OCR A Level Maths: Mechanics

Topic Questions

3.1 Forces

1
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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 total unbalanced force acting on the avocado is.

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2
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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
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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:

straight P space plus space 8 Q space equals 21
3 straight P space minus space 2 Q space equals space 11

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

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5a
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2 marks

A large steel wrecking ball with a weight of 24500 N is being lifted by a crane.  The tension in the lifting cable is T.  Other than the weight of the ball and the tension in the cable, all other influences on the motion of the ball can be ignored.

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

Given that the wrecking ball is moving vertically upwards at a constant velocity, use Newton’s First Law of Motion to find the value of T.  Be sure to give an explanation for your answer.

5b
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2 marks

The tension in the cable is now increased by 3000 N.  Using Newton’s First Law of Motion to justify your answer, describe the resulting motion of the wrecking ball.

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6a
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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.
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

For forces given in vector form, the resultant of the forces is the vector sum of the forces. So for two forces bold F subscript 1 and bold F subscript 2,  for example, the resultant R is given by the vector sum bold R space equals space bold F subscript 2 space plus space bold F subscript 1.

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

 bold F subscript 1 equals left parenthesis 14 bold i plus 3 bold j right parenthesis space straight N space space space space space space space space space and space space space space space space space space space bold F subscript 2 equals left parenthesis negative 12 bold i plus 7 bold j right parenthesis space straight N

The resultant of bold F subscript 1 and bold F subscript 2 is bold R.

Find R and calculate its magnitude.

7b
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3 marks

By drawing a sketch and using trigonometry, show that the angle between R and the vector i is  78.7° to 3 significant figures.

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8a
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2 marks

A particle is acted upon by three forces,  bold F subscript 1bold F subscript 2 and bold F subscript 3,  given in vector form as

bold F subscript 1 space space equals space open parentheses table row 9 row cell negative 2 end cell end table close parentheses space straight N space space space space space space space space space space bold F subscript bold 2 bold space equals space open parentheses table row cell negative 14 end cell row 5 end table close parentheses space straight N space space space space space space space space space space bold F subscript 3 space equals space open parentheses table row straight a row straight b end table close parentheses space straight N

where a and b are constants.

The resultant of forces  bold F subscript 1bold F subscript 2 and bold F subscript 3 is R

Find R in terms of a and b.

8b
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2 marks

An object is said to be in equilibrium when all the forces acting on it are balanced – i.e., when the resultant of all the vector forces acting on it is equal to zero.

Given that the particle acted upon by  bold F subscript 1bold F subscript 2 and bold F subscript 3 is in equilibrium,

find the values of the constants a and b.

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1
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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 space 0.5 space straight m space straight s to the power of negative 1 end exponent.  State, with a reason, what the resultant force acting on the pallet is.

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2
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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
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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

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4
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3 marks

The following force diagram depicts a particle moving with constant velocity, v :

q4-h-a-level-maths-mechanics

Determine the values of P and Q.

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5a
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2 marks

A pallet of large cheeses is being lowered into a cave for storage.  The total weight of the pallet and cheeses is 17000 N, and to make sure that the pallet remains level while it is being lowered, the same tension, T,  is maintained in each of the two support ropes.  Other than the weight of the pallet and cheeses, and the tension in the ropes, all other influences on the motion of the pallet can be ignored.

q5-h-a-level-maths-mechanics

Given that the pallet is moving vertically downwards at a constant speed, work out the tension, T,  in each rope.

5b
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2 marks

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

Describe the resulting motion of the pallet immediately following the change in tension, being sure to explain the physical reasoning behind your answer.

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6a
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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.

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

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

bold F subscript 1 equals left parenthesis negative 7 bold i minus 2 bold j right parenthesis space straight N space and space bold F subscript 2 equals left parenthesis 5 a bold i plus a bold j right parenthesis space straight N

where a is a constant.

Find the angle between bold F subscript 1 and negative bold i.

7b
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4 marks

The resultant of bold F subscript 1and bold F subscript 2 is R.

Given that R is parallel to the vector 6 bold i bold space plus bold space bold j, find the value of a.

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8a
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3 marks

A particle is acted upon by three forces, F1, F2 and F3, given in vector form as

bold F subscript 1 equals open parentheses table row a row cell negative 3 end cell end table close parentheses space straight N space space space space space space space space space space bold F subscript 2 equals open parentheses table row cell negative 9 end cell row b end table close parentheses space straight N space space space space space space space space space space bold F subscript 3 equals open parentheses table row cell 2 a end cell row 8 end table close parentheses space straight N

where a and b are constants.

Given that the particle is in equilibrium,

find the values of the constants a and b.

8b
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5 marks

The magnitude of force F3 is now halved, while its direction remains the same.  The resultant of the three forces after this change is R.

Find the magnitude of R and the angle that it makes with the vector  open parentheses table row cell negative 1 end cell row cell negative 1 end cell end table close parentheses.

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1
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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
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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
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3 marks

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

The following force diagram depicts a particle in equilibrium:

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

By setting up and solving a pair of simultaneous equations, find the values of P and Q.

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5a
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2 marks

On a building site, a pallet of large stone blocks is being raised by a crane.  The total weight of the pallet and blocks is 22000 N, and to make sure that the pallet remains level while it is being raised, the same tension, T,  is maintained in each of the two lifting cables.  Other than the weight of the pallet and blocks, and the tension in the cables, all other influences on the motion of the pallet can be ignored.

