Moments

A playground see-saw is modelled as a uniform plank AB of length 4.0 m and mass 20 kg, pivoted at its midpoint O. A child P of mass 32 kg sits at A. A second child Q, of mass 40 kg, sits on the plank between O and B, at a distance d metres from O. The children are modelled as particles and the plank rests in equilibrium in a horizontal position.

A workbench top is modelled as a uniform plank AB of length 3.2 m and weight 150 N. It rests horizontally on two trestles, one at A and one at B. A toolbox of weight 90 N is placed on the plank at C, where AC = 0.8 m. The toolbox is modelled as a particle.

A hanging decoration is made from a light rod PQ of length 60 cm. A bead of mass 0.30 kg is fixed at P and a bead of mass 0.50 kg is fixed at Q. The rod hangs in equilibrium in a horizontal position from a single light vertical thread attached to the rod at the point R. The beads are modelled as particles.

A uniform rod AB has length 1.8 m and mass 4 kg. It hangs in a horizontal position from two light vertical wires attached at C and D, where AC = 0.3 m and DB = 0.3 m. A lantern of mass 2 kg hangs from B. The lantern is modelled as a particle.

Figure 1 shows a light rod AB of length 2.5 m. Four vertical forces act on the rod at A, C, D and B, with the magnitudes, directions and distances shown in the figure.

Figure 1 shows a wheelbarrow, modelled as a horizontal rod AB of length 1.5 m. The ground exerts a vertical force on the barrow at the wheel axle A. The barrow has mass 15 kg and its centre of mass is at G. It carries bricks of total mass 60 kg, modelled as a particle at L. A gardener holds the barrow at rest, with AB horizontal, by applying a vertical force of magnitude F newtons at B. The distances are shown in the figure.

Figure 1 shows a steelyard used for weighing sacks. It is modelled as a light rod AB of length 80 cm, hanging from a hook at O, where AO = 10 cm. A pan of mass 0.40 kg hangs from A. A counterweight of mass 1.2 kg can be hung from any point of OB, which is marked every 10 cm measured from O. A sack is placed in the pan, and the rod rests in equilibrium in a horizontal position with the counterweight at the mark shown. The pan, sack and counterweight are modelled as particles.

Figure 1 shows a car of mass 1300 kg at rest on horizontal ground. The car is modelled as a rigid body whose weight acts at G. The forces from the road on the car act vertically at the front axle F and at the rear axle R. The distances are shown in the figure.

A diving board is modelled as a uniform rod AB of length 4.0 m and mass 30 kg, held in a horizontal position. The end A is bolted to the ground, and the bolt can exert a vertical force on the board in either direction. The board also rests on a support at C, where AC = 1.2 m. A diver of mass 65 kg stands at rest at B. The diver is modelled as a particle.

Two gardeners carry a log, modelled as a non-uniform rod AB of length 2.8 m and mass 54 kg, in a horizontal position. One gardener supports the log at A and the other at B, each with a vertical force. The force at A is 1.25 times the force at B.

A gymnastics training beam is modelled as a uniform rod AB of length 3.6 m and mass 18 kg. It rests in a horizontal position on two blocks at C and D, where AC = 0.6 m and DB = 0.9 m. A gymnast of mass 72 kg stands on the beam. The gymnast is modelled as a particle.

In an experiment, a non-uniform rod AB of length 1.5 m and mass m kg balances in a horizontal position on a knife-edge placed at C, where AC = 0.9 m. A small weight of mass 0.4 kg is then fixed to the rod at A, and the rod now balances in a horizontal position when the knife-edge is at D, where AD = 0.7 m. The small weight is modelled as a particle.

A short road bridge is modelled as a uniform beam AB of length 8 m and mass 600 kg, resting in a horizontal position on supports at A and at B. A vehicle of mass 1200 kg is on the bridge, at a distance x metres from A. The vehicle is modelled as a particle.

A car park barrier is modelled as a uniform rod AB of length 4.5 m and mass 12 kg, free to turn about a smooth horizontal axle at C, where AC = 0.5 m. A counterweight of mass 18 kg is fixed at A. When the barrier is closed, it rests in a horizontal position with the end B on a fixed post. The counterweight is modelled as a particle.

A metre rule is modelled as a uniform rod of length 1.00 m and mass m kg, with a scale marked in centimetres. Figure 1 shows the rule resting in equilibrium in a horizontal position on a knife-edge, with a particle of mass 100 g hanging from it by a light vertical string. The positions of the knife-edge and the string are read from the scale in the figure.

A non-uniform plank AB has length 3.0 m. It rests in a horizontal position with the end A on one set of bathroom scales and the end B on another, as shown in Figure 1. Each set of scales displays the magnitude of the vertical force on it divided by g, in kilograms. The scales are modelled as exerting vertical forces at A and at B.

