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Machines cheet sheets, Cheat Sheet of Physics

Physics

an immerse document that helps u ace a machines exam

Typology: Cheat Sheet

2025/2026

Uploaded on 03/22/2026

srihaas-yeluri 🇺🇸

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SCIENCE OLYMPIAD — MACHINES

Complete Binder Cheat Sheet

Division B & C | 2025–2026 | All Formulas · Concepts · Worked Examples

QUICK FORMULA FLASH⚡

W = F·d | MA = F_out/F_in | IMA = d_in/d_out

η = (AMA/IMA)×100% | τ = F·d_⊥ | ΣF = ma

KE = ½mv² | PE = mgh | p = mv

Page 1: Core Physics & Kinematics Page 2: Simple Machines Overview

Page 3: Levers & Torque In-Depth Page 4: Pulleys & Wheel/Axle

Page 5: Compound Machines, Gears & Efficiency Page 6: Energy, Momentum & Lever Device Tips

Partial preview of the text

Download Machines cheet sheets and more Cheat Sheet Physics in PDF only on Docsity!

SCIENCE OLYMPIAD — MACHINES

Complete Binder Cheat Sheet

Division B & C | 2025–2026 | All Formulas · Concepts · Worked Examples

⚡ QUICK FORMULA FLASH

W = F·d | MA = F_out/F_in | IMA = d_in/d_out

η = (AMA/IMA)×100% | τ = F·d_⊥ | ΣF = ma

KE = ½mv² | PE = mgh | p = mv

Page 1: Core Physics & Kinematics Page 2: Simple Machines Overview Page 3: Levers & Torque In-Depth Page 4: Pulleys & Wheel/Axle Page 5: Compound Machines, Gears & Efficiency Page 6: Energy, Momentum & Lever Device Tips

PAGE 1 — CORE PHYSICS: FORCES, KINEMATICS & NEWTON'S LAWS Units & Measurement Quantity SI Unit Symbol / Notes

Force Newton N = kg·m/s²

Mass Kilogram kg

Distance / Length Meter m

Work / Energy Joule J = N·m = kg·m²/s²

Power Watt W = J/s

Torque Newton-meter N·m

Velocity / Speed Meters/second m/s

Acceleration Meters/second² m/s²

Momentum kg·m/s p = mv

Frequency Hertz Hz = 1/s

Kinematics — 1D Motion Key quantities: position (x), displacement (Δx), velocity (v), acceleration (a), time (t)

Velocity (avg) v = Δx / Δt^ Δx = final − initial position

Acceleration (avg) a = Δv / Δt^ Units: m/s² — change in velocity over time

Big Four #1 v = v^ ₀^ + a·t Final velocity — no displacement needed

Big Four #2 Δx = v^ ₀ t + ½at²^ Displacement — used when no final v

Big Four #3 v² = v^ ₀ ² + 2aΔx^ No time needed — great for ramps/slides

Big Four #4 Δx = ½(v^ ₀ +v)t^ Avg velocity × time

Free Fall g = 9.8 m/s²^ (≈10)^ Downward acceleration due to gravity

Tip: On inclined planes the acceleration down the slope = g·sin(θ). Normal force = mg·cos(θ). Newton's Three Laws of Motion Law Statement Formula / Key Point

PAGE 2 — SIMPLE MACHINES: ALL 6 TYPES + MECHANICAL ADVANTAGE The Six Simple Machines — Master Reference IMA (Ideal MA) = assumes no friction/energy loss. AMA (Actual MA) = F_out / F_in, measured in real world. AMA ≤ IMA always. Efficiency = AMA/IMA × 100% Machine IMA Formula Examples + Key Facts

LEVER IMA = d_effort / d_load Seesaw, crowbar, scissors. 3 classes

(see p.3).

INCLINED PLANE IMA = L / h L = slope length, h = vertical height.

Ramp/slide.

WEDGE IMA ≈ L / w L = wedge length, w = widest thickness.

Knife, axe, zipper.

WHEEL & AXLE IMA = R_wheel / R_axle Doorknob, steering wheel, screwdriver.

Radii matter.

PULLEY IMA = # of supporting

ropes

Fixed: changes direction only (IMA=1). Movable: IMA=2+.

