Moments

A uniform rod AB has mass M and length 2a. A particle of mass 2M is attached to the rod at the point C, where AC = 0.5a The rod rests with end A on rough horizontal ground and end B against a vertical wall. The rod lies in a vertical plane which is perpendicular to the wall. The rod is in equilibrium at an angle α to the wall, as shown in Figure 4. In an initial model

• the vertical wall is modelled as being smooth

• the magnitude of the normal reaction of the ground on the rod at A is R

• the magnitude of the force exerted on the rod by the wall at B is S

Using the model,

find R in terms of M and g

show that S = Mg tanα In a refined model

• the vertical wall is modelled as being rough

• the magnitude of the normal reaction of the ground on the rod at A is R₁

State which is greater, R or R₁, giving a reason for your answer.

A second particle of mass 3M is now attached to the rod at B. The rod again rests with end A on rough horizontal ground and end B against the vertical wall. The rod lies in a vertical plane which is perpendicular to the wall. The rod is now in limiting equilibrium at an angle β to the wall, as shown in Figure 5. The vertical wall is again modelled as being smooth. The coefficient of friction between the rod and the ground is μ

use the model to show that μ =

TOTAL FOR MECHANICS IS 50 MARKS

Figure 5 shows a uniform rod AB of mass M and length 2a.

• the rod has its end A on rough horizontal ground

• the rod rests in equilibrium against a small smooth fixed horizontal peg P

• the point C on the rod, where AC = 1.5 a, is the point of contact between the rod and the peg

• the rod is at an angle θ to the ground, where tanθ =

The rod lies in a vertical plane perpendicular to the peg. The magnitude of the normal reaction of the peg on the rod at C is S.

The coefficient of friction between the rod and the ground is μ. Given that the rod is in limiting equilibrium,

find the value of μ.

TOTAL FOR MECHANICS IS 50 MARKS

A rod AB has mass M and length 2a. The rod has its end A on rough horizontal ground and its end B against a smooth vertical wall. The rod makes an angle θ with the ground, as shown in Figure 3. The rod is at rest in limiting equilibrium.

State the direction (left or right on Figure 3 above) of the frictional force acting on the rod at A. Give a reason for your answer. The magnitude of the normal reaction of the wall on the rod at B is S. In an initial model, the rod is modelled as being uniform. Use this initial model to answer parts (b), (c) and (d).

By taking moments about A, show that

The coefficient of friction between the rod and the ground is μ

find the value of μ

find, in terms of M and g, the magnitude of the resultant force acting on the rod at A. In a new model, the rod is modelled as being non-uniform, with its centre of mass closer to B than it is to A. A new value for S is calculated using this new model, with tanθ =

State whether this new value for S is larger, smaller or equal to the value that S would take using the initial model. Give a reason for your answer.

TOTAL FOR MECHANICS IS 50 MARKS

A beam AB has mass m and length 2a.

The beam rests in equilibrium with A on rough horizontal ground and with B against a smooth vertical wall.

The beam is inclined to the horizontal at an angle θ, as shown in Figure 2.

The coefficient of friction between the beam and the ground is μ

The beam is modelled as a uniform rod resting in a vertical plane that is perpendicular to the wall.

Using the model,

A horizontal force of magnitude kmg, where k is a constant, is now applied to the beam at A.

This force acts in a direction that is perpendicular to the wall and towards the wall. Given that tanθ = , μ = and the beam is now in limiting equilibrium,

use the model to find the value of k.

A beam AB has mass m and length 2a. The beam rests in equilibrium with A on rough horizontal ground and with B against a smooth vertical wall. The beam is inclined to the horizontal at an angle θ, as shown in Figure 2. The coefficient of friction between the beam and the ground is μ The beam is modelled as a uniform rod resting in a vertical plane that is perpendicular to the wall.

Using the model,

A horizontal force of magnitude kmg, where k is a constant, is now applied to the beam at A. This force acts in a direction that is perpendicular to the wall and towards the wall. Given that tanθ = , μ = and the beam is now in limiting equilibrium,

use the model to find the value of k.

