4.1 Moments

 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  as shown in the diagram below:

Calculate the resultant moment of the three forces about point .

 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  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 .

is a non-uniform rod of mass 10 kg and length 3 m.  rests horizontally on two supports placed at points  and , where  as shown in the diagram below:

Esmerelda attaches a 9 kg mass to the rod at a point 0.4 m to the left of , and measures the reaction force at . She then removes the mass and reattaches it to the rod at a point 0.4 m to the right of , and again measures the reaction force at . She finds that in her second measurement the reaction force at  is only one third the size of the reaction force at  in her first measurement.

By modelling the attached mass as a particle, use the above information to determine the position of the centre of mass of rod . Give the position in your answer as the distance of the centre of mass from point .

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

A weight of mass 15 kg is attached to the rod at point  and the rod is at the point of tilting about point . The weight is then removed.

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

The 15 kg weight is then 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.

A garden decoration consists of a wooden beam  of mass 8 kg and length 2 m supported at the point by a pole. There is a mass of  kg at the point which is designed to help keep the rod in horizontal equilibrium, and a hanging basket of mass 0.3 kg at the point where a plant can be placed. The rod can be moved back and forth on top of the pole so that the pole supports it at different places depending on the mass of the plant that is placed in the basket.

Modelling the mass as a particle and the beam as a uniform rod that contacts the vertical pole at a single point, draw a sketch of the decoration and add the forces acting on the beam.

Show that if no extra mass is added to the hanging basket and if the distance  is 50 cm, then the mass required at point to keep  in horizontal equilibrium is 12.22 kg.

For , find the distance  necessary to keep  in horizontal equilibrium if a plant of mass 3.2 kg is to be added to the hanging basket.

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

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 , 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  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.

Given that the reaction force at  is 9 N, find:

the location of the centre of mass of

the reaction force at .

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  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  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 the rod is on the point of tilting about , determine the location of the centre of mass of the rod. Give your answer as the value of , the distance of the centre of mass from 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 .

A model crane consists of a rod  of mass 0.8 kg and length 1.2 m supported at the point by a pole. The pole is not fixed to the rod, but rather rod  can be slid back and forth on top of the pole until point becomes such that the rod is balanced. 

There is a mass of 0.7 kg at the point  which is designed to help keep the rod in horizontal equilibrium, and a hanging bucket of mass kg at the point  where small toys can be placed.

Modelling the mass as a particle and the beam as a uniform rod that contacts the vertical pole at a single point, complete the diagram below by adding the missing forces acting on the toy crane.

Find the mass of the bucket if the rod is in equilibrium when the bucket is empty and . 

Given that the maximum possible length for  is 50 cm, find the maximum possible extra mass that can be added to the bucket with rod  still able to remain in horizontal equilibrium.

 is a triangular lamina in which the size of angle  is indicated by .  is the point on such that .  A 165 N force acts on point  in a direction perpendicular to  and a 420 N force acts on point  in a direction parallel to , as shown in the diagram below:

Given that the resultant force about point is in the clockwise direction, show that . 

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  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 .

A manager at the company Rods-We-Are has invented a device for locating the centre of mass of the 5 metre long barge poles that the company sells. He has connected force metres to two smooth supports located 2.5 m apart at the same horizontal level. A barge pole is placed on the supports so that it is held horizontally in equilibrium, with the supports located at points  and  as indicated in the diagram below:

The pole is then slid back and forth on the supports until a buzzer sounds, which indicates that the reaction force at  is exactly forty-nine times the reaction force at . The distance  from the left end of the pole in the diagram to is then measured. Finally, by modelling the barge pole as a rod, the value of  is used to calculate the distance between the left end of the pole in the diagram and the centre of mass of the pole.

A barge pole is placed on the device described above, and the buzzer sounds when . By first finding an expression for  in terms of , determine the location of the centre of mass of the barge pole.

The manager claims that although he has seen many non-uniform 5 metre barge poles, he has never found one for which the centre of mass could not be determined using his device. A new trainee manager claims that her calculations show that there could be 5 metre barge poles for which the device will not be able to determine the centre of mass.

Explain why both the manager and the trainee manager could be correct, supporting your answer with precise mathematical reasoning.

 is a non-uniform rod of length  and mass . A particle of mass  is attached to the rod at point , and the rod is then set to rest horizontally on two supports placed at points  and , with  and  as shown in the diagram below:

Given that the rod is at the point of tilting, find the two possible locations of the centre of mass of the rod.  In your answers give the distance of the centre of mass from point , with the values given in terms of .

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  and . 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.

A mathematics teacher has built a decoration for her classroom to celebrate ‘Pi Day’ on March 14^th. The decoration is in the shape of the letter , made of two identical poles and a non-uniform wooden beam of mass 1.4 kg and length 4 m. She attaches two identical baskets to the beam, one at point  hanging by a light inextensible string and the other fixed to the beam at point . The two poles are vertical and are placed 2 m apart, and the beam rests horizontally on top of them as shown in the diagram below.

When the beam is placed onto the poles with both baskets empty, and such that the midpoint of the beam is halfway between the two poles, then the beam is at the point of tilting about the pole closest to . With the beam in the same position a mass of 3.2 kg is placed into the basket at , and the beam is then at the point of tilting about the pole closest to .

Use the above information to find the distance of the centre of mass of the beam from , as well as the empty mass of each basket. In your solution you may treat the basket at , along with anything it contains, as being a point mass located at .