A LevelMathsEdexcelMechanicsExam QuestionsForces & Newton's LawsForces & Equilibrium

Forces & Equilibrium (Edexcel A Level Maths: Mechanics): Exam Questions

Exam code: 9MA0

2 hours33 questions
Easy
Medium
Hard
Very Hard
1
2 marks

Newton's First Law of Motion states that an object at rest will remain at rest, and an object moving with constant velocity will continue to move with constant velocity, unless an unbalanced force acts on the object.

An avocado with a mass of 170 g is lying motionless on top of a table. State, with a reason, what the resultant force acting on the avocado is.

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2
3 marks

The normal reaction is the force that acts in a direction perpendicular to a surface when an object is in contact with the surface. In practical terms, it is what stops things from falling straight through solid objects!

A large maths textbook with a weight of 25 N is sitting at rest on a horizontal table. State, with a reason, the size and direction of the normal reaction force acting on the book.

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3
2 marks

The resultant force on an object in a particular direction is the total force acting on the object in that direction.

Given that the resultant forces in the following force diagrams are all zero, work out the value of P:

(i)

2-6-q3-a-level-maths-mechanics

(ii)

2-6-e-q3-ii-a-level-maths-mechanics

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4
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3 marks

An object is said to be in equilibrium when all the forces acting on it are balanced — i.e., when for any force acting in a given direction there is a force of equal magnitude acting in the opposite direction.

In the following force diagram, the particle shown is in equilibrium:

2-6-q4-a-level-maths-mechanics

(i) Explain why the following two equations must be satisfied:

P + 8Q = 21

3P - 2Q = 11

(ii) By solving the simultaneous equations in part (i), determine the values of P and Q.

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5a
2 marks

A wrecking ball of weight 24500 N is being lifted vertically by a crane using a cable. The tension in the cable is T newtons.

2-6-q5-a-level-maths-mechanics

The wrecking ball is modelled as a particle and the cable is modelled as a light inextensible string. Air resistance is modelled as being negligible.

Given that the wrecking ball is moving vertically upwards at a constant velocity, find the value of T. Give a reason for your answer.

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5b
2 marks

The tension in the cable is now increased by 3000 N.

Describe the subsequent motion of the wrecking ball, giving a reason for your answer.

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6a
4 marks

A car with a weight of 12000 N is moving along a horizontal level road. The car's engine provides a forward thrust of 8500 N. The total resistance is modelled by a constant force of 1000 N. All other forces acting on the car may be ignored.

(i) Write down the magnitude and direction of the normal reaction force exerted on the car by the road.

(ii) Modelling the car as a particle, draw a force diagram to show the forces acting on the car.

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6b
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2 marks

Calculate the magnitude and direction of the resultant force acting on the car.

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7a
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3 marks

A particle is acted upon by two forces, \mathbf{F}_1 and \mathbf{F}_2, where

\mathbf{F}_1 = (14\mathbf{i} + 3\mathbf{j})\text{ N}

\mathbf{F}_2 = (-12\mathbf{i} + 7\mathbf{j})\text{ N}

The resultant force of \mathbf{F}_1 and \mathbf{F}_2 is \mathbf{R}.

Find \mathbf{R} and calculate the magnitude of \mathbf{R}.

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7b
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3 marks

By drawing a sketch, or otherwise, show that the angle between \mathbf{R} and the vector \mathbf{i} is 78.7^{\circ} to 3 significant figures.

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8a
2 marks

A particle P is acted upon by three forces, \mathbf{F}_1, \mathbf{F}_2 and \mathbf{F}_3, given by

\mathbf{F}_1 = (9\mathbf{i} - 2\mathbf{j})\text{ N}

\mathbf{F}_2 = (-14\mathbf{i} + 5\mathbf{j})\text{ N}

\mathbf{F}_3 = (a\mathbf{i} + b\mathbf{j})\text{ N}

where a and b are scalar constants.

The resultant of the three forces is \mathbf{R}.

Find \mathbf{R} in terms of a and b.

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8b
2 marks

Given that P is in equilibrium,

find the value of a and the value of b.

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1
2 marks

During a weightlifting competition, one of the competitors successfully lifts a barbell with a total mass of 135 kg and holds it stationary above her head with her elbows locked.

