Statics of rigid bodies

A mechanic loosens a wheel nut with a wheel brace. The nut is at O and she pushes at the point A on the handle, where OA = 0.40 m. All forces act in the plane in which the brace turns. Model the brace as a light rod.

A light rectangular frame ABCD has AB = 0.6 m and AD = 0.4 m. It lies in a vertical plane with AB horizontal and D vertically above A, and it is free to turn about a fixed smooth axis through A, perpendicular to the frame. Three forces act on the frame: a force of 20 N at B acting vertically upwards, a force of 15 N at D acting horizontally in the direction from D to C, and a force of 25 N at C acting at 30° above the horizontal, directed away from the side AD.

A diving board AB is modelled as a uniform rod of length 4.0 m and mass 30 kg. It is held horizontal by a bolt at A, which exerts a vertical force on the board, and by a support at C, where AC = 1.2 m, which pushes vertically upwards. A diver of mass 60 kg stands on the board at B. Model the diver as a particle. g = 9.8 m s⁻².

A uniform ladder of length 4 m and mass 18 kg rests in equilibrium with its foot on rough horizontal ground and its top against the smooth vertical wall of a shed. The ladder makes an angle of 70° with the ground and lies in a vertical plane perpendicular to the wall. g = 9.8 m s⁻².

Figure 1 shows a light lamina PABC on a coordinate grid, with distances in metres. The lamina lies in a vertical plane and is free to turn about a fixed smooth axis through P, perpendicular to the lamina. Forces act on the lamina at A, B and C as shown. The force at C has magnitude Q newtons and acts in the positive x-direction. i and j are unit vectors in the positive x- and y-directions.

A trapdoor is modelled as a uniform rod AB of length 1.2 m and mass 15 kg, freely hinged at A to a horizontal floor. The door is held open at an angle of 50° to the floor by a force of magnitude P newtons applied at B perpendicular to AB, as shown in Figure 1. The force and the door lie in a vertical plane. g = 9.8 m s⁻².

A uniform ladder AB has length 5 m and mass 20 kg. Its foot A rests on rough horizontal ground and its top B rests against a smooth vertical wall. A bag of mass 12 kg hangs from the top of the ladder, so that its weight acts at B. The ladder lies in a vertical plane perpendicular to the wall. The lengths are shown in Figure 1. Model the bag as a particle. g = 9.8 m s⁻².

A shelf AB is modelled as a uniform rod of length 0.9 m and mass 4 kg, freely hinged at A to a vertical wall. The shelf is held horizontal, perpendicular to the wall, by a light rigid strut CD, freely jointed to the shelf at C and to the wall at D, as shown in Figure 1. A box of mass 6 kg, modelled as a particle, rests on the shelf at B. The shelf and the strut lie in a vertical plane. g = 9.8 m s⁻².

A non-uniform pole AB has length 3.0 m and mass 12 kg. It rests horizontally on two trestles, one at A and one at B. The trestle at A exerts a vertical force of 70 N on the pole. Model the pole as a non-uniform rod. g = 9.8 m s⁻².

A shop sign hangs from a bracket. The bracket is a uniform rod AB of length 1.5 m and mass 6 kg, freely hinged at A to a vertical wall. The rod is held horizontal, perpendicular to the wall, by a light inextensible string attached to the rod at C, where AC = 1.0 m, and to a point on the wall vertically above A. The string makes an angle of 40° with the rod. The sign has weight 40 N and hangs from B. The rod and the string lie in a vertical plane. g = 9.8 m s⁻².

A car-park barrier arm AB is modelled as a uniform rod of length 3.0 m and mass 40 kg. The end A is freely hinged to a fixed point on horizontal ground. The arm points up and to the right of A, making an angle of 35° with the horizontal. It is held in equilibrium by a light rope attached at B. The rope is horizontal, lies in the vertical plane containing the arm, and pulls B to the left. g = 9.8 m s⁻².

A uniform ladder AB has length 5 m and mass 20 kg. Its foot A is on rough horizontal ground and its top B rests against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall and makes an angle θ with the ground, where tan θ = 3. A painter of mass 70 kg climbs the ladder. When she is 4 m from A, the ladder is on the point of slipping. Model the painter as a particle. g = 9.8 m s⁻².

A uniform ladder of length 6 m and mass 25 kg stands on a smooth horizontal floor with its top against a smooth vertical wall. It makes an angle of 65° with the floor and lies in a vertical plane perpendicular to the wall. The foot of the ladder is tied to the bottom of the wall by a light inextensible rope, which is horizontal. A librarian of mass 70 kg stands on the ladder 4.5 m from its foot. Model the librarian as a particle. g = 9.8 m s⁻².

A uniform pole AB of length 3 m and mass 8 kg rests with its end A on rough horizontal ground and with the pole resting on a smooth horizontal rail at C. The rail is perpendicular to the vertical plane containing the pole, and is 1.5 m above the ground. AC = 2.5 m. The pole is in limiting equilibrium. g = 9.8 m s⁻².

A non-uniform beam AB of length 3 m and mass 24 kg hangs in equilibrium with AB horizontal, supported by two light inextensible strings attached to the beam at A and B. The strings make the angles with the vertical shown in Figure 1, and lie in the same vertical plane as the beam. g = 9.8 m s⁻².

