Define a scalar quantity.

A scalar is a quantity that has a size (magnitude) only, with no direction attached to it. Distance, speed, time and mass are the scalars you meet in mechanics, and none of them can be negative.

Define a vector quantity.

A vector is a quantity that has a direction as well as a size (magnitude). Displacement, velocity, acceleration, force, weight and momentum are all vectors, and their components can be positive or negative.

A ball rolls forwards 60 cm before stopping. Is that a scalar or a vector quantity, and why?

It is a vector . The number 60 cm on its own would be a distance , which is a scalar. The word forwards attaches a direction to it, and a quantity with both a size and a direction is a vector.

Define fundamental (S.I.) units .

Fundamental units , or S.I. units , are the internationally standardised units used to measure the basic quantities. The three needed in mechanics are the metre for length, the second for time and the kilogram for mass, and every other unit is built from them.

True or False? A time of 2 hours 30 minutes is 2.30 hours.

False. There are 60 minutes in an hour rather than 100, so 30 minutes is half an hour and the time is 2.5 hours .

Why must every quantity be converted into S.I. units before it is substituted into a mechanics formula?

The mechanics formulae relate metres , seconds and kilograms to one another, so they are only correct when every quantity is measured in those units. Substituting a distance in kilometres alongside a time in seconds gives an answer that is in no consistent unit at all, and so means nothing.

A force is a push or a pull on an object. It is a vector , so it has both a magnitude and a direction, and it is measured in newtons (N) .

True or False? Friction acts on an object in the direction in which it is moving.

False. Friction is a resistive force: it always acts to oppose the motion, so it points in the opposite direction to the way the object is moving. Air resistance behaves the same way, which is why a falling object meets a resistive force acting upwards.

In mechanics, define a particle .

A particle is an object modelled as having negligible dimensions , so that it occupies a single point in space. Because it has no size, air resistance cannot act on it and can be left out of the model.

A string is modelled as light . What does that mean, and what does it let you assume about the tension?

Light means the string is modelled as having zero mass . There is then no weight anywhere along it to be supported, so the tension is the same at every point of the string.

Two particles are joined by an inextensible string. What does inextensible mean, and what does it let you assume about the two particles?

Inextensible means the string cannot be stretched , so its length is fixed. Whatever distance one particle moves, the other must move the same distance, so at every moment the two have the same speed and accelerations of equal magnitude .

A pulley is modelled as light and smooth . What does the smooth part let you assume about the tension?

That the tension is the same on both sides of the pulley. A smooth pulley exerts no friction on the string passing over it, so nothing acts to make the tension on one side differ from the tension on the other.

What does it mean to model a rod as uniform , and what does that let you assume about its weight?

Uniform means the rod's mass is distributed evenly along its length. Its whole weight can therefore be taken to act at the midpoint of the rod, rather than being spread along it.

A rod is modelled as rigid . What does that let you assume about it?

That it does not bend or change shape under the forces acting on it. Its length stays fixed and the points at which the forces act stay in the same places, which is what makes it possible to work with distances measured along it.

International A Level (IAL)MathsEdexcelMechanics 1FlashcardsMathematical Models & VectorsQuantities, Units & Modelling

Quantities, Units & Modelling (Edexcel International A Level (IAL) Maths: Mechanics 1): Flashcards

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Define a scalar quantity.

