Moments (A Level only) (AQA A Level Maths: Mechanics)

In each of the following examples the force indicated is acting on a lamina. Calculate the moment about the point  in each case.

The diagram below shows a set of forces acting on a light rod. Calculate the resultant moment about the point P.

The diagram below shows a set of forces acting on a lamina. Calculate the resultant moment about the point .

is a uniform rod of length 1.7 m and weight 50 N.  rests horizontally on supports placed at points  and , with  m,  m and  m as shown in the diagram below:

Calculate the magnitude of the reaction force at each of the support points.

is a non-uniform rod of length 1 m and weight 30 N.  rests horizontally on supports placed at points  and , with  and  m as shown in the diagram below:

Given that the centre of mass of  is 0.45 m from point , calculate the reaction force at each of the support points.

In the diagram below  is a uniform plank of length 8 m. It rests horizontally on two supports, one of which is placed at point B and the other of which is placed 2.4 m from point  as shown:

A man with a weight of 728 N stands on the plank at point  and begins to walk towards point . When he has gone a distance of 0.6 m, the plank is on the point of tilting.

By modelling the plank as a uniform rod and the man as a particle, use the information above to calculate the weight of the plank.

The man would like to be able to stand at point  without the plank tilting. In order to allow him to do this, he decides to place a large rock on the plank at point .

Given that the rock may also be modelled as a particle, find the minimum weight of the rock that the man would need.

is a non-uniform rod of mass 10 kg and length 3 m, with a load of mass 26 kg attached at point .  is held horizontally in equilibrium by two vertical wires attached at points  and , such that  m as shown in the diagram below:

The position of the centre of mass of the rod is indicated by point . The load at  may be modelled as a particle.

Given that  m,

show that the rod is on the point of tilting about point .

The load is then removed from point , and the rod is left suspended in horizontal equilibrium from the two wires.

Determine the tensions in the wires at  and  after the load is removed, giving your answer in terms of the gravitational constant of acceleration .

is a triangular lamina in which angle  is a right angle, and the lengths of sides and  are 58 cm and 42 cm respectively. Three forces are applied to the lamina at points  and  as shown in the diagram below:

Calculate the resultant moments of the three forces about each of the points  and .

 is a uniform rod of mass 6 kg and length 2 m, with a load of mass kg attached at point .  is held horizontally in equilibrium by two vertical wires attached at points  and , such that  m as shown in the diagram below:

The tension in the wire at  is found to be eight times the tension in the wire at . By modelling the load at  as a particle, find:

the value of

the tensions in the wires at and .

Your answers to (ii) should be given as multiples of the gravitational constant of acceleration .

In the diagram below  is a uniform beam of length 4 m. It rests horizontally on two supports placed at points and , such that and  m as shown:

A stone of mass 10 kg is placed at point  and the beam is on the point of tilting. That stone is removed, and another stone of mass  kg is placed at point  which causes the beam to begin tilting.

Given that the stones may be modelled as particles, show that , where  is the largest possible constant for which that inequality must be true.

is a non-uniform rod of mass 12 kg and length 4 m.  is held horizontally in equilibrium by a support placed at point  and a vertical wire attached to point  such that  m and  m as shown in the diagram below:

The distance from point  to the centre of mass of the rod is 1.75 m.

Find the ratio of the reaction force at  to the tension in the wire at . Give your answer in the form  where  and  are integers with no common factors other than 1.

A weight of mass 15 kg is attached to the rod between points  and .

Find the greatest distance to the left of point  that the weight can be attached without the rod beginning to tilt.

 is a uniform rod of mass 4 kg and length 1.2 m. is held horizontally in equilibrium by two vertical wires attached 0.8 m apart at points  and , where  is 0.3 m from  as shown in the diagram below.

A particle of mass  kg is attached to  at the point , such that  remains in horizontal equilibrium and the tensions in the wires at  and  are equal.

Given that point  is in between points  and , show that .

After laying one of the stone blocks for his new pyramid, the architect Hemiunu realises that his wife's favourite scarab pendant has been left on the ground underneath the block. Therefore he decides to tilt the block up on one of its edges so that the pendant may be retrieved.

The block is a cuboid with weight 98 kN, but it may be modelled as a rectangular lamina with side lengths  m and  m. The centre of mass may be assumed to be at the intersection of the diagonals  and . The block is tilted by means of a horizontal rope attached at point , with tension in the rope causing the block to pivot around point . As the block is being tilted side  makes an angle of  with the ground as shown in the diagram below:

The frictional force between the block and the ground at  is at all times sufficient to prevent the block from slipping.

The block is raised until point  is a vertical distance of 1.7 m from the ground. The rope is then used to hold the block stationary while the pendant is retrieved.

Given that the rope remains horizontal, find the tension  in the rope while the block is being held stationary.