State, with a reason, what the resultant force acting on the barbell is during the time that it is being held stationary above the weightlifter's head.

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2
3 marks

As part of an acrobatics display, one acrobat with a weight of 735 N is standing motionless on a horizontal floor. Another acrobat with a weight of 495 N is standing motionless on the first acrobat's shoulders.

State, with a reason, what the total normal reaction force exerted by the floor on the first acrobat is, and in what direction it is acting.

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3
3 marks

Given that the particle in each of the following diagrams is stationary, work out the value of P:

(i)

2-6-m-q3-a-level-maths-mechanics

(ii)

2-6-m-q3-ii-a-level-maths-mechanics

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4
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3 marks

Figure 1 shows a set of coplanar forces acting on a particle.

2-6-m-q4-a-level-maths-mechanics
Figure 1

Given that the particle is in equilibrium,

find the value of P and the value of Q.

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5a
2 marks

A pallet of stone blocks of total weight 22000 N is being raised by a crane using two vertical cables. The tension in each cable is T newtons.

2-6-m-q5-a-level-maths-mechanics

The pallet and stone blocks are modelled as a single particle and the cables are modelled as light inextensible strings. Air resistance is modelled as being negligible.

Given that the pallet is moving vertically upwards at a constant speed, find the value of T.

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5b
2 marks

The tension in each cable is now increased by 1000 N.

Describe the subsequent motion of the pallet, giving a reason for your answer.

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6a
3 marks

A truck with a total weight of 27000 N is moving along a horizontal level road. The truck's engine provides a forward thrust of 13500 N. The total resistance is modelled by a constant force of 1700 N. All other forces acting on the truck may be ignored.

Modelling the truck as a particle, draw a force diagram to show the forces acting on the truck.

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6b
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2 marks

Calculate the magnitude and direction of the resultant force acting on the truck.

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7a
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2 marks

A particle P is acted upon by two forces, \mathbf{F}_1 and \mathbf{F}_2, where

\mathbf{F}_1 = (5\mathbf{i} + 7\mathbf{j})\text{ N}

\mathbf{F}_2 = (-3\mathbf{i} + b\mathbf{j})\text{ N}

and b is a constant.

Find the angle between the direction of \mathbf{F}_1 and the vector \mathbf{j}. Give your answer in degrees to one decimal place.

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7b
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3 marks

The resultant of the two forces is \mathbf{R}.

Given that \mathbf{R} is parallel to the vector \mathbf{i} - 13\mathbf{j}, find the value of b.

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8a
3 marks

[In this question, \mathbf{i} and \mathbf{j} are perpendicular horizontal unit vectors.]

A particle P is acted upon by three forces, \mathbf{F}_1, \mathbf{F}_2 and \mathbf{F}_3, given by

\mathbf{F}_1 = (-3\mathbf{i} + 5\mathbf{j})\text{ N}

\mathbf{F}_2 = (12\mathbf{i} + 2\mathbf{j})\text{ N}

\mathbf{F}_3 = (a\mathbf{i} + b\mathbf{j})\text{ N}

where a and b are scalar constants.

Given that P is in equilibrium, find the value of a and the value of b.

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8b
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5 marks

The force \mathbf{F}_2 is now removed. The resultant of the remaining forces acting on P is \mathbf{R}.

Find the magnitude of \mathbf{R} and the size of the angle between the direction of \mathbf{R} and the vector \mathbf{i}. Give your angle in degrees to one decimal place.

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9a
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3 marks

[In this question, bold i and bold j are perpendicular unit vectors in a horizontal plane.]

A particle is being acted upon by two forces, F subscript 1 and F subscript 2, given in vector form

F subscript 1 equals open parentheses 5 bold i plus 8 bold j close parentheses bold space straight N space and space F subscript 2 equals open parentheses 2 a bold i plus a bold j close parentheses space straight N

where a is a constant.

Find the angle F subscript 1makes with the vector bold j. Give your answer to the nearest degree.

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9b
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3 marks

The resultant force acting on the particle, R, is acting in a parallel direction to vector 4 bold i plus 3 bold j. Find the value of constant a.

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1
2 marks

On a building site, a crane is being used to lift a pallet of bricks. The pallet has a total mass of 1600 kg, and it is moving vertically upwards at a constant speed of 0.5\text{ m s}^{-1}.

