Mechanics & Energy

Mechanical Advantage of Pulley Systems

"Movable Pulleys Multiply, Fixed Pulleys Fix Direction"

Example:

A system with 3 movable pulleys has MA = 3

Adding fixed pulleys changes direction but MA stays the same

Combustion & Chemical Energy Release

"Break High, Make Low — Energy Will Flow"

Break high-energy bonds

Make low-energy bonds

Then energy is released

Spring Potential Energy

"Spring Energy is Half K X-Squared"

Power in Translational Motion

"Power Flows from Force and Speed"

Torque Direction & Sign Convention

"Counter is Positive — Clock is Not"

Translational Kinetic Energy

The energy of an object due to its motion in a straight line (translation through space).

Static Rotational Equilibrium

No Torque, No Twist

A state where an object is not rotating or rotating at a constant angular velocity, with no net torque acting on it.

Gravity - Conservative Force

Gravity's Work is Path-Free

A conservative force is a force where the work done moving an object between two points is independent of the path taken.

Mechanical Advantage

"Machines Amplify"

Mechanical Advantage is a measure of how much a machine multiplies input force to make work easier.

Static vs. Kinetic Friction

“Static Stops, Kinetic Keeps Sliding"

Static = Sticking

Kinetic = Keeps moving

Torque and the Sine Function

Sine Spins Strongest at 90

Torque is strongest when the force is perpendicular to the lever arm

Mechanical Advantage of a Lever

The mechanical advantage (MA) of a lever is how much the lever amplifies the input force to move a load.

Work

Work = Force × Distance × Cosine

Work is done when a force causes displacement of an object in the direction of the force.

Weight

Weight Means Gravity

The force due to gravity acting on an object's mass.

Tough Rotations Require Force and Radius

A measure of how much a force causes an object to rotate around an axis.

Conservation of Total Energy

Energy Can Change Clothes, But Never Leaves the Party

The total energy of an isolated system remains constant—it can change forms (e.g., kinetic ↔ potential), but cannot be created or destroyed.

Gravitational Potential Energy

U Might Go High

The energy an object has due to its position above the ground in a gravitational field

Heat of Fusion

"Heat of fusion is the quiet power that melts a solid—without making it hotter, just looser."

Snell’s Law (Law of Refraction)

“Snell’s Law says light changes its stride—bending its path to match the ride."

Law of Reflection (“In = Out, Angle by Angle!”)

States that the angle of incidence (incoming ray) equals the angle of reflection (outgoing ray) on smooth surfaces

Doppler Effect (“Closer = Higher, Farther = Lower”)

The change in frequency or wavelength of a wave (sound, light, etc.) when the source or observer is moving relative to the other

Elastic Force & Equilibrium

• Stretch it, it pulls back

• Compress it, it pushes out

  
  Elastic force always acts toward the equilibrium position

Electric Force in an Electric Field

"Electric fields whisper a push to any nearby charge—just multiply and feel the nudge."

Place a charge in a field—force pops out directly from the combo of q and E

Force in Direction of Motion

When net force acts in the same direction as an object’s motion, it causes the object to accelerate (speed up)

Kinetic Energy and Speed Relationship

"When energy leaps twice as high, speed grows by root two—not sky high."

Period (T)

“Time for One Turn = T”

Frequency–Period Formula

“Flip the F and T!”

They're reciprocals—each defines how fast or slow cycles occur

Velocity Equation (Average Speed)

“D Over T Is How Fast You’ll Be!”

Circumference of a Circle

“2πR Rolls the Rim”

Finding Period (T) Using Circumference and Speed

"Period is how long it takes to loop the circle once—distance around divided by speed you roll."

What does a friction vs. applied force graph show, and how does static friction transition to kinetic friction?

Static climbs, kinetic cruises.

