ME 161 Assignment 1 (PDF)

ME 161 (CE & EE) ASSIGNMENT I 1. Determine the maginitude and direction of the resultant FR = F1 + F2 + F3 and specify its direction measured counter-clockwise from the x-axis....

ME 161 (CE & EE) ASSIGNMENT I 1. Determine the maginitude and direction of the resultant FR = F1 + F2 + F3 and specify its direction measured counter-clockwise from the x-axis. Fig 1 2. Determine P and Ɵ so that the three forces shown in Fig. 2 are equivalent to the single force R = 85i + 20j kN Fig. 2 3. Replace the three forces acting on the bracket in Fig 3 with a single equivalent force. Fig. 3 4. The 400 N engine block is suspended by the cables AB and AC. If you don’t want either TAB or TAC to exceed 400 N, what is the smallest acceptable value of the angle α? 400 Fig 4 5. The man wants the 200 N crate to start sliding toward the righ. To achieve this, the horizontal component of force exerted on the crate must be 0.3 times the weight of the crate. What is the magnitude of the force the man must exert on the rope in each of the cases presented in Figure 5? Fig 5 6. The man exerts a force P of magnitude 250 N on the handles of the wheelbarrow. Knowing that the resultant of the two forces P, Q and W is the force R = 50i, determine W. Fig 6 7. Replace the three forces acting on the guy wires by a single, equivalent force acting on the flagpole. Use T1 = 1000 N, T2 = 2000 N, and T3 = 1750 N. Fig 7 8. Determine the resultant force R that is equivalent to the forces exerted by the three tugboats as they manoeuvre the barge. Specify the coordinate of the point on the x-axis through which R passes. (Hint: First determine the resultant force for the two forces at point A, and then combine this result with the force at point B.)  Fig. 8 9. In order to move a wrecked truck, two cables are attached at A and pulled by winches B and C as shown. Knowing that the tension is 10 kN in cable AB and 7.5 kN in cable AC, determine the magnitude and direction of the resultant of the forces exerted at A by the two cables. Fig. 9 10. Determine the magnitude and direction of the smallest force F3 so that the resultant of all three forces has a magnitude of 20N. Fig. 10 11. The 600 N box I held in place on the smooth bed of the dump truck by the rope AB. Determine the tension in the rope if α = 25o. If the maximum allowable tension in the rope is 400 N, what is the maximum allowable value of α? Fig. 11 12. Resolve the 250 N force shown in Figure 12 into components acting along the u and v axes and determine the magnitudes of these components. Figure 12  FORCES & MOMENTS Moment of a Force Forces have the tendency to cause two types motions in rigid bodies; translational and rotational motions. The tendency of a force to rotate a body is referred to as moment. A moment may occur about a point; the Moment Centre. F MO    A Mo  r  F r θ rsin  d O d M o  Fr sin   Fd DDEK/2015/ME 161 - BASIC MECHANICS # 76  FORCES & MOMENTS Moment of a force d is always perpendicular to the Force’s line of action, so, the Force is treated as a sliding vector, due to the principle of transmissibility in rigid body mechanics. F F’.... The direction of the moment is determined by the right hand rule. DDEK/2015/ME 161 - BASIC MECHANICS # 77  FORCES & MOMENTS Principle of Moments (Varignon’s Theorem )  States that the moment about a given point O of the resultant of several concurrent forces is equal to the sum of the moments of the various forces about the same point O.  