State, with a reason, what the resultant force acting on the pallet is.

How did you do?

2
3 marks

A yoga practitioner with a weight of 686 N is standing motionless on a horizontal floor with both feet on the ground. He changes to a new pose where one leg is lifted in the air, so that he is now standing motionless with only one foot on the ground.

State, with a reason, what the difference is between the total normal reaction force exerted by the floor on the yoga practitioner when he has both feet on the ground, and the total normal reaction force when he has only one foot on the ground.

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3
3 marks

Given that the particle in each of the following diagrams is in equilibrium, work out the value of P:

(i)

q3-h-a-level-maths-mechanics

(ii)

q3-ii-h-a-level-maths-mechanics

How did you do?

4
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3 marks

Figure 1 shows a set of coplanar forces acting on a particle.

Diagram of forces acting on a point with arrows: up (P+3Q) N, down (11Q−2P) N, left (2P+13) N, right (15Q−2P) N; velocity v upwards.
Figure 1

Given that the particle is moving with constant velocity,

find the value of P and the value of Q.

How did you do?

5a
2 marks

A pallet of cheeses of total weight 17000 N is being lowered into a cave using two vertical ropes. The tension in each rope is T newtons.

q5-h-a-level-maths-mechanics

The pallet and cheeses are modelled as a single particle and the ropes are modelled as light inextensible strings. Air resistance is modelled as being negligible.

Given that the pallet is moving vertically downwards at a constant speed, find the value of T.

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5b
2 marks

The tension in each rope is now increased by 500 N.

Describe the motion of the pallet immediately following this increase, giving a reason for your answer.

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6a
3 marks

A train locomotive is moving along a horizontal level track. The locomotive's engine provides a forward thrust of 75\,000 N. The total resistance is modelled by a constant force of F N. The locomotive is not pulling any other train cars.

Modelling the locomotive as a particle, draw a force diagram to show the forces acting on the locomotive.

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6b
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2 marks

Given that the resultant force acting on the locomotive is 71\,800 N in the direction of the locomotive's motion, find the value of F.

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7a
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2 marks

[In this question, \mathbf{i} and \mathbf{j} are horizontal unit vectors due east and due north respectively.]

A particle P is acted upon by two forces, \mathbf{F}_1 and \mathbf{F}_2, where

\mathbf{F}_1 = (-7\mathbf{i} - 2\mathbf{j})\text{ N}

\mathbf{F}_2 = (5a\mathbf{i} + a\mathbf{j})\text{ N}

and a is a scalar constant.

Find the angle between the direction of \mathbf{F}_1 and the vector -\mathbf{i}. Give your answer in degrees to one decimal place.

How did you do?

7b
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4 marks

The resultant of the two forces is \mathbf{R}.

Given that \mathbf{R} is parallel to the vector 6\mathbf{i} + \mathbf{j}, find the value of a.

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8a
3 marks

[In this question, \mathbf{i} and \mathbf{j} are perpendicular horizontal unit vectors.]

A particle P is acted upon by three forces, \mathbf{F}_1, \mathbf{F}_2 and \mathbf{F}_3, given by

\mathbf{F}_1 = (a\mathbf{i} - 3\mathbf{j})\text{ N}

\mathbf{F}_2 = (-9\mathbf{i} + b\mathbf{j})\text{ N}

\mathbf{F}_3 = (2a\mathbf{i} + 8\mathbf{j})\text{ N}

where a and b are scalar constants.

Given that P is in equilibrium, find the value of a and the value of b.

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8b
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5 marks

The magnitude of the force \mathbf{F}_3 is now halved, but its direction remains unchanged. The resultant of the three forces acting on P is now \mathbf{R}.

Find the magnitude of \mathbf{R} and the size of the angle between the direction of \mathbf{R} and the vector -\mathbf{i} - \mathbf{j}. Give your angle in degrees to one decimal place.

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1
3 marks

A communications satellite with a mass of 3600 kg orbits the earth in a circular orbit at an altitude of 1400 km. The satellite moves at a constant speed.

State, with a reason, whether there is any resultant force acting on the satellite.

