AQA A Level Maths: Mechanics

Topic Questions

4.1 Moments (A Level only)

1
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4 marks

The moment of a force about a given point is found by multiplying the magnitude of the force by the perpendicular distance from the point to the line of action of the force:

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Note that the direction, clockwise or anticlockwise, must be specified when talking about a moment. The standard units for moments are newton metres (N m).

Calculate the moment about P of the forces indicated in each of the following diagrams:

q1-1-4-1-easy-aqa-a-level-maths-mechanics

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2
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4 marks

If a number of forces acting on a body all act in the same plane, it is possible to calculate the resultant moment of the forces about a given point. The resultant of a number of moments about a point is the total moment about that point.

To calculate the resultant moment, a 'positive' direction - clockwise or anticlockwise - must first be chosen. If clockwise is the positive direction, then the resultant moment in the clockwise direction is the sum of all the clockwise moments minus the sum of all the anticlockwise moments. If anticlockwise is the positive direction, then the resultant moment in the anticlockwise direction is the sum of all the anticlockwise moments minus the sum of all the clockwise moments. A negative result means that the resultant moment is in the 'negative' direction.

Calculate the resultant moment about P of the forces indicated in each of the following diagrams:

AqQDXNz5_q2-4-1-easy-aqa-a-level-maths-mechanics

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3a
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4 marks

A rigid body is said to be in equilibrium when the total force in any direction is zero and the total moment about any point is zero.

In problems involving rigid bodies, a judicious choice of which point to take the moments about can often simplify the problem.

In the following diagram A B is a light rod held in equilibrium by the three forces indicated:

q3a-4-1-easy-aqa-a-level-maths-mechanics

(i)
By considering the total force perpendicular to A B and the total moment about point B, show that the following simultaneous equations must hold:

P space plus space Q space equals space 32 space
10 P space plus space 2 Q space equals space 160

(ii)
Solve the simultaneous equations in part (i) to find the values of P and Q.
3b
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3 marks

Solve to find the values of P and Q by instead considering the total moment about point A.

3c
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1 mark

By comparing the methods used in parts (a) and (b), explain why it was more convenient to choose A as the point to take the moments about.

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4a
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3 marks

In rigid body problems involving rods, the weight of the rod may always be represented by a single force vector acting vertically downwards at the centre of mass of the rod. If the rod is a uniform rod then the centre of mass is at the midpoint. For a non-uniform rod, however, the centre of mass may lie anywhere along the length of the rod.

The following diagram depicts a rod A B of length 1 m and weight 30 N held horizontally in equilibrium by two supports at points C and D:

q4-1-4-1-easy-aqa-a-level-maths-mechanics

Besides the weight of the rod, the only forces acting on the rod are the reaction forces from the supports acting vertically upwards at points C and D.

For the case that A B is a uniform rod, calculate the magnitude of the reaction forces at points C andspace D.

4b
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3 marks

For the case that A B is a non-uniform rod with its centre of mass 0.1 m to the right of point C, calculate the magnitude of the reaction forces at points C and D.

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5a
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4 marks

When a rigid body is on the point of tilting or rotating about a pivot point, it means that the reaction force at any other support, or the tension in any other supporting wire or string, is zero.

In the diagram below A B is a uniform rod of length 5 m and weight 120 N. AB is held horizontally in equilibrium by two wires, one of which is attached at point B and the other of which is attached at point C where A C space equals space 2 m as shown:

GrbmXsj5_q5a-4-1-easy-aqa-a-level-maths-mechanics

A particle of weight 30 N is attached to the rod at point A, and the rod remains horizontally in equilibrium.

(i)
By considering moments around point C, show that the rod is on the point of tilting about C.
(ii)
Write down the tension in the wire attached at point C.
5b
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3 marks

In the diagram below A B is a uniform rod of length 1 m which rests horizontally on supports placed 0.3 m from either end at points C and D as shown:

q5b-4-1-easy-aqa-a-level-maths-mechanics

A particle of weight 24 N is placed at point B, and the rod is then at the point of rotating about D. By considering the moments about D, determine the weight of rod A B.