SCREW IMA = 2πr / pitch Pitch = distance per one full turn. Bolt,

jar lid, drill.

1. Inclined Plane — Deep Dive A ramp. Allows you to move a load a longer distance with less force to reach the same height.

IMA (ramp) IMA = L / h^ L = length of slope, h = vertical rise

Force needed (ideal)

F_effort = (m·g·h) /

L

= Weight × (1/IMA) Weight components

W_∥ = mg·sin(θ) |

W_⊥ = mg·cos(θ)

Parallel (down slope) | Perpendicular (into surface) Example: Ramp 5 m long, 1 m high. IMA = 5/1 = 5. To lift 500 N load ideally takes only 100 N over 5 m. Trick: sin(θ) = h/L, cos(θ) = base/L. The steeper the ramp, the lower the IMA and the more force needed.

2. Wedge A moving inclined plane (or two back-to-back). Force applied to blunt end; output forces push outward/apart.

IMA (wedge) IMA = L / w^

L = wedge length (along slope), w = widest width Key point: A longer, thinner wedge = higher IMA = more splitting force for the same input. Examples: Knife blade, axe head, nail, zipper teeth, doorstop, plow blade, scalpel.

3. Screw An inclined plane wrapped around a cylinder. Each full rotation moves the screw one 'pitch' forward.

IMA (screw) IMA = 2π·r /^ pitch^

r = radius of handle/lever; pitch = advance per turn Pitch

pitch = 1 / (threads

per cm)

More threads per cm = smaller pitch = larger IMA Example: r = 0.1 m handle, pitch = 0.005 m → IMA = 2π(0.1)/0.005 = 125. Self-locking: Screws with small pitch angle won't move backward when effort is removed (friction ≥ component of load). Summary: Trade-Off Rule (The Golden Rule of Machines)

GAIN in Force →

Input distance > Output distance IMA > 1 (most common case) Example: Wheelbarrow, block & tackle

GAIN in Speed/Distance →

Output distance > Input distance IMA < 1 (sacrifices force for speed) Example: Tweezers, human forearm

KEY LAW: W_in = W_out (ideal) → F_in × d_in = F_out × d_out

Machines NEVER create energy — they only redirect or redistribute it.

EXAMPLE 1 — Find Effort Force 1st class lever. Load = 300 N at 0.5 m from fulcrum. Effort arm = 2.0 m. Find effort.

τ_load = τ_effort

300 × 0.5 = F_e × 2.

150 = 2F_e

F_e = 75 N

IMA = 2.0/0.5 = 4

EXAMPLE 2 — Find Fulcrum Position A 4 m lever. 800 N load on left end. 200 N effort on right end. Where is fulcrum?

Let x = dist from load to fulcrum

800·x = 200·(4−x)

800x = 800 − 200x

1000x = 800

x = 0.8 m from load end

Lever Device Tips (for the Practical Part)

Set up the lever on its fulcrum before adding any masses
Record the position of EACH mass carefully (measure from fulcrum, not from end)
Torque balance: m ·d₁ ₁ = m ·d₂ ₂(for two masses — set it and solve for unknowns)
With 3 masses: m d₁ ₁ + m d₂ ₂ = m d₃ ₃ (pick one side vs. the other)
Mass ratio = m /m₁ ₂ = d /d₂ ₁(the INVERSE of the distance ratio)
Use the balance point of the lever itself to determine its center of mass (mark it!)
When all torques balance and lever is level, you can solve for mass ratios directly

PAGE 4 — PULLEYS, WHEEL & AXLE, AND FORCE ANALYSIS Pulleys — Complete Guide A pulley is a wheel with a grooved rim where a rope/cable runs. Changes direction and/or magnitude of force. Type Characteristics IMA & Notes Fixed Pulley Axle is attached to a fixed support. Only changes direction of force — pull down to lift up. Like a flagpole.

IMA = 1 No force advantage; 1

rope supports load. Movable Pulley Axle moves WITH the load. Load is supported by 2 rope segments. Halves the effort needed.

IMA = 2 Must pull rope twice the

load's distance. Block & Tackle Combination of fixed and movable pulleys. Used in cranes, elevators, sailing. Many rope segments support the load.

