CCEA GCE Mathematics (2210) · Quantities and units in mechanics
Mini-Lesson
Quantities and units in mechanics
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Mission 1Why can a 70 kg person be modelled as a dot?
The specification points and sources are listed under the lesson: see Sources.
Mission 1
Why can a 70 kg person be modelled as a dot?
Make a guess: Why can a 70 kg person be modelled as a dot?
Mechanics · quantities, units and modelling
SI units: base and derived
The SI base quantities are length (metre, m), mass (kilogram, kg) and time (second, s). They are independent of each other.
Derived units combine them: velocity m s−1⁻¹, acceleration m s−2⁻², force and weight in newtons (N).
From F=maF = ma, 1 N=1 kg m s−21 N = 1 kg m s⁻²: the force that gives 1 kg an acceleration of 1 m s−2⁻².
Moment is measured in N m and momentum in kg m s−1⁻¹ (both Unit A2 2).
Quick check
?Which of these is equivalent to 1 newton?
Worked exampleMechanics · quantities, units and modelling
Converting to SI units
Question
A car speeds up from 36 km h−1⁻¹ to 90 km h−1⁻¹ in 12 s. Find its average acceleration in m s−2⁻².
1
Convert km h−1⁻¹ to m s−1⁻¹: multiply by 1,000 and divide by 3,600, which is dividing by 3.6
2
Apply it: 36÷3.6=1036 ÷ 3.6 = 10 m s−1⁻¹ and 90÷3.6=2590 ÷ 3.6 = 25 m s−1⁻¹
3
Acceleration is change in velocity over time: a=1225−10a = (25 − 10)/12
Answer
a=1.25a = 1.25 m s−2⁻²
⚠️ Watch out
working in km h−1⁻¹ gives 1254=4.554/12 = 4.5, which is not in m s−2⁻². Convert first.
Calculate
?A runner covers 400 m in 50 s at a constant speed. Find the speed in km h−1⁻¹. Give your answer as a decimal.
km h−1⁻¹
Mechanics · quantities, units and modelling
Scalars, vectors and weight
A scalar has size only: distance, speed, mass, time. A vector has size and direction: displacement, velocity, acceleration, force.
Weight is the force of gravity on a mass, acting vertically downwards:
W=mgW = mg
Mass is in kg and never changes; weight is in N and depends on gg.
Use g=9.8g = 9.8 m s−2⁻² unless told otherwise (sometimes 9.81 or 10). gg is not a universal constant: it depends on location.
Sort it
?Is each quantity a scalar or a vector?
Tap a statement, then tap the group it belongs in. Then press Check.
Calculate
?A van has mass 3.2 tonnes. Taking g=9.8g = 9.8 m s−2⁻², find its weight. Give your answer in newtons to 3 significant figures.
N
Mechanics · quantities, units and modelling
The language of modelling
Word
Meaning in the model
What it lets you assume
Particle
all the mass acts at one point
size, shape and rotation are ignored
Light
mass negligible (string, rod, pulley)
tension is the same throughout a string
Inextensible
the string does not stretch
connected objects move with the same speed and acceleration
Smooth
no friction
contact forces act at right angles to the surface
Rigid
the body does not bend
it keeps its shape under forces
Uniform
mass spread evenly
the weight acts at the centre
A model gives predictions; compare them with reality and refine it, e.g. add air resistance.
Quick check
?A box on a smooth table is pulled by a light inextensible string that passes over a smooth pulley to a hanging weight. What does "inextensible" allow you to assume?
Mission check
?Why can a 70 kg person on a zip wire be modelled as a dot?
A particle model treats all of the mass as if it were concentrated at one point. That is reasonable when the object's size, shape and spinning do not noticeably change the motion you are studying, such as a person sliding along a zip wire. The person keeps their 70 kg of mass and their weight of 70×9.8=68670 × 9.8 = 686 N.
Recap
Big ideas
SI base units: m, kg, s. Derived: m s−1⁻¹, m s−2⁻², N == kg m s−2⁻². Convert km h−1⁻¹ to m s−1⁻¹ by dividing by 3.6.
Weight W=mgW = mg is a force (a vector, in N); mass is a scalar in kg. Use g=9.8g = 9.8 m s−2⁻² unless told otherwise.
Modelling words (particle, light, inextensible, smooth, rigid, uniform) each tell you what to ignore and what you may assume.
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Mission 1Why can a 70 kg person be modelled as a dot?
Specification points and sources for this lesson are listed at the bottom of the page.
Sources (1)
The specification points and sources behind this lesson.
1 CCEA GCE Mathematics (2018) specification, Unit AS 2 Applied Mathematics Section A (Quantities and units in mechanics; Forces and Newton's laws); overarching themes