A shelf is modelled as a uniform rod AB of length 1.6 m and mass 4 kg. It rests in a horizontal position on two brackets at C and D, which exert vertical forces on it; the shelf is not fixed to the brackets. A box of books of mass 9 kg stands on the shelf with its centre of mass vertically above E, and a vase of mass 2 kg stands at B. The distances are shown in Figure 1. The box and the vase are modelled as particles.

Figure 1 shows a mobile hanging in equilibrium with both of its rods horizontal. The rods PQ and RS are light, and all the threads are light and vertical. PQ hangs from a thread attached at O. An ornament of mass M kg hangs from P, and a thread from Q is attached to RS at T. Ornaments of masses 0.05 kg and m kg hang from R and S. The ornaments are modelled as particles and the distances are shown in the figure.

A non-uniform plank AB has length 4.0 m and weight 300 N. Its centre of mass is at G. The plank rests in a horizontal position on two supports at C and D, where AC = 0.5 m and AD = 3.0 m. A person of weight W newtons walks slowly along the plank, starting at A. The person is modelled as a particle at a distance x metres from A. The graph in Figure 1 shows how the magnitude R_C newtons of the reaction at C varies with x; the line ends at x = 3.75.

Figure 1 shows a uniform pole AB of length 2.0 m and mass 4 kg being carried in a horizontal position by two people. One person pushes vertically upwards on the pole at A with a force of magnitude F_A newtons, and the other at B with a force of magnitude F_B newtons. Buckets of mass 12 kg and 20 kg hang from the pole at C and D. The buckets are modelled as particles and the distances are shown in the figure.

A non-uniform plank AB has length 5.0 m and mass 30 kg. Its centre of mass is at G. The plank rests in a horizontal position on two supports at C and D, where AC = 1.0 m and DB = 1.0 m. When a boy of mass 36 kg stands at B, the plank is on the point of tilting about D. The boy, and later a girl, are modelled as particles.

A window cleaner's platform is modelled as a uniform rod AB of length 6.0 m and mass 80 kg. It hangs in a horizontal position from two light vertical cables, one attached at A and the other at C, where CB = 1.5 m. Each cable is safe only while its tension is at most 1200 N. A window cleaner of mass 70 kg stands on the platform at a distance x metres from A. The window cleaner is modelled as a particle.

A non-uniform rod AB has length 2.0 m and mass M kg. Its centre of mass G is at a distance d metres from A. The rod is held in a horizontal position by two light vertical strings attached at C and D, where AC = 0.2 m and DB = 0.4 m. In a first experiment, a particle of mass 2 kg is attached to the rod at A, and the tensions in the two strings are equal. In a second experiment, the 2 kg particle is removed and a particle of mass 6 kg is attached at B; the rod is then on the point of turning about D.

A plank is laid across a ditch. It is modelled as a uniform rod AB of length 4.8 m and mass 25 kg, resting in a horizontal position on the two banks at C and D, where AC = 0.6 m and DB = 1.2 m. A bag of mass 15 kg is placed on the plank at A. A man of mass 80 kg steps onto the plank at C and walks slowly towards B. The bag and the man are modelled as particles.

A non-uniform beam AB has length 5.0 m. It rests in a horizontal position on two supports, one at A and one at B. A load of weight W newtons is hung from the beam at a fixed point E. For different values of W, the magnitudes of the reactions at A and at B are measured. The graph in Figure 1 shows the results, which lie on two straight lines. The load is modelled as a particle.

Figure 1 shows a uniform plank AB, of length 4.0 m and mass 35 kg, resting in a horizontal position on two supports at C and D. A second uniform plank PQ, of length 2.0 m and mass 10 kg, lies in line with AB: its end Q rests on top of AB at B, and its end P rests on a ledge, so that PQ is horizontal. A person of mass 50 kg stands on PQ at a distance y metres from P. The person is modelled as a particle, and all the contact forces are vertical.

A scaffold board is modelled as a uniform rod AB of length 6.0 m and mass 30 kg. It rests in a horizontal position on two trestles at C and D, which exert vertical forces on it. A tin of paint of mass 10 kg stands at A, and a painter of mass 75 kg stands on the board at a distance x metres from A, as shown in Figure 1 (the painter's position in the figure is only one possibility). The tin and the painter are modelled as particles.

A workshop girder is modelled as a uniform rod AB of length 12 m and weight 24 kN, resting in a horizontal position on two columns, one at A and one at B. A hoist trolley runs along the top of the girder. Its rear wheel is at a distance x metres from A and its front wheel is 2 m further towards B. With its load, the trolley presses vertically down on the girder with the forces shown in Figure 1, which act at the wheels.