IMA = # supporting rope

segments Count ropes holding the

movable block. How to Count Supporting Rope Segments RULE: Count ONLY the rope segments that are directly supporting the movable pulley/load block. Do NOT count:

The free end of the rope being pulled
Rope segments attached to a fixed support (these don't support the load) Example: 4 rope segments hold the lower (movable) block → IMA = 4. Effort = Load ÷ 4. You must pull 4× the distance the load rises. Pulley System Worked Examples Example A — effort needed A block & tackle has 5 supporting ropes. Load = 1000 N. η = 80%.

IMA = 5

AMA = η × IMA = 0.8 × 5 = 4

F_effort = Load / AMA = 1000/4 =

250 N

Example B — rope pulled IMA = 3 system. Load raised 2 m. How far is rope pulled?

d_input = IMA × d_output

d_input = 3 × 2 m = 6 m

You pull 6 m of rope to raise load 2 m. Wheel and Axle Two cylinders of different radii sharing the same axis of rotation. Effort applied to wheel; output at axle (or vice versa). IMA (effort on wheel)

IMA = R_wheel /

R_axle

R_wheel > R_axle → force multiplier IMA (effort on axle)

IMA = R_axle /

R_wheel

R_axle < R_wheel → IMA < 1 → speed multiplier

PAGE 5 — COMPOUND MACHINES, GEARS & EFFICIENCY Compound Machines A compound machine is two or more simple machines working together. Overall IMA = product of individual IMAs. Total IMA

IMA_total = IMA ₁ ×

IMA ₂ × IMA ₃ ...

Multiply IMA of every simple machine component Example: Lever IMA=3, pulley system IMA=4 → total IMA = 12 Common Compound Machines Machine Components How IMA Works

Scissors 2 levers + 2 wedges Two 1st-class levers sharing a fulcrum (the

screw). Blade edges act as wedges.

Wheelbarrow 2nd-class lever +

wheel & axle

Lever (handles) + wheel (rolls over ground reducing friction).

Bicycle Wheel/axle + gears +

levers (brakes)

Pedal crank → chain gear → rear wheel. Multiple gear ratios.

Crane Pulley system +

lever arm

Block & tackle × boom arm. Very high total IMA.

Stapler 2nd-class lever +

wedge

Pressing handle compresses spring, staple wedge pierces paper.

Axe on handle Lever + wedge Handle = 3rd-class lever; blade = wedge.

Jackscrew Screw + lever Turning a lever arm drives a screw. Used to lift

cars.

Drill press Wheel/axle + wedge +

screw

Motor → rotary motion → drill bit (wedge) + feed screw. Gears (Wheel & Axle Variant) Gears are interlocking toothed wheels. They transmit torque and can change speed, force, and direction of rotation. Gear Ratio (IMA)

IMA = T_output /

T_input

T = number of teeth on each gear Speed Relationship

ω_in / ω_out = T_out

/ T_in

ω = angular speed (rev/s or RPM) Torque Relationship

τ_out / τ_in = T_out

/ T_in

More output teeth = more output torque Direction Rule: Two meshing gears rotate in OPPOSITE directions. Add an idler gear between them → same direction.

Gear Train Analysis FORCE Advantage (Speed Reducer) Small input gear → Large output gear

T_in = 10, T_out = 50

IMA = 50/10 = 5

Output torque = 5× input torque Output speed = 1/5 input speed Example: Car low gear, wrench socket drive SPEED Advantage (Torque Reducer) Large input gear → Small output gear

T_in = 60, T_out = 20

IMA = 20/60 = 0.

Output speed = 3× input speed Output torque = 1/3 input torque Example: Bicycle high gear, fan/propeller Efficiency Efficiency measures how much of the input work actually becomes useful output work. Always ≤ 100% due to friction. Efficiency

η = (W_out / W_in) ×

W in Joules; η as percentage (0–100%) Also

η = (AMA / IMA) ×

AMA = actual, IMA = ideal (no friction) Work Output

W_out = F_load ×

d_load

Useful work done on the load Work Input

W_in = F_effort ×

d_effort

Work done by you on the machine Energy Lost (heat)

W_friction = W_in −

W_out

Energy that went to heat/sound Example: Efficiency Calculation A ramp (IMA=4) is used to push 200 N box up a 4 m slope. Effort = 60 N.