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2
3 marks

A tourist of weight 1246 N is standing on the horizontal floor of a lift. The tourist is wearing a backpack of weight 352 N.

The lift is moving vertically upwards at a constant speed of 6.1\text{ m s}^{-1}.

The tourist and the backpack are modelled as a single particle.

State the magnitude and the direction of the normal reaction force exerted on the tourist by the floor of the lift. Give a reason for your answer.

How did you do?

3
3 marks

Given that the particle in each of the following diagrams is moving with constant velocity, v, work out the value of P:

(i)

mech-3-1-vh-q3i

(ii)

mech-3-1-vh-q3ii

How did you do?

4
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4 marks

The following force diagram depicts a particle in equilibrium:

mech-3-1-vh-q4

Given that 5F_4 = 9F_2 and 3F_1 + 5F_3 = 450, determine the values of F_1, F_2, F_3 and F_4.

How did you do?

5a
2 marks

A load of slates rests on a platform. The total weight of the platform and the slates is 21000 N. The platform is being raised vertically by two vertical cables. The tension in each cable is T newtons.

mech-3-1-vh-q5

The loaded platform is modelled as a single particle and the cables are modelled as light inextensible strings. Air resistance is modelled as being negligible.

Given that the platform is moving vertically upwards at a constant speed of 0.25\text{ m s}^{-1}, find the value of T.

How did you do?

5b
3 marks

As the platform rises, two workers with a combined weight of 1600 N jump onto the centre of the platform. The tension in the cables remains unchanged.

Describe the subsequent motion of the platform, considering both the time immediately after the workers jump on and the motion in the longer term. Give physical reasons for your answer.

How did you do?

6a
3 marks

A monorail train is moving along a horizontal level track. The train's engine provides a forward thrust of T N. The total resistance is modelled by a constant force of F N.

Modelling the train as a particle, draw a force diagram to show the forces acting on the train.

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6b
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3 marks

Given that

  • the train's speed is increasing in its direction of motion

  • the resultant force acting on the train is 4200 N

  • of T and F, 9 times the magnitude of the smaller force is equal to 2 times the magnitude of the larger force

find the values of T and F.

How did you do?

7a
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2 marks

[In this question, \mathbf{i} and \mathbf{j} are perpendicular horizontal unit vectors.]

A particle P is acted upon by two forces, \mathbf{F}_1 and \mathbf{F}_2, where

\mathbf{F}_1 = (-13\mathbf{i} + 3\mathbf{j})\text{ N}

\mathbf{F}_2 = (a\mathbf{i} + b\mathbf{j})\text{ N}

and a and b are scalar constants such that a \neq b.

Find the size of the angle between the direction of \mathbf{F}_1 and the vector -\mathbf{j}. Give your answer in degrees to one decimal place.

How did you do?

7b
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6 marks

The resultant of the two forces is \mathbf{R}.

Given that \mathbf{R} is parallel to the vector 3\mathbf{i} - \mathbf{j}, and that |\mathbf{R}| = \sqrt{160}, find the value of a and the value of b.

How did you do?

8a
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3 marks

A particle P is acted upon by three forces, \mathbf{F}_1, \mathbf{F}_2 and \mathbf{F}_3, given by

\mathbf{F}_1 = \begin{pmatrix} a \\ 2a \end{pmatrix}\text{ N}, \quad \mathbf{F}_2 = \begin{pmatrix} -7 \\ a + 2b \end{pmatrix}\text{ N}, \quad \mathbf{F}_3 = \begin{pmatrix} a + b \\ 12 \end{pmatrix}\text{ N}

where a and b are scalar constants.

Given that P is in equilibrium, find the value of a and the value of b.

How did you do?

8b
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6 marks

The force \mathbf{F}_1 is now removed, and another force \mathbf{F}_4 is applied to P in its place. The resultant of the three forces \mathbf{F}_2, \mathbf{F}_3 and \mathbf{F}_4 is \mathbf{R}.

Given that \mathbf{F}_4 is parallel to the vector \begin{pmatrix} 3 \\ 2 \end{pmatrix}, and that the angle between the direction of \mathbf{R} and the vector \begin{pmatrix} 0 \\ -1 \end{pmatrix} is 45^{\circ}, find the two possible exact values of the magnitude of \mathbf{R}.

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