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

Given that the pallet is moving vertically upwards at a constant speed, work out the tension, T,  in each cable.

5b
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2 marks

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

Describe the resulting motion of the pallet, being sure to explain the physical reasoning behind your answer.

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6a
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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.

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 is acted upon by two forces, bold F subscript 1 and bold F subscript 2 ,  given in vector form as

bold F subscript 1 equals left parenthesis 5 bold i plus 7 bold j right parenthesis space straight N space space space space space space space and space space space space space space space bold F subscript 2 equals left parenthesis negative 3 bold i plus b bold j right parenthesis space straight N

where b is a constant.

Find the angle between bold F subscript 1 and bold j.

7b
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3 marks

The resultant of bold F subscript 1 and bold F subscript 2 is bold R.

Given that R is parallel to the vector bold i space minus space 13 bold j,  find the value of b.

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8a
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3 marks

A particle is acted upon by three forces, bold F subscript 1bold F subscript 2 and bold F subscript 3,  given in vector form as

 bold F subscript 1 bold space equals space open parentheses table row cell negative 3 end cell row 5 end table close parentheses space straight N space space space space space space space space bold F subscript 2 bold space equals bold space open parentheses table row 12 row 2 end table close parentheses bold space straight N space space space space space space space space bold F subscript 3 bold space equals space open parentheses table row a row b end table close parentheses space straight N space space space space space space space space space space

where a and b are constants.

Given that the particle is in equilibrium,

find the values of the constants a and b.

8b
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5 marks

The force bold F subscript 2 is now removed.  The resultant of forces bold F subscript 1 and bold F subscript 3 is R.

Find the magnitude of R and the angle that it makes with the horizontal.

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1
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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
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3 marks

A tourist with a weight of 1246 N is riding in a lift to the top of the Empire State Building in New York City.  On his back he has a backpack full of  ‘I Love New York’ souvenirs, with a total weight of 352 N.  The lift is moving vertically upwards at a constant speed of 6.1 space straight m space straight s to the power of negative 1 end exponent , and the tourist is standing still on the horizontal floor of the lift waiting for it to reach the top. 

State, with a reason, the magnitude and direction of the total normal reaction force exerted on the tourist by the floor of the lift.

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3
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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

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4
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4 marks

The following force diagram depicts a particle at equilibrium:

mech-3-1-vh-q4

Given that space 5 F subscript 4 space equals space 9 F subscript 2and 3 F subscript 1 plus space 5 F subscript 3 space equals space 450,  determine the values of space F subscript 1 comma space F subscript 2 comma space F subscript 3 space and space F subscript 4.

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5a
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2 marks

In a slate quarry a load of slates is being raised to the surface for processing.  The total weight of the slates and the platform on which they rest is 21000 N, and to make sure that the platform remains level while it is being raised, the same tension, T,  is maintained in each of the two lifting cables.  Other than the weight of the platform and slates, and the tension in the cables, all other influences on the motion of the platform can be ignored.

mech-3-1-vh-q5

Given that the platform is moving vertically upwards at a constant speed of 0.25 space straight m space straight s to the power of negative 1 end exponent, work out the tension, T,  in each cable.

5b
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3 marks

As it passes one of the levels in the quarry, two workers with a combined weight of 1600 N jump onto the centre of the slow-moving platform, hoping to catch a ride to the surface at the end of their shift.

Given that the tension in the cables stays the same, and that the platform remains perfectly level, describe the resulting motion of the platform after the workers jump onto it. Consider both the time immediately after they jump on as well as the longer term implications for the motion of the platform.  Be sure to explain the physical reasoning behind your answer.

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6a
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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.

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.

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7a
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2 marks

A particle is acted upon by two forces, F1 and F2,  given in vector form as

bold F subscript 1 equals left parenthesis negative 13 bold i plus 3 bold j right parenthesis space straight N space space space space space and space space space space space bold F subscript 2 equals left parenthesis a bold i plus b bold j right parenthesis space straight N

where a and b are constants with a space not equal to space b.

Find the angle between F1 and −j .

7b
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6 marks

The resultant of F1 and F2 is R.

Given that R is parallel to the vector 3 bold i space minus space bold j comma space and that open vertical bar bold R close vertical bar space equals space square root of 160,  find the values of a and b.

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8a
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3 marks

A particle is acted upon by three forces, F1, F2 and F3,  given in vector form as

 bold F subscript 1 equals open parentheses table row a row cell 2 a end cell end table close parentheses space straight N space space space space space space space space space space bold F subscript 2 equals open parentheses table row cell negative 7 end cell row cell a italic space plus space 2 b end cell end table close parentheses space straight N space space space space space space space space space space bold F subscript 3 equals open parentheses table row cell a italic space plus italic space b end cell row 12 end table close parentheses space straight N

where a and b are constants.

Given that the particle is in equilibrium,

find the values of the constants a and b.

8b
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6 marks

Force F1 is now removed, and another force F4 is applied in its place. The resultant of the three forces F2, F3 and F4 is R.

Given that F4 is parallel to the vector open parentheses table row 3 row 2 end table close parentheses and that the angle between R and the vector open parentheses table row 0 row cell negative 1 end cell end table close parentheses is 45°, find the two possible values of the magnitude of R. Your answer in each case should be given as an exact value.

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