W_in = 60 N × 4 m = 240 J

h = 4/4 = 1 m (since IMA = L/h)

W_out = 200 N × 1 m = 200 J

η = 200/240 × 100% = 83.3%

Key Efficiency Facts

η = 100% only in ideal (frictionless) machines
η < 100% always in real life
Higher friction → lower η
AMA < IMA when η < 100%
Lubrication increases η
Smooth surfaces increase η Comparing IMA vs AMA IMA (Ideal Mech. Advantage) AMA (Actual Mech. Advantage) Based ONLY on geometry/measurements Based on measured forces: F_out / F_in No friction assumed Includes friction and losses = d_input / d_output = F_output / F_input

PAGE 6 — ENERGY, MOMENTUM, CIRCULAR MOTION & EXAM TIPS Energy Conservation & Machines The Law: Energy cannot be created or destroyed, only transformed. Total mechanical energy = KE + PE. Mechanical Energy

E_mech = KE + PE =

½mv² + mgh

Conserved when no friction With friction

E_mech,f = E_mech,i

− W_friction

W_friction = f_k × d (energy lost to heat) Ramp launch

mgh = ½mv² → v =

√(2gh)

Object sliding down frictionless ramp from height h

On an inclined plane with friction: KE_bottom = mgh − μ_k·mg·cos(θ)·L

Momentum & Impulse Momentum describes how hard it is to stop a moving object. Impulse is the change in momentum.

Momentum p = m·v^ kg·m/s — vector quantity (has direction)

Impulse J = F·Δt = Δp^

Larger time → smaller force for same Δp (e.g., airbag) Conservation of Momentum

m ₁ ₁ v + m ₂ ₂ v = m ₁ ₁ v '

+ m ₂ ₂ v '

Total momentum constant when no external forces Elastic collision

KE conserved:

½m ₁ ₁ v ²+½m ₂ ₂ v ² =

½m ₁ ₁ v '²+½m ₂ ₂ v '²

Billiard balls, ideal collisions Perfectly Inelastic

(m ₁ +m ₂ )v_f = m ₁ ₁ v +

m ₂ ₂ v

Objects stick together after collision Circular Motion & Centripetal Force Centripetal force is the net inward force required to keep an object moving in a circle. It is NOT a separate force — it's the label for whatever force (tension, gravity, normal, friction) points toward the center. Centripetal Force

F_c = m·v²/r =

m·ω²·r

Always points toward center of circle

Centripetal Acceleration a_c = v²/r^ m/s² — directed inward

Period T = 2πr/v =^ 2π/ω^ Time for one full revolution; ω in rad/s

Angular velocity ω = 2π/T = 2πf^ Radians per second; f = frequency in Hz

On a banked curve: tan(θ) = v²/(r·g) — no friction needed at this exact speed

Vertical circle (top): F_c = mg + N (or T) — gravity adds to centripetal force at top

Inertia & Rotational Inertia

Inertia I ∝ m (linear)^ More mass = harder to accelerate

Rotational Inertia (Moment of Inertia)

I = Σmr² Depends on mass distribution and axis

Point mass on rod I = mr²^ r = distance from rotation axis

Newton's 2nd (rotation) τ_net = I·α^ α = angular acceleration; analogous to F=ma

All Formulas Quick-Reference Card MECHANICS SIMPLE MACHINES ENERGY / MOMENTUM

F = ma

W = Fd·cos(θ)

v = v ₀ + at

Δx = v ₀ t + ½at²

v² = v ₀ ² + 2aΔx

τ = F·d_⊥

F_c = mv²/r

f_k = μ_k·N

N = mg (flat)

IMA = d_in/d_out

AMA = F_out/F_in

η = AMA/IMA × 100%

Lever: IMA=d_e/d_L

Ramp: IMA=L/h

Pulley: IMA=# ropes

W&A: IMA=R_w/R_a

Screw: IMA=2πr/p

Gears: IMA=T_out/T_in

KE = ½mv²

PE = mgh

W_net = ΔKE

p = mv

J = FΔt = Δp

P = W/t = F·v

ω = 2π/T

τ_net = Iα

v = √(2gh)