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Cards in this collection (30)

Define a scalar quantity.
A scalar is a quantity that has a size (magnitude) only, with no direction attached to it.
Distance, speed, time and mass are the scalars you meet in mechanics, and none of them can be negative.
Define a vector quantity.
A vector is a quantity that has a direction as well as a size (magnitude).
Displacement, velocity, acceleration, force, weight and momentum are all vectors, and their components can be positive or negative.
Fill in the blanks to complete the three vector quantities of motion:
$______$ is the distance moved in a given direction from a starting point.
$______$ is a speed in a given direction.
$______$ is the change in velocity over time.
The completed statements are:
Displacement is the distance moved in a given direction from a starting point.
Velocity is a speed in a given direction.
Acceleration is the change in velocity over time.
A ball rolls forwards 60 cm before stopping. Is that a scalar or a vector quantity, and why?
It is a vector.
The number 60 cm on its own would be a distance, which is a scalar. The word forwards attaches a direction to it, and a quantity with both a size and a direction is a vector.
True or False?
Two objects with velocities of $7 {m s}^{- 1}$ and $- 7 {m s}^{- 1}$ are travelling at the same speed.
True.
Velocity is a vector, so the minus sign records only the direction of travel: the two objects are moving in opposite directions.
Speed is the scalar that goes with velocity, so it keeps the size and drops the direction, and both objects have a speed of $7 {m s}^{- 1}$ .
How are vector quantities distinguished from scalars in printed text, and how should you distinguish them in handwriting?
In print a vector is set in bold, non-italic type, as in $F$ or $a$ .
You cannot write in bold, so in handwriting a vector is underlined instead.
Define fundamental (S.I.) units.
Fundamental units, or S.I. units, are the internationally standardised units used to measure the basic quantities.
The three needed in mechanics are the metre for length, the second for time and the kilogram for mass, and every other unit is built from them.
Convert 27.8 km and 3.8 hours into S.I. units.
Multiply each by the number of smaller units contained in one larger one:
$27.8 \times 1000 = 27800 m$
$3.8 \times 3600 = 13680 s$
There are 1000 metres in a kilometre, and $60 \times 60 = 3600$ seconds in an hour.
True or False?
A time of 2 hours 30 minutes is 2.30 hours.
False.
There are 60 minutes in an hour rather than 100, so 30 minutes is half an hour and the time is 2.5 hours.
A mass is given as 0.054 g. Fill in the blanks to convert it into the S.I. unit of mass:
$0.054 \div______=______kg$
The completed conversion is:
$0.054 \div 1000 = 0.000054 kg$
There are 1000 grams in a kilogram, so a mass in grams is divided by 1000. An answer this small is usually written as $5.4 \times 10^{- 5} kg$ .
Why must every quantity be converted into S.I. units before it is substituted into a mechanics formula?
The mechanics formulae relate metres, seconds and kilograms to one another, so they are only correct when every quantity is measured in those units.
Substituting a distance in kilometres alongside a time in seconds gives an answer that is in no consistent unit at all, and so means nothing.
Define a derived unit.
A derived unit is a unit made by combining the fundamental S.I. units of length, time and mass.
A density measured in ${kg m}^{- 3}$ is one example, since it combines the kilogram with the metre.
Force $=$ mass $\times$ acceleration. What does that make the derived S.I. unit of force, and what is that unit normally called?
A mass in kilograms multiplied by an acceleration in ${m s}^{- 2}$ gives a unit of ${kg m s}^{- 2}$ .
That combination is normally called the newton, written N, so $1 N = 1 {kg m s}^{- 2}$ .
An acceleration is measured in ${m s}^{- 2}$ . Fill in the blanks to read that unit back as a formula:
$acceleration = \frac{______}{______}$
The completed formula is:
$acceleration = \frac{velocity}{time}$
${m s}^{- 2}$ is ${m s}^{- 1}$ divided again by seconds, and ${m s}^{- 1}$ is the unit of velocity. Reading a unit this way will give you back a formula you have forgotten.
Convert $72 {km h}^{- 2}$ into S.I. units.
Convert the kilometres and the hours separately, remembering that the hours are squared:
$72 \times 1000 \div 3600^{2} = 0.00556 {m s}^{- 2}$
Because the unit is $h^{- 2}$ rather than $h^{- 1}$ , the conversion factor for hours has to be squared as well.
True or False?
Speed and velocity are measured in the same derived unit.