A jib AB is modelled as a uniform rod of length 4 m and mass 30 kg. The end A is freely hinged to a fixed point on horizontal ground. The jib is held at 60° to the ground by a light inextensible cable joining B to a point C on the ground, where CA = 3 m and C is on the opposite side of A from B, as shown in Figure 1. A load of mass 50 kg hangs from B. The jib, the cable and the load lie in one vertical plane. Model the load as a particle. g = 9.8 m s⁻².

A garden gate ABCD is modelled as a uniform rectangular lamina of weight 200 N with the dimensions shown in Figure 1. It hangs in a vertical plane on a vertical post, from two hinges at A and D, with AB horizontal. Model the hinge at D as exerting a horizontal force only, so that the hinge at A supports all the vertical load. g = 9.8 m s⁻².

A uniform rod AB of length 2.4 m and mass 6 kg is held at an angle of 30° to the horizontal by two fixed rough pegs, P and Q, which are perpendicular to the vertical plane containing the rod. The end A presses against the underside of P, and the rod rests on top of Q, where AQ = 0.8 m, as shown in Figure 1. The coefficient of friction between the rod and each peg is μ. The rod is about to slide down. g = 9.8 m s⁻².

A uniform ladder AB has mass 20 kg. Its foot A rests on rough horizontal ground and its top B rests against a smooth vertical wall, in the position shown in Figure 1. The ladder lies in a vertical plane perpendicular to the wall. The coefficient of friction between the ladder and the ground is 0.2. A horizontal force of magnitude P newtons, acting towards the wall in the same vertical plane, is applied to the ladder at A. g = 9.8 m s⁻².

A uniform log AB of length 3 m and mass 40 kg has its end A on rough horizontal ground. It is held at 20° to the ground by a light rope attached at B, which pulls at 75° to the horizontal, as shown in Figure 1. The log and the rope lie in a vertical plane. Model the log as a rod. The log is on the point of slipping. g = 9.8 m s⁻².

A hanging basket is supported by a bracket. The bracket is a uniform rod AB of length 2 m and mass 5 kg, freely hinged at A to a vertical wall. The rod points away from the wall and slopes upwards at 20° above the horizontal. A light inextensible string joins B to a point C on the wall vertically above A, so that angle ABC = 70°. The basket, of mass 8 kg, hangs from B. The rod, the string and the basket lie in a vertical plane perpendicular to the wall. Model the basket as a particle. g = 9.8 m s⁻².

A uniform ladder AB has length 5 m and mass 20 kg. The foot A is on rough horizontal ground and the top B rests against a rough vertical wall. The coefficient of friction between the ladder and the wall is 0.25. The ladder lies in a vertical plane perpendicular to the wall and makes an angle α with the ground, where tan α = 4/3. A window cleaner of mass 80 kg stands on the ladder at a point 3.5 m from A, and the ladder is about to slip. Model the window cleaner as a particle. g = 9.8 m s⁻².

A uniform rod AB has length 2 m and mass 4 kg. The end A rests against a rough vertical wall, and the rod is held horizontal and perpendicular to the wall by a light inextensible string joining B to the point D on the wall, 1.5 m vertically above A. The coefficient of friction between the rod and the wall is 0.6. A particle of mass 3 kg is attached to the rod at P, where AP = x m. The rod and the string lie in a vertical plane. g = 9.8 m s⁻².

A uniform plank AB of length 4 m and weight 60 N is placed with its end A on rough horizontal ground and the plank resting on a smooth peg C fixed to a garden fence, where AC = 2.5 m. The plank lies in a vertical plane perpendicular to the peg and makes an angle of 25° with the ground. The coefficient of friction between the plank and the ground is 0.3. Model the plank as a rod.

A uniform ladder AB has mass 20 kg. Its foot A rests on rough horizontal ground and its top B rests against a rough vertical wall, in the position shown in Figure 1. The ladder lies in a vertical plane perpendicular to the wall. The coefficient of friction is 0.4 between the ladder and the ground and 0.3 between the ladder and the wall. A person of mass 70 kg climbs the ladder. When she reaches the point C, a distance d metres from A, the ladder is about to slip. Model the person as a particle. g = 9.8 m s⁻².

A uniform rod AB of length 2.5 m and mass 4 kg is freely hinged at A to a vertical wall. The rod is held in equilibrium, sloping downwards away from the wall, by a light inextensible string joining B to the point D on the wall, vertically above A. The distances are shown in Figure 1. A particle of mass 6 kg is attached to the rod at P, where AP = x metres. The rod and the string lie in a vertical plane perpendicular to the wall. g = 9.8 m s⁻².

A uniform rod AB of length 2 m and mass 5 kg has its end A against a rough vertical wall. The rod rests on a smooth horizontal peg C, where AC = 0.8 m, and rises away from the wall at an angle θ to the horizontal, where tan θ = 3/4, as shown in Figure 1. The peg is parallel to the wall and the rod lies in a vertical plane perpendicular to the wall. At first no particle is attached; the particle shown hanging from B is used from part (d) onwards. g = 9.8 m s⁻².

A crate is modelled as a uniform rectangular lamina ABCD of mass 50 kg, with the dimensions shown in Figure 1. It stands on rough horizontal ground with AB on the ground. A rope attached at C pulls with a force of magnitude P newtons at 30° above the horizontal, in the plane ABCD, as shown. The coefficient of friction between the crate and the ground is 0.4. P is increased gradually from zero. g = 9.8 m s⁻².