• Static friction resists until it can't

• Kinetic friction keeps a steady drag once sliding starts

What are the 4 main kinematic equations? ("Big 4 SUVAT equations")

S = displacement
U = initial velocity (same as v_0_​)
V = final velocity
A = acceleration
T = time

• v = v_0_+ at
  → Final velocity

• x = v_0_t + ½ at^2^
  → Displacement

• v^2^ = v_0_^2^ + 2a(x−x_0_)
  → Final velocity squared

• x = ((v+v_0_)/2) ⋅ t
  → Displacement using average velocity

What is sin⁡(90∘)?

Think of the unit circle:
At 90∘, the point is (0, 1) → sine = y-coordinate = 1

What does Newton’s Third Law state?

So, when body A exerts a force on body B,
then body B exerts an equal and opposite force on body A.

F_AB​ = − F_BA​

How do you find the horizontal and vertical components of velocity?

If an object is launched with speed v at angle θ above the horizontal, then:

• Horizontal component:

  v_x = v⋅cos⁡(θ)

• Vertical component:

  v_y = v⋅sin⁡(θ)

How do you calculate displacement using a velocity-time graph? (“Area under v–t = how far you get!”)

• For straight-line segments (constant acceleration):
  Use area formulas for rectangle and triangle.

• Displacement is the area under the velocity-time graph during the given time interval.

  • If velocity is positive, area = positive displacement

  • If velocity is negative, area = negative displacement

  • If velocity changes sign, subtract negative area from positive to find net displacement

What are the values of sin(60°) and cos(60°)?

Use the 30°–60°–90° triangle to remember these:

• Hypotenuse = 1

• Opposite 60° = (root 3)/2 = 0.866

• Adjacent to 60° = 1/2

What are the major SI units?

"My Kind Teacher Always Keeps Moles Counted"

• Meter (m)

• Kilogram (kg)

• Time: second (s)

• Ampere (A)

• Kelvin (K)

• Mole (mol)

• Candela (cd)

What are vectors in physics and math?

• Vectors are quantities that have both magnitude (size) and direction.

• They are often represented by arrows, where the length = magnitude and the arrow points in the direction.

What are the main vector quantities?

• Displacement

• Velocity

• Force

• Acceleration

What are scalars in physics and math?

• Scalars are quantities that have magnitude only — no direction.

What kinds of quantities can scalars be?

They may be:

1. Magnitude of vectors — e.g., speed (magnitude of velocity)

2. Dimensionless numbers — e.g., coefficients of friction, refractive index

How can vectors be added?

Graphical methods:

1. Tip-to-tail method: Place the tail of the second vector at the tip of the first. The resultant vector goes from the tail of the first to the tip of the last.

2. Analytical methods:

   • Component method: Break vectors into x and y components, add all the x components and then add all the y components, then use Pythagoras’ theorem and trigonometry to find the resultant.

How do you subtract vectors?

1. Graphical methods:

   • Graphical method:

     • Reverse the direction of the vector and use tip-to-tail method.

2. Component method:

   • Subtract x and y components separately:

What happens when you multiply a vector by a scalar?

• Magnitude changes: The vector gets longer or shorter depending on the scalar.

• Direction may reverse: If the scalar is negative, the vector points in the opposite direction.

• Direction unchanged if the scalar is positive.

What is the dot product of two vectors?

• The dot product is a way to multiply two vectors that gives a scalar (a number), not another vector.

• Formula:

A⃗ ⋅ B⃗ =∣A⃗∣∣B⃗∣cos⁡θ

• ∣A⃗∣| and ∣B⃗∣ are the magnitudes of the vectors

• θ is the angle between them

What happens when you multiply two vectors using the cross product?

• The cross product (also called vector product) of two vectors results in a vector, not a scalar.

• Formula:

A⃗ × B⃗ = ∣A⃗∣∣B⃗∣sin⁡θn^

• ∣A⃗∣ and ∣B⃗∣| = magnitudes of the vectors

• θ = angle between vectors

• n^ = unit vector perpendicular to the plane formed by A and B (direction given by the right-hand rule)

What is displacement in physics?