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1
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5 marks

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

q1-4-1-easy-aqa-a-level-maths-mechanics

Calculate the resultant moments of the three forces about each of the points A comma space B and C.

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2
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6 marks

A B is a uniform rod of mass 6 kg and length 2 m, with a load of mass m subscript B kg attached at point BA B is held horizontally in equilibrium by two vertical wires attached at points A and C, such that A C space equals space 1.5 m as shown in the diagram below:

q2-4-1-easy-aqa-a-level-maths-mechanics


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

(i)
the value of m subscript B
(ii)
the tensions in the wires at A and C.

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

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3
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6 marks

In the diagram below A B is a uniform beam of length 4 m. It rests horizontally on two supports placed at points C and D, such that A C space equals space 1.5 space m spaceand D B space equals space 1.2 m as shown:

q3-4-1-easy-aqa-a-level-maths-mechanics


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

Given that the stones may be modelled as particles, show that m subscript A greater than space k, where k is the largest possible constant for which that inequality must be true.

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4a
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4 marks

A B is a non-uniform rod of mass 12 kg and length 4 m. A B is held horizontally in equilibrium by a support placed at point C and a vertical wire attached to point D such that A C space equals space 0.8 m and D B space equals space 1 m as shown in the diagram below:

q4a-4-1-easy-aqa-a-level-maths-mechanics

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

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

4b
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3 marks

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

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

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5
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7 marks

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

q5a-4-1-easy-aqa-a-level-maths-mechanics

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

Given that point E is in between points D and B, show that 0.8 less than m subscript E less than 1.

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6
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7 marks

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 A B C D with side lengths A B space equals space C D space equals space 2.5 m and B C space equals space A D space equals space 1.5 m. The centre of mass may be assumed to be at the intersection of the diagonals A C and B D. The block is tilted by means of a horizontal rope attached at point A, with tension in the rope causing the block to pivot around point D. As the block is being tilted side D C makes an angle of theta with the ground as shown in the diagram below:

q6-4-1-easy-aqa-a-level-maths-mechanics

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

The block is raised until point C 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 T in the rope while the block is being held stationary.

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1
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4 marks

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

q1-4-1-medium-aqa-a-level-maths-mechanics

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2a
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3 marks

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

q2a-4-1-medium-aqa-a-level-maths-mechanics

2b
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3 marks

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

q2b-4-1-medium-aqa-a-level-maths-mechanics

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3
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4 marks

A B is a uniform rod of length 1.7 m and weight 50 N. A B rests horizontally on supports placed at points C and D, with A C space equals space 0.4 m, C D space equals space 1 m and D B space equals space 0.3 m as shown in the diagram below:

q3-4-1-medium-aqa-a-level-maths-mechanics

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

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4
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3 marks

A B is a non-uniform rod of length 1 m and weight 30 N. A B rests horizontally on supports placed at points C and D, with A C space equals space 0.35 space straight m and D B space equals space 0.25 m as shown in the diagram below:

q4-4-1-medium-aqa-a-level-maths-mechanics

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

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5a
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4 marks

In the diagram below A B 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 A as shown:

q5a-4-1-medium-aqa-a-level-maths-mechanics


A man with a weight of 728 N stands on the plank at point C and begins to walk towards point A. 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.

5b
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3 marks

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

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

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6a
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3 marks

A B is a non-uniform rod of mass 10 kg and length 3 m, with a load of mass 26 kg attached at point BA B is held horizontally in equilibrium by two vertical wires attached at points A and C, such that C B space equals space 0.5 m as shown in the diagram below:

q6a-4-1-medium-aqa-a-level-maths-mechanics

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

Given that A D space equals space 1.2 m,

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

6b
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4 marks

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

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

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