True.
A direction is not something a unit can record, so attaching one to a speed leaves the unit unchanged: both are measured in ${m s}^{- 1}$ .
What tells you which of the two you are looking at is the sign or the vector notation, never the unit.
When converting $14 {g cm}^{- 3}$ into ${kg m}^{- 3}$ , why do you multiply by $100^{3}$ rather than divide?
Because the centimetres are in the denominator, so making the length unit larger makes the number larger.
There are $100^{3}$ cubic centimetres in a cubic metre, so a quantity measured per cubic centimetre becomes $100^{3}$ times as much per cubic metre:
$14 \div 1000 \times 100^{3} = 14000 {kg m}^{- 3}$
A cyclist takes 15 minutes to travel 2.54 km. Why is it worth converting both values into S.I. units before dividing?
Dividing kilometres by minutes would give an answer in ${km min}^{- 1}$ , which is not an S.I. unit, so it would have to be converted afterwards anyway.
Converting first gives 2540 m and 900 s, and the division then lands straight in the right unit:
$\frac{2540}{900} = 2.82 {m s}^{- 1}$
Define a force.
A force is a push or a pull on an object.
It is a vector, so it has both a magnitude and a direction, and it is measured in newtons (N).
Tension and thrust are both labelled $T$ N. What is the difference between them?
Tension is a pulling force, so it acts away from the object it is attached to, along the string or rod.
Thrust, also called compression, is a pushing force, so it acts towards the object.
Define the weight of an object.
An object's weight is the force on it produced by its mass and gravity, and it always acts vertically downwards.
It is found from $W = m g$ and, being a force, is measured in newtons.
True or False?
Friction acts on an object in the direction in which it is moving.
False.
Friction is a resistive force: it always acts to oppose the motion, so it points in the opposite direction to the way the object is moving.
Air resistance behaves the same way, which is why a falling object meets a resistive force acting upwards.
An astronaut takes a 12 kg toolbox to the Moon. Which of its mass and its weight changes, and why?
Its mass does not change.
Mass is a scalar measure of the amount of matter in an object, in kilograms, and it is the same everywhere in the universe.
Its weight does change, because weight depends on $g$ , the acceleration due to gravity, and $g$ is smaller on the Moon than on Earth.
In mechanics, define a particle.
A particle is an object modelled as having negligible dimensions, so that it occupies a single point in space.
Because it has no size, air resistance cannot act on it and can be left out of the model.
A string is modelled as light. What does that mean, and what does it let you assume about the tension?
Light means the string is modelled as having zero mass.
There is then no weight anywhere along it to be supported, so the tension is the same at every point of the string.
Fill in the blanks to complete the two surface assumptions:
A $______$ surface exerts no frictional force on an object in contact with it, while a $______$ surface does.
The completed statement is:
A smooth surface exerts no frictional force on an object in contact with it, while a rough surface does.
Modelling a surface as smooth is what allows friction to be left out of the equations altogether.
Two particles are joined by an inextensible string. What does inextensible mean, and what does it let you assume about the two particles?
Inextensible means the string cannot be stretched, so its length is fixed.
Whatever distance one particle moves, the other must move the same distance, so at every moment the two have the same speed and accelerations of equal magnitude.
A pulley is modelled as light and smooth. What does the smooth part let you assume about the tension?
That the tension is the same on both sides of the pulley.
A smooth pulley exerts no friction on the string passing over it, so nothing acts to make the tension on one side differ from the tension on the other.
What does it mean to model a rod as uniform, and what does that let you assume about its weight?
Uniform means the rod's mass is distributed evenly along its length.
Its whole weight can therefore be taken to act at the midpoint of the rod, rather than being spread along it.
A rod is modelled as rigid. What does that let you assume about it?
That it does not bend or change shape under the forces acting on it.
Its length stays fixed and the points at which the forces act stay in the same places, which is what makes it possible to work with distances measured along it.

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