## Question 1 > Two men, A and B, are holding the ends of a light pole which is 3 m long. A load of 1 kN is suspended from the light pole at a point 1.2 m from A, What load is each man carrying? > [!HELP]- Solution > Vertical force equilibrium: >$ R_{A}+R_{B}=1000 >$ >Taking moments about point $A$ >$ >L=3m \qquad W=1000N \qquad d=1.2m >$ >$ >W \times d = R_{B} \times L \qquad 1000\times 1.2=R_{B} \times 3 >$ >Solve for $R_{B}$ and $R_{A}$ >$ >R_{B}=\frac{1000 \times 1.2}{3} \qquad R_{A}=1000-R_{B} >$ >$ >\boxed{R_{B}=400N} \qquad \boxed{R_{A}=600N} >$ ## Question 2 >A man of mass 200 kg wants to weigh himself, but the bathroom scales only register up to 180 kg. However he has two sets of bathroom scales. So he arranges a plank, of mass 20 kg, with one set of scales under each end. However he stands on the plank only a quarter of the way along from the left-hand scales. What masses are measured on each set of scales? > [!HELP]- Solution >$ >F_{A}+F_{B}=2200 \text{ N} >$ >$ >2000 \times 0.25L + 200 \times 0.5L = F_{B} \times L >$ >$ >500L + 100L = F_{B}L \qquad \Rightarrow \qquad F_{B}=600\text{ N} >$ >$ >F_{A}=2200-600=1600\text{ N} >$ >$ >m_{A}=\frac{F_A}{g}=\frac{1600}{10}=160\text{ kg} \qquad m_{B}=\frac{F_B}{g}=\frac{600}{10}=60\text{ kg} >$ >$ >\boxed{m_{A}=160\text{ kg}} \qquad \boxed{m_{B}=60\text{ kg}} >$ ## Question 3 > A uniform rod of mass 10 kg is hinged at one end, and supported by a cord attached to the other end, as illustrated below. The cord is secured to a point at the same level as the hinge. The rod and the string are inclined at the same angle (30º) to the horizontal. Find the tension in the cord. > [!HELP]- Solution > $ > \text{Weight of rod: } W = mg = 10 \times 10 = 100 \text{ N} > $ > $ > \text{Clockwise moment (weight)} = \text{Counterclockwise moment (tension)} > $ > $ > W \times \frac{L}{2}\cos(30°) = T \times L\sin(120°) > $ > $ > 100 \times \frac{1}{2}\cos(30°) = T\sin(120°) > $ > $ > T = \frac{50\cos(30°)}{\sin(120°)}=\boxed{50 \text{ N}} > $ ## Question 4 >A uniform beam AB of weight W can turn in a vertical plane about a hinge at A. The other end B is tied to a rope which passes over a smooth pulley C, which is vertically above A so that AB = AC. Find the tension in the rope necessary to keep the beam at an angle of 60° to the horizontal. > [!HELP]- Solution > $ > AB=AC \qquad \Rightarrow \qquad \angle ABC=\angle BCA > $ > $ > \angle CAB=90^{\circ}-60^{\circ}=30^{\circ} \qquad \angle ABC=\frac{180-30}{2}=75^{\circ} > $ > Taking moments about point $A$ > $ > M_{W}=W\times \frac{L}{2}\cos 60=\frac{WL}{4} \qquad M_{T}=T\times L \sin 75 > $ > For equilibrium: $M_{T}=M_{W}$ > $ > T\times L \sin 75 = \frac{WL}{4} \qquad T=\frac{W}{4\sin 75}=\boxed{0.259W} > $ ## Question 5 >A “sandwich board” advertising sign is constructed as shown in the diagram following. The sign’s mass is 8.00 kg. Calculate the tension in the chain, assuming that there is no friction between the bottoms of the legs and the ground. > >The weight of each side of the sign can be assumed to act through the centres of gravity (CG in the diagram) halfway up the height of each side.