• Displacement is a vector representing the change in position of an object.

• Path-independent: Only depends on the starting and ending points, not the route taken.

• Magnitude: Straight-line distance between start and end positions.

• Direction: Points from start to end.

  "Displacement = straight arrow from start to finish" — ignores the path traveled.

What is distance in physics?

• Distance is a scalar quantity — it has magnitude only, no direction.

• Reflects the total path traveled, not just the start and end points.

• Always positive and path-dependent.

  "Distance = all the steps you take" — counts the path, ignores direction.

What is velocity in physics?

• Velocity is a vector that represents the rate of change of displacement with respect to time.

• Formula:

v⃗ = Δx⃗/Δt

• Δx⃗ = displacement

• Δt = time interval

• Has both magnitude (speed) and direction.

Average velocity:

• Definition: A vector that is the total displacement divided by the total time.

v_avg = Δx_total/Δt_total

Average speed:

• Definition: Total distance ÷ total time

speed_avg = distance_total/Δt_total​​

• Scalar → has magnitude only.

• Path dependent — cares about the full route taken.

Instantaneous velocity:

• The limit of the change in displacement over time as the change in time approaches zero.

• Formula:

v = lim⁡ Δx/Δt

Δt→0

• Represents the velocity at a specific instant.

• Vector quantity — includes both magnitude and direction.

Instantaneous speed:

• The magnitude of the instantaneous velocity vector.

• Scalar quantity — has no direction.

• Always non-negative.

Force:

• Any push or pull that can cause an acceleration.

• Vector quantity — has both magnitude and direction.

• SI unit: newton (N), where 1 N = 1 kg·m/s².

Gravity:

• Attractive force between two objects due to their masses.

• Always acts along the line joining the centers of the objects.

• Formula: F_g = Gm₁m₂/r²

• G = gravitational constant 6.67×10⁻¹¹ Nm²/kg².

Friction:

• Force that opposes motion between two surfaces in contact.

• Caused by electrostatic interactions at the surfaces.

• Can be static (no motion yet) or kinetic (sliding in progress).

Static Friction:

• Exists between two objects not moving relative to each other.

• Prevents motion from starting.

• Magnitude ranges from 0 up to a maximum value fs ≤ μsN, where μs​ is the static friction coefficient and N is the normal force.

Kinetic Friction:

• Exists between two objects sliding past each other.

• Magnitude is fk = μkN, where μk​ is the kinetic friction coefficient and N is the normal force.

• Value is constant for given surfaces and does not depend on speed.

Difference between static and kinetic friction:

• Static friction: Varies from 0 up to a maximum depending on the applied force; prevents motion from starting.

• Kinetic friction: Constant once objects are sliding; opposes motion.

• Key idea: static adjusts, kinetic is fixed.

Coefficient of Friction:

• Depends on the two materials in contact.

• Static friction coefficient (μs​) is always higher than kinetic friction coefficient (μk​).

• Explains why it’s harder to start moving an object than to keep it sliding.

Mass vs Weight:

• Mass: Amount of matter in an object; scalar; constant; SI unit = kg.

• Weight: Force due to gravity acting on mass; vector; changes with gravitational field; SI unit = N.

• Relationship: W = mg, where g = acceleration due to gravity.

Mass

• Measure of an object’s inertia — its resistance to changes in motion.

• Represents the amount of material in the object.

• Scalar quantity, SI unit = kg.

Weight:

• The force experienced by a mass due to gravity.

• Formula: W = mg

  • m = mass

  • g = acceleration due to gravity (~9.8 m/s² on Earth)

• Vector quantity — points toward the center of the Earth.

• SI unit = newton (N).

Acceleration:

• Vector quantity representing the change in velocity over time.

• Can be average or instantaneous:

  • Average: a⃗_avg = Δv⃗/Δt

  • Instantaneous: a⃗ = dv⃗/dt

• Has magnitude and direction.