Applied physics-1 homework

SCALAR: Magnitude only
- Distance: 5 km
- Speed: 60 km/h
- Mass: 50 kg
- Temperature: 25°C
- Time: 2 seconds

VECTOR: Magnitude + Direction
- Displacement: 5 km North
- Velocity: 60 km/h West
- Force: 100 N upward
- Weight: 50 kg downward
- Acceleration: 9.8 m/s² downward

Representation:
- Arrow notation: →v (arrow over v)
- Bold: **v**
- Component form: (x, y) components

FIRST LAW (Inertia):
"An object at rest stays at rest, and an object in motion stays in motion unless acted upon by a net force."

Formula: ∑F = 0 means a = 0 (no acceleration)

Examples:
- Car seatbelts protect when car stops suddenly
- Spinning ice skater: no friction = keeps spinning
- Shuttle in space: no friction = travels forever

SECOND LAW (F = ma):
"The acceleration of an object is proportional to the force applied and inversely proportional to its mass."

Formula: F = ma
Where:
- F = Force (Newtons = N = kg·m/s²)
- m = Mass (kilograms = kg)
- a = Acceleration (m/s²)

Implications:
- Same force → heavier object accelerates less
- Same mass → larger force gives larger acceleration
- a = F/m (rearranged)

Example Calculation:
A 1000 kg car accelerates at 5 m/s²
What force is needed?
F = ma = 1000 × 5 = 5000 N

THIRD LAW (Action-Reaction):
"For every action, there's an equal and opposite reaction."

Formula: F_action = -F_reaction

Examples:
- Rocket pushes exhaust down → exhaust pushes rocket up
- You push ground with feet → ground pushes you up
- Car hits wall with force F → wall hits car with force F

Critical point:
- Forces act on different objects
- That's why objects can accelerate despite equal forces

WORK:
Definition: Force applied over distance
Formula: W = F × d × cos(θ)
Units: Joules (J) = N·m

Where:
- F = Force applied
- d = Distance moved
- θ = Angle between force and motion

Examples:
- Push box horizontally 10 m with 50 N force
  W = 50 × 10 × cos(0°) = 500 J
  
- Person holds bag (force up) while walking (motion horizontal)
  W = 0 (force perpendicular to motion)

KINETIC ENERGY:
Energy due to motion
Formula: KE = ½ mv²
Units: Joules (J)

Examples:
- Car (1000 kg) at 20 m/s:
  KE = ½ × 1000 × 20² = 200,000 J = 200 kJ

POTENTIAL ENERGY:
Stored energy due to position
Formula: PE = mgh (for gravity)
Units: Joules (J)

Where:
- m = Mass
- g = 9.8 m/s² (gravity)
- h = Height above reference

Example:
- 100 kg person on 10 m high building:
  PE = 100 × 9.8 × 10 = 9,800 J = 9.8 kJ

ENERGY CONSERVATION:
Total energy = KE + PE = Constant (if no friction)

Example: Ball thrown upward with initial KE
- As it rises: KE decreases, PE increases
- At top: All PE, zero KE
- Coming down: PE decreases, KE increases
- At bottom: Same KE as start

POWER:
Rate of doing work
Formula: P = W / t
Units: Watts (W) = J/s

Alternative: P = F × v

Examples:
- Lifting 100 kg (force = weight = 980 N) 10 m in 10 seconds:
  P = (980 × 10) / 10 = 980 W ≈ 1 kW

- Lightbulb rated 60W uses 60 joules per second

ANGULAR VELOCITY (ω):
How fast something is spinning
Formula: ω = θ / t
Units: Radians per second (rad/s)

Relation to linear velocity:
v = ω × r
(On a circle, linear speed depends on radius)

CENTRIPETAL ACCELERATION:
Acceleration toward center of circle
Formula: a_c = v² / r = ω² × r
Units: m/s²

Required force (centripetal force):
F_c = m × a_c = m × v² / r

Example: Car turning
- Car (1000 kg) at 20 m/s on curved road (radius 50 m)
- Centripetal acceleration: a = 20² / 50 = 8 m/s²
- Force needed: F = 1000 × 8 = 8000 N
- This force comes from friction with road!