In effect, moments of force components can be taken to get the components of a resultant moment. z z     F1 F2  MO   r  F  Fz  MO   r  F      Fy    A     r  F1  r  F2  r  F3  r A     r  Fx  r  Fy  r  Fz      Fx r F3     O O  r (F1  F2  F3 ) y  r (Fx  Fy  Fz ) y      r R x C  r R x C DDEK/2015/ME 161 - BASIC MECHANICS # 78  FORCES & MOMENTS Calculating the Moment of a force  Scalar Approach Mo  d  F  Only the magnitude of the moment is calculated using only the magnitudes of the force and the moment arm, d, defined as the perpendicular distance between the line of action of the force and the moment centre.  Used when the moment, d can easily be determined. The sense of the moment is determined by inspection.  Vector Approach  The position vector for the point of application of the force is multiplied by the components of the force to get the components of the Resultant moment. DDEK/2015/ME 161 - BASIC MECHANICS # 79  FORCES & MOMENTS Calculating the Moment of a force Example A 100-lb vertical force is applied to the end of a lever which is attached to a shaft at O. A Determine: a) moment about O, 100 lb b) horizontal force at A which creates the same moment, c) smallest force at A which produces the 60o same moment, O d) location for a 240-lb vertical force to produce the same moment, DDEK/2015/ME 161 - BASIC MECHANICS # 80  FORCES & MOMENTS Calculating the Moment of a force Solution FBD Moment about O is equal to the product of the force and the A perpendicular distance between the line of action of the force and O. M O  Fd 100 lb d  24 in.  cos 60   12 in. M O  100 lb 12 in.   Since the force tends to rotate the lever clockwise, the moment 60o O vector is into the plane of the paper. d Mo DDEK/2015/ME 161 - BASIC MECHANICS # 81  FORCES & MOMENTS Calculating the Moment of a force Solution FBD Horizontal force at A that produces the same moment, A F d  24 in.sin 60  20.8 in. M O  Fd d 1200 lb  in.  F 20.8 in. 1200 lb  in. F 60o 20.8 in.  O Mo DDEK/2015/ME 161 - BASIC MECHANICS # 82  FORCES & MOMENTS Calculating the Moment of a force Solution FBD The smallest force A to produce the same moment occurs when A the perpendicular distance is a maximum or when F is perpendicular to OA. F M O  Fd 1200 lb  in.  F 24 in. 1200 lb  in. 60o F 24 in. O  Mo DDEK/2015/ME 161 - BASIC MECHANICS # 83  FORCES & MOMENTS Calculating the Moment of a force Solution The point of application of a 240 lb force to produce the same moment, FBD A M O  Fd 1200 lb  in.  240 lb d 240 lb 1200 lb  in. d  5 in. 240 lb OB  5cos -1 60  60o O d Mo DDEK/2015/ME 161 - BASIC MECHANICS # 84  FORCES & MOMENTS Calculating the Moment of a force Example Determine the moment of the force F about point A. DDEK/2015/ME 161 - BASIC MECHANICS # 85  FORCES & MOMENTS Calculating the Moment of a force Solution (Scalar Approach) OR The vector is treated as a sliding vector and moved to point C DDEK/2015/ME 161 - BASIC MECHANICS # 86  FORCES & MOMENTS Calculating the Moment of a force Example Determine the moment of the 800 N force about point A. 800 N 38o 0.6 m Ans: M A  131.99 Nm A 0.5 m DDEK/2015/ME 161 - BASIC MECHANICS # 87  FORCES & MOMENTS Calculating the Moment of a force Example 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 centerline of AB. Determine the moment of the force about C. Determine the moment of that same force about C if it were being exerted at B, parallel to the centreline AC. DDEK/2015/ME 161 - BASIC MECHANICS # 88  FORCES & MOMENTS Calculating the Moment of a force Example A crowbar is used to remove a nail as shown. Determine the moment of the 60 N force about the point O of contact between the crowbar and the small support block. N DDEK/2015/ME 161 - BASIC MECHANICS # 89  FORCES & MOMENTS Calculating the Moment of a force  Multiplying Vectors  Vectors are expressed in components, arranged in a matrix form, and the determinant of the matrix taken.         