Equilibrium of rigid bodies

The diagram shows a non-uniform rod AB of mass 4 kg and length 2 m. The end A of the rod rests against a rough vertical wall. The rod is held in a horizontal position, perpendicular to the wall, by a light inextensible string attached to the rod at B. The other end of the string is attached to the wall at a point vertically above A. The string is inclined at an angle of θ° to the horizontal, where

The rod rests in equilibrium in a vertical plane perpendicular to the wall. The rod’s weight acts at the point G on AB.

You are given that the magnitude of the moment of the rod’s weight about A is 47.04 N m.

Show that the distance AG is 1.2 m.

Find the tension in the string.

Determine the magnitude of the contact force exerted on the rod by the wall at A.

By taking moments, show that the magnitude of the normal contact force between the lamina

and the wall is 1.24 N, correct to 3 significant figures.

Determine the coefficient of friction between the lamina and the ground.

A uniform rod AB of mass 4 kg and length 3 m rests in a vertical plane with A on rough horizontal ground.

A particle of mass 1 kg is attached to the rod at B. The rod makes an angle of 60° with the horizontal and is held in limiting equilibrium by a light inextensible string CD. D is a fixed point vertically above A and CD makes an angle of 60° with the vertical. The distance AC is x m (see diagram).

Find, in terms of g and x, the tension in the string.

Determine the value of x.

A beam, AB, has length 4 m and mass 20 kg. The beam is suspended horizontally by two vertical ropes. One rope is attached to the beam at C, where AC = 0.5 m. The other rope is attached to the beam at D, where DB = 0.7 m (see diagram).

The beam is modelled as a non-uniform rod and the ropes as light inextensible strings.

It is given that the tension in the rope at C is three times the tension in the rope at D.

Determine the distance of the centre of mass of the beam from A.

A particle of mass m kg is now placed on the beam at a point where the magnitude of the moment of the particle’s weight about C is 3.5 mg N m. The beam remains horizontal and in equilibrium.

Determine the largest possible value of m.

The diagram shows a wall-mounted light. It consists of a rod AB of mass 0.25 kg and length 0.8 m which is freely hinged to a vertical wall at A, and a lamp of mass 0.5 kg fixed at B. The system is held in equilibrium by a chain CD whose end C is attached to the midpoint of AB. The end D is fixed to the wall a distance 0.4 m vertically above A. The rod AB makes an angle of 60° with the downward vertical.

The chain is modelled as a light inextensible string, the rod is modelled as uniform and the lamp is modelled as a particle.

By taking moments about A, determine the tension in the chain.

Determine the magnitude of the force exerted on the rod at A.

Calculate the direction of the force exerted on the rod at A.

Suggest one improvement that could be made to the model to make it more realistic.

The diagram shows a ladder AB, of length 2 a and mass m, resting in equilibrium on a vertical wall of height h. The ladder is inclined at an angle of 30° to the horizontal. The end A is in contact with horizontal ground. An object of mass 2 m is placed on the ladder at a point C where AC = d.

The ladder is modelled as uniform, the ground is modelled as being rough, and the vertical wall is modelled as being smooth.

Hence find, in terms of a, the greatest possible value of d.

State one improvement that could be made to the model.

the magnitude of the reaction of the support on the plank at D,

the value of x.

A stone block, which is modelled as a particle, is now placed at the end of the plank at B and the plank is on the point of tilting about D.

Find the weight of the stone block.

Explain the limitation of modelling

the stone block as a particle,

the plank as a rigid rod.