Tutorial Sheet 1 (Beer & Johnston)

Problem 2.73: To stabilize a tree partially uprooted in a storm, cables AB and AC are attached to the upper trunk of the tree and then are fastened to steel rods anchored in the ground. Knowing that the tension in cable AB is 950 N, determine (a) the components of the force exerted by this cable on the tree, (b) the angles

θx,θy, and θz that the force forms with axes at A which are parallel to the coordinate axes. Solve by answering the following

• Let OA=h. What the coordinates of A, B in terms of h?

• What is the position vector rAB?

• What does the vector F=950nAB N represent?

• Find (F.i)/|(|F||i|), (F.j)/|(|F||j|), (F.k)/|(|F||k|),. What do these values represent?

PROBLEM 2.97: For the tree of Prob. 2.73, knowing that the tension in cable AB is 760 N and that the

resultant of the forces exerted at A by cables AB and AC lies in the yz plane, determine (a) the tension in AC, (b) the magnitude and direction of the resultant of the two forces. Solve by answering the following

• What the coordinates of A, B, C in terms of h?

• What are the position vectors rAB, rAC?

• Let FAB=760N and FAC an unknown number representing the tension in cable AC. Write down the vector sum of FAB and FAC. What is the component of this vector sum along y?

Problem 3.25: In an arm wrestling contest, a 150-N force P is applied to the hand of one of the contestants by his opponent. Knowing that AB = 15.2 cm and BC = 16 cm, determine the moment of the force about C. Solve by answering the following

• What the coordinates of A, B, C?

• What are the position vectors rBA, rBC? What is the vector rCA?

• Write the direction cosines of the vector P. Write P in terms of unit vectors I, j, k.

• Write down the cross product rCA

X P. What does this represent? PROBLEM 3.31: In Prob. 3.25, determine the perpendicular distance from point C to the action of force P. Solve by answering the following

• What is the magnitude of the moment?

• If the perpendicular distance is h, then what is the magnitude of the moment in terms of h and P?

Problem 3.53: The frame ACD is hinged at A and D and is supported by a cable that passes through a ring at B and is attached to hooks at G and H. Knowing that the tension in the cable is 1125 N, determine the moment about the diagonal AD of the force exerted on the frame by portion BH of the cable. Solve by answering the following

• What are the coordinates of A, B, C, D, H?

• Is tension the same in BC and BH? Why?

• What is the unit vector along BH?

• Express the tension T in vector form.

• Find the unit vector nAD along the line AD?

• What does the triple product [T AB nAD] represent and why?

PROBLEM 3.62: In Prob. 3.53, determine the perpendicular distance between portion BH of the cable and

the diagonal AD. Solve by answering the following

• What is the component of the tension T in a direction normal to the line AD?

• If the perpendicular distance between BH and AD is d, then what is the expression for the moment of the tension T about AD?

PROBLEM 3.83: A landscaper tries to plumb the tree shown in Fig. P3.82 by applying a 54-lb force. (a) Replace that force with an equivalent force-couple system at C. (b) Two helpers attempt to plumb the same tree, with one applying a horizontal force at C and the other pulling at B. Determine these two forces if they are to be equivalent to the single force of part a. Solve by answering the following

• Set up your coordinate system with origin at C. What is the vector sum of forces acting on the the tree in the first case ?

• What is the vector sum of moments of all forces about C in the first case?

• Assuming that the two forces in the second case have unknown magnitudes P and Q, write them in vector form in terms of P and Q

respectively.

• What is the sum of forces and moments about C in the second case in terms of P and Q?

PROBLEM 3.107: When the couple M is applied to the link in a mechanism, the resulting forces exerted on the link from a guide and the connecting links are as shown. Determine (a) the values of M and α so that the applied forces and couple can be reduced to a single equivalent force whose line of action passes through points B and D, (b) the equivalent force. Solve by answering the following

• Set up your coordinate system with origin at B. What are the coordinates of the points A, B, C, D?

• What is the vector sum of forces acting on the link? What is the vector sum of couples and moments of forces about B?

• Find the unit vector AB. Assuming the unknown force has magnitude F, write down

the force in vector form. Write down the sum of forces and sum of moments of forces about B in this case.

PROBLEM 3.69: A piece of plywood in which several holes are being drilled successively has been secured to a workbench by means of two nails. Knowing that the drill exerts a 12 N⋅m couple on the piece of plywood, determine the magnitude of the resulting forces applied to the nails if they are located (a) at A and B, (b) at B and C, (c) at A and C.

PROBLEM 3.119: A portion of the flue for a furnace is attached to the ceiling at A. While supporting the free end of the flue at F, a worker pushes in at E and pulls out at F to align end E with the furnace. Knowing that the 50-N force at F lies in a plane parallel

to the yz plane, determine (a) the angle α the force at F should form with the horizontal if duct AB is not to tend to rotate about the vertical, (b) the force-couple system at B equivalent to the given force system when this condition is satisfied. PROBLEM 3.120: A portion of the flue for a furnace is attached to the ceiling at A. While supporting the free end of the flue at F, a worker pushes in at E and pulls out at F to align end E with the furnace. Knowing that the 50-N force at F lies in a plane parallel to the yz plane and that α = 60°, (a) replace the given force system with an equivalent force-couple system at C, (b) determine whether duct

CD will tend to rotate clockwise or counterclockwise relative to elbow C, as viewed from D to C.