If r  xi  yj  zk and F  Fx i  Fy j  Fz k Expressing as a matrix,    z i j k Fzk    MO  r  F  x y z A (x,y,z) zk Fx Fy Fz Fyj r A Taking the determinant of the matrix, O Fxi     yj M O   yFz  zFy i   xFz  zFx  j  xFy  yFx k y xi     M xi  M y j  M z k C x DDEK/2015/ME 161 - BASIC MECHANICS # 90  FORCES & MOMENTS Calculating the Moment of a force  Multiplying Vectors  Alternatively,   the vector components are multiplied. MO  r  F          xi  yj  zk  Fx i  Fy j  Fz k                      xi  yj  zk  Fx i  xi  yj  zk  Fy j  xi  yj  zk  Fz k   But   z i  i  0 j  i  k k  i  j Fzk i i  j  k j  j  0 k  j  i k x A (x,y,z) ik   j jk  i k k  0 zk j Fyj Therefore, r A       O Fxi M O   yFx k  zFx j  xFy k  zFy i  xFz j  yFz i yj    y   yFz  zFy i   xFz  zFx  j  xFy  yFx k xi    C  M xi  M y j  M z k x DDEK/2015/ME 161 - BASIC MECHANICS # 91  FORCES & MOMENTS Calculating the Moment of a force DDEK/2015/ME 161 - BASIC MECHANICS # 92  FORCES & MOMENTS Calculating the Moment of a force Example Determine the moment of the force F about point A. DDEK/2015/ME 161 - BASIC MECHANICS # 94  FORCES & MOMENTS Calculating the Moment of a force Solution (Vector Approach)  4 3 F   200i   200 j 5 5  160i  120 j   r  rAB  rB  rA  0  4 i  6  0  j  4i  6 j i j k      M A  r  F  rAB  F   4 6 0  160 120 0  480k lb.in DDEK/2015/ME 161 - BASIC MECHANICS # 95  FORCES & MOMENTS Calculating the Moment of a force Example Determine the moment of the force F, about point C the perpendicular distance between C and the line of action of F. z 2m B F = 500 N A 3m y rCA 4m C x DDEK/2015/ME 161 - BASIC MECHANICS # 96  FORCES & MOMENTS Calculating the Moment of a force    Solution M C  rA / C  F       rA / C  rA  rC  0 m - 2 m i  4 m - 4 m  j  0 m - 0 m k  2 mi z    2i  4 j  3k  F  F  500 N    2 2   4   32  2   2m     (185.53 N) i  371.06 N  j  278.29 N k B F = 500 N    A i j k 3m  y MC   2 0 0 rCA 185.53  371.06 278.29 4m C x  556.58 j  742.12k M C  (556.58) 2  (742.12) 2  927.64 Nm M 927.64 Nm The perpendicular distance, d    1.86 m F 500 N DDEK/2015/ME 161 - BASIC MECHANICS # 97  FORCES & MOMENTS Calculating the Moment of a force Example A tension of magnitude 10 kN is applied to the cable attached to the top A of the rigid mast and secured to the ground at B. Determine the moment of T about the point O. DDEK/2015/ME 161 - BASIC MECHANICS # 98  FORCES & MOMENTS Calculating the Moment of a force Example The turn buckle is tightened until the tension is cable AB is 2.4 kN. Determine the moment about point O of the cable force acting on point A and the magnitude of this moment. DDEK/2015/ME 161 - BASIC MECHANICS # 99  FORCES & MOMENTS Calculating the Moment of a force Example The rectangular plate is supported by the brackets at A and B and by a wire CD. Knowing that the tension in the wire is 200 N, determine the moment about A of the force exerted by the wire at C. DDEK/2015/ME 161 - BASIC MECHANICS # 100  FORCES & MOMENTS Calculating the Moment of a force Solution    M A  rC A  F    rC A  0.3 m i  0.08 m  j       0.3 m i  0.24 m  j  0.32 m k F  F  200 N  0.5 m     120 N  i  96 N  j  128 N k    i j k  M A  0.3 0 0.08  120 96  128           M A   7.68 N  m i  28.8 N  m j  28.8 N  m k DDEK/2015/ME 161 - BASIC MECHANICS # 101