Exam Strategy & Common Mistakes ⚠ COMMON MISTAKES

Forgetting to label units — ALWAYS write N, m, J, etc.
Confusing IMA and AMA — IMA uses distances, AMA uses forces
Wrong lever class — always identify position of fulcrum, effort, load
Counting wrong rope segments in

✅ WINNING TIPS

Draw a diagram and FBD for EVERY problem
For levers: ALWAYS use torque balance about the fulcrum
For pulleys: count supporting ropes, then multiply effort by IMA = distance you pull
For efficiency: check if η < 100% before

Machines cheet sheets, Cheat Sheet of Physics

Related documents

Partial preview of the text

Download Machines cheet sheets and more Cheat Sheet Physics in PDF only on Docsity!

SCIENCE OLYMPIAD — MACHINES

Complete Binder Cheat Sheet

Division B & C | 2025–2026 | All Formulas · Concepts · Worked Examples

W = F·d | MA = F_out/F_in | IMA = d_in/d_out

η = (AMA/IMA)×100% | τ = F·d_⊥ | ΣF = ma

KE = ½mv² | PE = mgh | p = mv

Force Newton N = kg·m/s²

Mass Kilogram kg

Distance / Length Meter m

Work / Energy Joule J = N·m = kg·m²/s²

Power Watt W = J/s

Torque Newton-meter N·m

Velocity / Speed Meters/second m/s

Acceleration Meters/second² m/s²

Momentum kg·m/s p = mv

Frequency Hertz Hz = 1/s

Velocity (avg) v = Δx / Δt^ Δx = final − initial position

Acceleration (avg) a = Δv / Δt^ Units: m/s² — change in velocity over time

Big Four #1 v = v^ ₀^ + a·t Final velocity — no displacement needed

Big Four #2 Δx = v^ ₀ t + ½at²^ Displacement — used when no final v

Big Four #3 v² = v^ ₀ ² + 2aΔx^ No time needed — great for ramps/slides

Big Four #4 Δx = ½(v^ ₀ +v)t^ Avg velocity × time

Free Fall g = 9.8 m/s²^ (≈10)^ Downward acceleration due to gravity

LEVER IMA = d_effort / d_load Seesaw, crowbar, scissors. 3 classes

INCLINED PLANE IMA = L / h L = slope length, h = vertical height.

WEDGE IMA ≈ L / w L = wedge length, w = widest thickness.

WHEEL & AXLE IMA = R_wheel / R_axle Doorknob, steering wheel, screwdriver.

PULLEY IMA = # of supporting

ropes

SCREW IMA = 2πr / pitch Pitch = distance per one full turn. Bolt,

IMA (ramp) IMA = L / h^ L = length of slope, h = vertical rise

F_effort = (m·g·h) /

L

W_∥ = mg·sin(θ) |

W_⊥ = mg·cos(θ)

IMA (wedge) IMA = L / w^

IMA (screw) IMA = 2π·r /^ pitch^

pitch = 1 / (threads

per cm)

GAIN in Force →

GAIN in Speed/Distance →

KEY LAW: W_in = W_out (ideal) → F_in × d_in = F_out × d_out

τ_load = τ_effort

300 × 0.5 = F_e × 2.

150 = 2F_e

F_e = 75 N

IMA = 2.0/0.5 = 4

Let x = dist from load to fulcrum

800·x = 200·(4−x)

800x = 800 − 200x

1000x = 800

x = 0.8 m from load end

IMA = 1 No force advantage; 1

IMA = 2 Must pull rope twice the

IMA = # supporting rope

segments Count ropes holding the

IMA = 5

AMA = η × IMA = 0.8 × 5 = 4

F_effort = Load / AMA = 1000/4 =

250 N

d_input = IMA × d_output

d_input = 3 × 2 m = 6 m

IMA = R_wheel /

R_axle

IMA = R_axle /

R_wheel

IMA_total = IMA ₁ ×

IMA ₂ × IMA ₃ ...

Scissors 2 levers + 2 wedges Two 1st-class levers sharing a fulcrum (the

Wheelbarrow 2nd-class lever +

wheel & axle

Bicycle Wheel/axle + gears +

levers (brakes)

Crane Pulley system +

lever arm

Stapler 2nd-class lever +