A ladder AB has mass M and length 6a. The end A of the ladder is on rough horizontal ground. The ladder rests against a fixed smooth horizontal rail at the point C. The point C is at a vertical height 4a above the ground. The vertical plane containing AB is perpendicular to the rail. The ladder is inclined to the horizontal at an angle α, where sinα = , as shown in Figure 1. The coefficient of friction between the ladder and the ground is μ. The ladder rests in limiting equilibrium. The ladder is modelled as a uniform rod. Using the model,

show that the magnitude of the force exerted on the ladder by the rail at C is

Hence, or otherwise, find the value of μ.

A non-uniform rod AB, of mass 5kg and length 4m, rests with one end A on rough horizontal ground. The centre of mass of the rod is d metres from A. The rod is held in limiting equilibrium at an angle to the horizontal by a force P, which acts in a direction perpendicular to the rod at B, as shown in Figure 2. The line of action of P lies in the same vertical plane as the rod.

Find, in terms of d, g and ,

the magnitude of the vertical component of the force exerted on the rod by the ground,

the magnitude of the friction force acting on the rod at A.

Given that tan θ and that the coefficient of friction between the rod and the ground

find the value of d.

A non-uniform rod, AB, of mass m and length 2 l, rests in equilibrium with one end A on a rough horizontal floor and the other end B against a rough vertical wall. The rod is in a vertical plane perpendicular to the wall and makes an angle of 60° with the floor as shown in Figure 1. The coefficient of friction between the rod and the floor is and the coefficient of friction between the rod and the wall is . The rod is on the point of slipping at both ends.

Find the magnitude of the vertical component of the force exerted on the rod by the floor.

The centre of mass of the rod is at G.

Find the distance AG.

A uniform rod AB, of mass m and length 2 a, is freely hinged to a fixed point A. A particle of mass m is attached to the rod at B. The rod is held in equilibrium at an angle to the horizontal by a force of magnitude F acting at the point C on the rod, where AC = b, as shown in Figure 3. The force at C acts at right angles to AB and in the vertical plane containing AB.

Find, in terms of a, b, g, m and ,

the horizontal component of the force acting on the rod at A,

the vertical component of the force acting on the rod at A.

Given that the force acting on the rod at A acts along the rod,

Figure 1

A uniform rod AB, of mass 5 kg and length 4 m, has its end A smoothly hinged at a fixed point. The rod is held in equilibrium at an angle of 25° above the horizontal by a force of magnitude F newtons applied to its end B. The force acts in the vertical plane containing the rod and in a direction which makes an angle of 40° with the rod, as shown in Figure 1.

Find the value of F.

Find the magnitude and direction of the vertical component of the force acting on the rod at A.

A uniform rod AB, of mass 3 m and length 4 a, is held in a horizontal position with the end A against a rough vertical wall. One end of a light inextensible string BD is attached to the rod at B and the other end of the string is attached to the wall at the point D vertically above A, where AD = 3 a. A particle of mass 3 m is attached to the rod at C, where AC = x. The rod is in equilibrium in a vertical plane perpendicular to the wall as shown in Figure 3. The

Show that

x = 3 a,

the horizontal component of the force exerted by the wall on the rod has magnitude 5 mg.

The coefficient of friction between the wall and the rod is —. Given that the rod is about to slip,

find the value of —.

A triangular frame is formed by cutting a uniform rod into 3 pieces which are then joined to form a triangle ABC, where AB = AC = 10 cm and BC = 12 cm, as shown in Figure 1.

Find the distance of the centre of mass of the frame from BC.

The frame has total mass M. A particle of mass M is attached to the frame at the mid-point of BC. The frame is then freely suspended from B and hangs in equilibrium.

Find the size of the angle between BC and the vertical.

