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MM Applied Cheat Sheet – AS Mech
Notations, Modelling, Rules and Reminders
- 𝑛 = 𝑛𝑎𝑣𝑒
A particle is a body whose entire weight acts through a single point
...

A lamina is a two dimensional body, it is the 2-dimensional equivalent of a particle
...

A body is said to be uniform if it has constant density
...

A smooth surface is one which has no friction
...

We say an object is light if it has no mass
...

An inextensible string is a string which has a fixed length- it is impossible to stretch
...


- Use variables before putting in the values to analyse the general case easier and make workings
less confusing
...

- When question asks for a quantity which has a clear equation, don’t start jumping into other
related ones, write down the equation directly relating to/is the conclusion of the question
asked
...


MM Applied Cheat Sheet – AS Mech
1
...
B
...

if accn = 0 from equation, check
graphically if at stationary point the
curve is smooth then defined, but if
it is a maximum/minimum then
undefined as gradient from both
sides different signs even though
function is continuous
...
2 Constant accn
1
...
𝑠 = 𝑢𝑡 + 2 𝑎𝑡 2
3
...

5
...
B
...
use
calculus instead of
Suvat,
although
calculus can be used
for both constant and
variable
...
B
...


⑤= 𝑣𝑡

1
...
3 Kinematics

N
...
: Change signs for
displacement, velocities
or accn if they are defined
to be different directions
...
B
...


1
...


Same magnitude & direction = same vector
Magnitude-direction: ⃗⃗⃗⃗⃗
𝑂𝐴 (or a) = (r, )
𝑟 cos 𝜃
⃗⃗⃗⃗⃗
Component: 𝑂𝐴 = ( 𝑟 sin 𝜃 )
𝑎𝑥
𝑥
a + b = (𝑎𝑎𝑥 +𝑏
); a= (𝑎
)
+𝑏
𝑦

𝑦

𝑦

|a| = √(𝑎𝑥 )2 + (𝑎𝑦 )

=

1
...
Remember the ‘+c’s when integrating
...

Just the Tip™: greatest [scalar]:

𝑑[𝑣𝑒𝑐𝑡𝑜𝑟]
𝑑𝑡

=0

N
...
: To find the scalar counterpart of a vector use
Pythagorean addition 𝑥⨁𝑦: √𝑥 2 + 𝑦 2
...


MM Applied Cheat Sheet – AS Mech
1
...
B
...
Eg
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𝑢=(

𝑢 cos 𝜃
)
𝑢 sin 𝜃

0
𝑎=( )
−𝑔

Trajectory:
𝑥

𝑥 = 𝑟𝑥 = 𝑢 cos 𝜃 𝑡 [dst] -> 𝑡 = 𝑢 cos 𝜃
1

𝑦 = 𝑟𝑦 = 𝑢 sin 𝜃 𝑡 + 2 (−𝑔)𝑡 2 [Suvat]
2
𝑥
1
𝑥
= 𝑢 sin 𝜃
− 𝑔(
)
𝑢 cos 𝜃 2 𝑢 cos 𝜃
𝑔
= tan 𝜃 𝑥 −
𝑥2
2
2𝑢 cos 2 𝜃
Optimum release angle = 45°

∵ no linear cartesian +c graphs look like the
above 𝑟𝑦 function
∴ it is a trajectory

MM Applied Cheat Sheet – AS Mech
2
...
2 Newton’s First and Third Laws of Motion
First Law

Third Law

2
...
Friction oppose relative motion
2
...
𝑓𝑠 ≤ 𝑓𝑠−𝑚𝑎𝑥 = 𝜇𝑠 𝑁
Between 2 obj
...
𝑓𝑠 = 𝑓𝑠−𝑚𝑎𝑥
When sliding about to happen
3
...
moving relative to e/o
N
...

•
Friction may happen in direction of motion of obj
...

Just the Tip™: if it is unknown which way friction acts, use
a dotted line pointing both direction
...
t
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2
...


Just the Tip™: triangle
rules (cosine/sine) can
be used to resolve
components
...
5 Newton’s Second Law
Forces

Suvat

∑ 𝐹 = 𝑚𝑎

2
...
7 Connected Bodies
𝐹

1

2

3

4

5

𝐹4,5

𝐹4,5 = 𝑚𝑎
𝐹
= 𝑚( )
5𝑚
𝐹
=
5

5

𝐹 = (5𝑚)𝑎
𝐹
𝑎=
5𝑚
𝐹2,3

3

4

5

Tension: Pulling, between connections
Thrust: Pushing, compression between connections
N
...
: A string can only be in tension
...
1 Moments
Moments (↻/↺) = 𝐹 ×⊥ 𝑑
Equilibrium: ∑ M↻ − ∑ M↺ = 0 (𝐹𝑛𝑒𝑡 = 0)

A couple: 𝐹𝑛𝑒𝑡 = 0 but has rotation

3
...
edge of side), then toppling happens first,
o/w sliding first
...

1
...

2
...
If TRUE, then
not sliding
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Check position of N from pivot using
moments
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then toppling
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1 Gravitational Potential and Kinetic Energy in 1D
All forces are derived from potential energy function 𝑉(𝑥)
𝑥
1
...


𝑥

Work-energy principle: ∆𝐸𝑃 (work done) = 𝑉(𝑥1) − 𝑉(𝑥2) = ∫𝑥 1 𝐹(𝑠) 𝑑𝑠
1

2

1

2

2

3
...


Mechanical energy conservation: 𝑚𝑔∆ℎ = 2 𝑚𝑣 2 − 2 𝑚𝑢 2 [𝐹(𝑆) = 𝐹 = 𝑚𝑔] (no external
work done, only weight acting on obj
...
Total energy is all energy
...
𝐸𝑚𝑒𝑐ℎ only conserved if external forces ignored
...


1

1

𝐸𝑃 = 𝑚𝑔ℎ
1
𝐸𝐾 = 𝑚𝑣 2
2
1
𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 = 𝐹𝑠: −𝑚𝑔(ℎ𝑓 − ℎ𝑖 ) = 2 𝑚(𝑣 2 − 𝑢2 )
[mechanical energy conserved]
𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 = 𝐹𝑠 = ∆𝐸𝑃 + ∆𝐸𝐾 + 𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 𝑣 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒
[mechanical energy not conserved]
𝑛

∑ 𝐹(𝑠)𝑖 = ∆𝐸𝐾 (all forces directional to x)
𝑖=1

5
...

𝑊 𝐹𝑠
𝑃=
=
= 𝐹𝑣
𝑡
𝑡
∆𝑊
𝑃=
=𝐹×𝑣
∆𝑡
Just the Tip™: use obj
...
Rolling (still touching ground level but
CoM rising and falling)



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