$$T_{B} = 9352.9 \, N$$

## Exercise $$\PageIndex{11}$$

An 8-foot ladder sits propped up against a wall at a 60-degree angle as shown below. It has a weight of 50 lbs acting at its center point and supports a 120-lb woman 6 feet from the bottom. Assume that friction acts at the bottom of the ladder, but not the top. What are the normal forces acting at the bottom and top of the ladder, and what is the friction force acting at the bottom of the ladder? Figure $$\PageIndex{11}$$: problem diagram for Exercise $$\PageIndex{11}$$; an 8-ft ladder weighing 50 lbs and supporting a 120-lb woman who has climbed 75% of the way up leans, at 60° above the horizontal, against a wall.
Solution

$$F_{N \, Top} = 66.4 \, lbs$$

$$F_{N \, Bottom} = 170 \, lbs$$

$$F_f = 66.4 lbs$$

## Exercise $$\PageIndex{12}$$

An SUV with a weight of 4200 lbs and a center of mass located as shown below is parked pointed downhill on a 10-degree incline. The parking is engaged, locking up the back wheels but not the front wheels. What is the expected normal force at the front wheels, the expected normal force at the back wheels, and the expected friction force at the back wheels assuming the SUV does not slip? Figure $$\PageIndex{12}$$: problem diagram for Exercise $$\PageIndex{12}$$; an SUV with front and rear wheels 6 feet apart, and a center of mass 2 feet above the ground and 2 feet behind the front wheel, is parked pointing downhill on a 10° incline.
Solution

$$F_f = 729.3 \, lbs$$

$$F_{N \, front} = 3000.6 \, lbs$$

$$F_{N \, back} = 1135.6 \, lbs$$

## Exercise $$\PageIndex{13}$$

A cart with a mass of 3500 kg sits on an inclined surface as shown below. Determine the reaction forces acting on each wheel of the cart as well as the tension in the cable supporting the cart. Figure $$\PageIndex{13}$$: a two-wheeled cart is parked facing uphill on a 30° slope. A cable stretches from the front of the cart to a support on the incline, making a 38° angle with the plane of the incline.
Solution

$$T = 21786 \, N$$

$$F_A = 7222 \, N; \, F_B = 35925 \, N$$

## Exercise $$\PageIndex{14}$$

The lighting rig above a stage consists of two 100-lb, uniform beams joined together in a T as shown below (assume the weight acts in the center of each beam). The rig is supported by three cables at A, B, and C. Determine the tension in each of the three cables. Figure $$\PageIndex{14}$$: problem diagram for Exercise $$\PageIndex{14}$$; two beams attached perpendicularly to each other lie in the $$xy$$ plane in a T shape experience upward tension forces from 3 cables, one attached to each free end of the T.
Solution

$$T_A = 50 \, lbs; \, \, T_B = 66.7 \, lbs; \, T_C = 83.3 \, lbs$$

## Exercise $$\PageIndex{15}$$

A 9-meter-long pole with a mass of 100 kg is suspended horizontally, 4 meters from the ceiling with three cables as shown below. Assuming the center of mass of the pole is at the center point of the pole, what is the expected tension in each of the three cables? Figure $$\PageIndex{15}$$: problem diagram for Exercise $$\PageIndex{15}$$; a pole hangs 4 meters below the ceiling, suspended by 3 cables attached to the pole at different locations and angles.
Solution

$$T_A = 357.66 \, N; \, T_B = 357.66 \, N; \, T_C = 408.75 \, N$$

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