A wooden beam AB, of mass 150 kg and length 9 m, rests in a horizontal position supported by two vertical ropes. The ropes are attached to the beam at C and D, where AC = 1.5 m and BD = 3.5 m. A gymnast of mass 60 kg stands on the beam at the point P, where AP = 3 m, as shown in Figure 2. The beam remains horizontal and in equilibrium. By modelling the gymnast as a particle, the beam as a uniform rod and the ropes as light inextensible strings,

find the tension in the rope attached to the beam at C.

The gymnast at P remains on the beam at P and another gymnast, who is also modelled as a particle, stands on the beam at B. The beam remains horizontal and in equilibrium. The mass of the gymnast at B is the largest possible for which the beam remains horizontal and in equilibrium.

Find the tension in the rope attached to the beam at D.

A plank AB has length 6m and mass 30kg. The point C is on the plank with CB = 2m. The plank rests in equilibrium in a horizontal position on supports at A and C. Two people, each of mass 75kg, stand on the plank. One person stands at the point P of the plank, where AP = x metres, and the other person stands at the point Q of the plank, where AQ = 2 x metres. The plank remains horizontal and in equilibrium with the magnitude of the reaction at C five times the magnitude of the reaction at A. The plank is modelled as a uniform rod and each person is modelled as a particle.

Find the value of x.

State two ways in which you have used the assumptions made in modelling the plank as a uniform rod.

A non-uniform plank AB has length 6 m and mass 30 kg. The plank rests in equilibrium in a horizontal position on supports at the points S and T of the plank where AS = 0.5 m and TB = 2 m. When a block of mass M kg is placed on the plank at A, the plank remains horizontal and in equilibrium and the plank is on the point of tilting about S. When the block is moved to B, the plank remains horizontal and in equilibrium and the plank is on the point of tilting about T. The distance of the centre of mass of the plank from A is d metres. The block is modelled as a particle and the plank is modelled as a non-uniform rod. Find

the value of d,

the value of M.

A non-uniform beam AD has weight W newtons and length 4 m. It is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. The ropes are attached to two points B and C on the beam, where AB = 1 m and CD = 1 m, as shown in Figure 3. The tension in the rope attached to C is double the tension in the rope attached to B. The beam is modelled as a rod and the ropes are modelled as light inextensible strings.

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

A small load of weight kW newtons is attached to the beam at D. The beam remains in equilibrium in a horizontal position. The load is modelled as a particle. Find

an expression for the tension in the rope attached to B, giving your answer in terms of k and W,

the set of possible values of k for which both ropes remain taut.

A beam AB has length 15 m. The beam rests horizontally in equilibrium on two smooth supports at the points P and Q, where AP = 2 m and QB = 3 m. When a child of mass 50 kg stands on the beam at A, the beam remains in equilibrium and is on the point of tilting about P. When the same child of mass 50 kg stands on the beam at B, the beam remains in equilibrium and is on the point of tilting about Q. The child is modelled as a particle and the beam is modelled as a non-uniform rod.

Find the mass of the beam.

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

When the child stands at the point X on the beam, it remains horizontal and in equilibrium. Given that the reactions at the two supports are equal in magnitude,

find AX.

A uniform rod AB has length 2 m and mass 50 kg. The rod is in equilibrium in a horizontal position, resting on two smooth supports at C and D, where AC = 0.2 metres and DB = x metres, as shown in Figure 5. Given that the magnitude of the reaction on the rod at D is twice the magnitude of the reaction on the rod at C,

find the value of x.

The support at D is now moved to the point E on the rod, where EB = 0.4 metres. A particle of mass m kg is placed on the rod at B, and the rod remains in equilibrium in a horizontal position. Given that the magnitude of the reaction on the rod at E is four times the magnitude of the reaction on the rod at C,

find the value of m.

A steel girder AB, of mass 200 kg and length 12 m, rests horizontally in equilibrium on two smooth supports at C and at D, where AC = 2 m and DB = 2 m. A man of mass 80 kg stands on the girder at the point P, where AP = 4 m, as shown in Figure 1.

The man is modelled as a particle and the girder is modelled as a uniform rod.

Find the magnitude of the reaction on the girder at the support at C.

