M201 assessment Moments

Transcription

1 Do the questions as a test circle questions you cannot answer Red 1) A light see-saw is 10 m long with the pivot 3 m from the left. a) A 4 kg weight is placed on the left-hand end of the see-saw. Write down the anticlockwise moment about the pivot [3] b) A force of magnitude F N is applied to the right-hand end of the see-saw. The force acts vertically downwards. Write down the clockwise moment about the pivot due to this weight. [1] c) Find the value of F for which the system is in equilibrium [3] 1

2 2) A uniform rod AB of length 8 m and weight 180 N is held in horizontal equilibrium by two vertical wires. One wire is 1 m from A and the other 2 m from B wires A 1 m 2 m B 8 m a) Draw a diagram showing all the forces acting on the rod. [1] b) Calculate the tensions in the wires. [5] 2

3 4) An injured climber is tied to a stretcher AB of length 2.5 m. The total mass of the climber and the stretcher is 90 kg. In each part of this question you should make the following modelling assumptions: the centre of mass, G, of the stretcher with climber is a distance m from the end A of the stretcher, as shown in the diagram; all the forces acting on the system are in the same vertical plane. lifting force lifting force A m 2.5 m G The lifting forces are each vertically upwards at the ends A and B of the stretcher, and the stretcher is held in horizontal equilibrium. Calculate the values of the lifting forces. B [5] 3

4 Amber 5) A uniform rod AB has length 1.5 m and mass 8 kg. A particle of mass m kg is attached to the rod at B. The rod is supported at the point C, where AC = 0.9 m, and the system is in equilibrium with AB horizontal, as shown in Figure 2. a) Show that m = 2 [4] b) A particle of mass 5 kg is now attached to the rod at A and the support is moved from C to a point D of the rod. The system, including both particles, is again in equilibrium with AB horizontal. [3] c) Find the distance AD [2] 4

5 6) A 5000 kg bus hangs 12 m over the edge of a cliff and has 1000 kg of gold at the front. The gold sits on a wheeled cart. A group of n people, each weighing 70 kg, stands at the other end. The bus is 20 m long. a) Write down the total clockwise moment about the cliff edge in terms of n. [7] b) Find the smallest number of people needed to stop the bus falling over the cliff. [2] c) One person needs to walk to the end of the bus to retrieve the gold. Find the smallest number of people needed to stop the bus falling over the cliff in this situation, including the one retrieving the gold. [4] 5

6 7) Alice, who weighs 50 kg, sits on the right-hand end of a light see-saw. Bob, who weighs 80 kg, stands on the opposite side at a distance x m from the end. The length of the see-saw is 4 m and it pivots about its centre. a) Draw a diagram showing the forces acting on the see-saw due to the two people. Label the value of each force in newtons [2] b) Write down the total clockwise moment about the centre in terms of x [5] c) Find the value of x for which the see-saw is in equilibrium [2] d) Given that Bob remains on the opposite side to Alice, describe with inequalities the range of x for which the see-saw tilts towards Alice [2] e) Describe one limitation of this model. [1] 6

7 8) A uniform plank AB has weight 120 N and length 3 m. The plank rests horizontally in equilibrium on two smooth supports C and D, where AC = 1 m and CD = x m, as shown in the diagram. The reaction of the support on the plank at D has magnitude 80 N. Modelling the plank as a rod, a) show that x = 0.75 [3] A rock is now placed at B and the plank is on the point of tilting about D. Modelling the rock as a particle, find b) the weight of the rock, [4] c) the magnitude of the reaction of the support on the plank at D [2] d) State how you have used the model of the rock as a particle. [1] 7

8 Green 9) Two identical 5 m light see-saws are joined at their ends. Robert, who weighs 80 kg, stands on top of the joint. The distance between Robert and each of the pivots is 2 m. Poppy and Quentin stand on the two remaining ends of the see-saws. Poppy weighs p kg and Quentin weighs q kg. The system is in equilibrium. Show that, to the nearest whole number, p + q = 53 [8] 8

9 10) 0.4 m 1.0 m 0.2 m C A A uniform, horizontal, rigid shelf CD has a weight of 40 N and length 1.6 m. It is resting on two thin brackets A and B which are 0.4 m and 0.2 m respectively from C and D, as shown in the diagram above. a) Calculate the reaction forces of the brackets on the shelf. [4] An object is placed on the shelf so that its weight, W N, acts on the shelf at a distance x m from C. b) Show that the vertical reaction force on the shelf at A is (24 W(x 1.4)) N. Find a similar [4] expression for the vertical reaction force on the shelf at B. c) For what values of x will the shelf not tip up if W = 200? [4] B D 9

10 11) A plank AB has mass 12 kg and length 2.4 m. A load of mass 8 kg is attached to the plank at the point C, where AC = 0.8 m. The loaded plank is held in equilibrium, with AB horizontal, by two vertical ropes, one attached at A and the other attached at B, as shown in Figure 2. The plank is modelled as a uniform rod, the load as a particle and the ropes as light inextensible strings. a) Find the tension in the rope attached at B. [4] b) The plank is now modelled as a non-uniform rod. With the new model, the tension in the rope attached at A is 10 N greater than the tension in the rope attached at B. Find the distance of the centre of mass of the plank from A [6] 10

11 12) C A B 1 5 A beam AB has mass 12 kg and length 5 m. It is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. One rope is attached to A, the other to the point C on the beam, where BC = 1 m, as shown in Figure 2. The beam is modelled as a uniform rod, and the ropes as light strings. a) Find (i) the tension in the rope at C, (ii) the tension in the rope at A A small load of mass 16 kg is attached to the beam at a point which is y metres from A. The load is modelled as a particle. Given that the beam remains in equilibrium in a horizontal position, b) Find, in terms of y, an expression for the tension in the rope at C. [3] The rope at C will break if its tension exceeds 98 N. The rope at A cannot break. c) Find the range of possible positions on the beam where the load can be attached without the rope at C breaking [5] [3] 11

12 13) R N T N x m P Q b m W N a A cuboid of weight W N is in equilibrium m on a horizontal table. The cuboid is a m long and the centre of mass is b m from end P. A vertical force, T N, acts on it at end Q. The reaction force, R N, of the table on the cuboid acts x m from end P, as shown in the diagram a) By resolving and taking moments, write down two equations for the equilibrium of the cuboid. Wb Ta Hence show that x W T. [5] b) Find an expression for R in terms of W, a and b when the cuboid is on the point of [2] turning about the edge through P. TOTAL

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