Definition:
Motion where acceleration is proportional to displacement, opposite direction
Formula: a = -ω² × x

Characteristics:
- Periodic (repeats)
- Periodic (repeats)
- Sinusoidal (follows sine wave)
- Examples: Pendulum, mass on spring, tuning fork

KEY PARAMETERS:
- Amplitude (A): Maximum displacement from center
- Period (T): Time for one complete cycle
- Frequency (f): Cycles per second (Hz)
  Relation: T = 1/f
- Angular frequency (ω): ω = 2πf = 2π/T

DISPLACEMENT EQUATION:
x(t) = A × sin(ωt + φ)

Where:
- A = Amplitude
- ω = Angular frequency
- φ = Phase constant
- t = Time

EXAMPLE: Mass on spring
- Amplitude: 0.1 m
- Period: 2 seconds
- ω = 2π/2 = π rad/s
- Displacement: x(t) = 0.1 sin(πt)

SIMPLE PENDULUM:
- Small mass on light string
- Swings back and forth
- Period formula: T = 2π√(L/g)

Where:
- L = Length of string
- g = 9.8 m/s²

Example:
- Pendulum length 1 meter
- T = 2π√(1/9.8) = 2π × 0.319 = 2.0 seconds

IMPORTANT INSIGHT:
- Period depends only on length!
- Independent of mass
- Independent of amplitude (for small angles)

MASS ON SPRING:
Period: T = 2π√(m/k)

Where:
- m = Mass
- k = Spring constant

Energy: E = ½ kA²
(Proportional to amplitude squared)

TORSIONAL OSCILLATIONS:
Period: T = 2π√(I/κ)

Where:
- I = Moment of inertia
- κ = Torsional constant

MECHANICAL WAVES:
- Require medium (air, water, solid)
- Examples: Sound, water ripples, seismic waves
- Speed depends on medium properties

ELECTROMAGNETIC WAVES:
- Don't require medium
- Travel through vacuum at speed of light
- Examples: Light, radio, X-rays, microwaves

TRANSVERSE WAVES:
- Particles vibrate perpendicular to wave motion
- Example: Light, water waves
- Can be polarized

LONGITUDINAL WAVES:
- Particles vibrate parallel to wave motion
- Example: Sound waves
- Cannot be polarized

WAVE EQUATION:
v = f × λ

Where:
- v = Wave speed
- f = Frequency (Hz)
- λ = Wavelength (distance between waves)

Example: Sound in air
- Speed: 330 m/s (in air at sea level)
- Frequency: 440 Hz (musical A note)
- Wavelength: λ = v/f = 330/440 = 0.75 m

WAVE EQUATION (Mathematical):
y(x,t) = A sin(kx - ωt)

Where:
- A = Amplitude
- k = 2π/λ (wave number)
- ω = 2πf (angular frequency)
-c = x,t = Position and time

REFLECTION:
- Wave bounces off barrier
- Angle of incidence = Angle of reflection
- Used in: Mirrors, radar, sonar

REFRACTION:
- Wave bends entering different medium
- Speed changes, direction changes
- Snell's Law: n₁ sin(θ1) = n₂ sin(θ2)
- Used in: Lenses, prisms, mirages

INTERFERENCE:
- Two waves combine
- Constructive: Amplitudes add (louder, brighter)
- Destructive: Amplitudes cancel (quieter, darker)
- Creates patterns

DIFFRACTION:
- Wave bends around obstacle
- More pronounced with longer wavelengths
- Sound bends around corners more than light
- Used in: Diffraction gratings for spectrum analysis

RESマANCE:
- Object vibrates when driven at natural frequency
- Amplitude becomes very large
- Dangerous: Bridge collapse, glass breaking
- Useful: Radio tuning, MRI machines

Problem: A car accelerates from 0 to 30 m/s in 6 seconds.
Find acceleration and distance traveled.

Solution:
a = Δv / Δt = 30 / 6 = 5 m/s²

Distance using: s = ut + ½ at²
s = 0 + ½(5)(6)² = 90 m

Or using: v² = u² + 2as
30² = 0 + 2(5)s
s = 900/10 = 90 m (✓ confirms)

Problem: A ball (0.5 kg) is thrown upward with velocity 20 m/s.
Find maximum height reached.

Solution:
Using energy conservation:
Initial KE = Final PE
½ mv² = mgh
v² / (2g) = h
20² / (2 × 9.8) = h
400 / 19.6 = h
h = 20.4 m

KINEMATICS:
- v = u + at
- s = ut + ½ at²
- v² = u² + 2as

DYNAMICS:
- F = ma
- Weight = mg
- Frictional force = μ × N

WORK & ENERGY:
- W = Fd cos(θ)
- KE = ½ mv²
- PE = mgh
- P = W/t = Fv

CIRCULAR MOTION:
- v = ωr
- a_c = v²/r
- F_c = mv²/r

SIMPLE HARMONIC:
- T = 2π√(m/k) [spring]
- T = 2π√(L/g) [pendulum]
- ω = 2πf = 2π/T

WAVES:
- v = fλ
- v = (wavelength) × (frequency)