The support at D is now moved to the point X on the girder, where XB = x metres. The man remains on the girder at P, as shown in Figure 2.

Given that the magnitudes of the reactions at the two supports are now equal and that the girder again rests horizontally in equilibrium, find

the magnitude of the reaction at the support at X,

the value of x.

A non-uniform rod AB has length 3 m and mass 4.5 kg. The rod rests in equilibrium, in a horizontal position, on two smooth supports at P and at Q, where AP = 0.8 m and QB = 0.6 m, as shown in Figure 1. The centre of mass of the rod is at G. Given that the magnitude of the reaction of the support at P on the rod is twice the magnitude of the reaction of the support at Q on the rod, find

the magnitude of the reaction of the support at Q on the rod,

the distance AG.

A plank PQR, of length 8 m and mass 20 kg, is in equilibrium in a horizontal position on two supports at P and Q, where PQ = 6 m. A child of mass 40 kg stands on the plank at a distance of 2 m from P and a block of mass M kg is placed on the plank at the end R. The plank remains horizontal and in equilibrium. The force exerted on the plank by the support at P is equal to the force exerted on the plank by the support at Q. By modelling the plank as a uniform rod, and the child and the block as particles,

find the magnitude of the force exerted on the plank by the support at P,

find the value of M.

State how, in your calculations, you have used the fact that the child and the block can be modelled as particles.

A uniform beam AB has mass 20 kg and length 6 m. The beam rests in equilibrium in a horizontal position on two smooth supports. One support is at C, where AC = 1 m, and the other is at the end B, as shown in Figure 1. The beam is modelled as a rod.

Find the magnitudes of the reactions on the beam at B and at C.

A boy of mass 30 kg stands on the beam at the point D. The beam remains in equilibrium. The magnitudes of the reactions on the beam at B and at C are now equal. The boy is modelled as a particle.

Find the distance AD.

A beam AB has length 6 m and weight 200 N. The beam rests in a horizontal position on two supports at the points C and D, where AC 1 m and DB 1 m. Two children, Sophie and Tom, each of weight 500 N, stand on the beam with Sophie standing twice as far from the end B as Tom. The beam remains horizontal and in equilibrium and the magnitude of the reaction at D is three times the magnitude of the reaction at C. By modelling the beam as a uniform rod and the two children as particles, find how far Tom is standing from the end B.

A pole AB has length 3 m and weight W newtons. The pole is held in a horizontal position in equilibrium by two vertical ropes attached to the pole at the points A and C where AC = 1.8 m, as shown in Figure 2. A load of weight 20 N is attached to the rod at B. The pole is modelled as a uniform rod, the ropes as light inextensible strings and the load as a particle.

Show that the tension in the rope attached to the pole at C is W N.

Find, in terms of W, the tension in the rope attached to the pole at A.

Given that the tension in the rope attached to the pole at C is eight times the tension in the rope attached to the pole at A,

find the value of W.

A beam AB is supported by two vertical ropes, which are attached to the beam at points P and Q, where AP = 0.3 m and BQ = 0.3 m. The beam is modelled as a uniform rod, of length 2 m and mass 20 kg. The ropes are modelled as light inextensible strings. A gymnast of mass 50 kg hangs on the beam between P and Q. The gymnast is modelled as a particle attached to the beam at the point X, where PX = x m, 0 < x < 1.4 as shown in Figure 2. The beam rests in equilibrium in a horizontal position.

Show that the tension in the rope attached to the beam at P is (588 – 350 x) N.

Find, in terms of x, the tension in the rope attached to the beam at Q.

Hence find, justifying your answer carefully, the range of values of the tension which could occur in each rope.

Given that the tension in the rope attached at Q is three times the tension in the rope attached at P,

find the value of x.

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.

Find the tension in the rope attached at 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.

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.

Find

the tension in the rope at C,

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,

find, in terms of y, an expression for the tension in the rope at C.

The rope at C will break if its tension exceeds 98 N. The rope at A cannot break.

Find the range of possible positions on the beam where the load can be attached without the rope at C breaking.