Worked Examples
Learn by watching a problem solved properly — every step shown and checked.
Easy
Angular Velocity — Spinning CD (Worked Example)
A CD spins at 300 rpm. Convert to angular velocity in rad/s.…
Angular Velocity — Ceiling Fan Speed (Worked Example)
A ceiling fan spins at 240 rpm. Convert revolutions per minute to radians per second.…
Angular Velocity — Merry-Go-Round (Worked Example)
Edge speed 3 m/s at radius 2 m. Find angular velocity.…
Archimedes' Principle — Buoyant Force on a Boat (Worked Example)
A boat displaces 2.5 m³ of seawater. Find the buoyant force and the mass it can support.…
Buoyant Force — Cargo Ship (Worked Example)
A ship displaces 50,000 m³ of seawater. Find the buoyant force.…
Buoyant Force — Life Jacket (Worked Example)
0.015 m³ of foam in seawater. Find the buoyant force.…
Capacitance — Camera Flash (Worked Example)
A 100 μF capacitor charged to 300 V. Find the stored charge.…
Capacitance — Supercapacitor (Worked Example)
A 0.5 F supercapacitor at 5 V. Find stored charge.…
Carnot Efficiency — Solar Thermal Plant (Worked Example)
Th = 600 K, Tc = 300 K. Find max efficiency.…
Electric Field — Parallel-Plate Capacitor (Worked Example)
12 V across 2 mm plates. Find the uniform field with E = V/d.…
Electric Field — From Measured Force (Worked Example)
A proton feels 3.2×10⁻¹⁵ N in a field. Find E with E = F/q.…
Electric Potential — Battery Voltage (Worked Example)
18 J moves 1.5 C through a battery. Find the voltage with V = W/q.…
Electric Potential — Near a Nanocoulomb (Worked Example)
A 1 nC charge; find the potential 5 cm away.…
Flow Rate — Garden Hose (Worked Example)
Hose diameter 2 cm, water speed 3 m/s. Find the volume flow rate.…
Flow Rate — Water Main (Worked Example)
Pipe diameter 0.5 m, flow speed 2 m/s. Find the flow rate.…
Flow Rate — River Discharge (Worked Example)
A river 20 m wide, 2 m deep, flowing 1.5 m/s. Find discharge.…
Friction Force — Pushing a Crate at Constant Speed (Worked Example)
A 40 kg crate slides at constant velocity with μk = 0.35. Find the friction force and the push needed — step b…
Newton's Law of Gravitation — Apple and Earth (Worked Example)
Compute Earth's gravitational pull on a 0.18 kg apple and check it equals mg.…
Half-Life — Dating with Carbon-14 (Worked Example)
A sample holds 1/8 of its original carbon-14. With a 5730-year half-life, find its age.…
Half-Life — Iodine-131 Decay (Worked Example)
100 g of I-131 (t½ = 8 days) after 24 days. Find the remaining mass.…
Half-Life — Tritium in Exit Signs (Worked Example)
40 g of tritium (t½ = 12.3 y) after 24.6 years. Find remaining mass.…
Heat Transfer — Cooling Coffee (Worked Example)
0.3 kg coffee cools 85°C → 60°C. Find heat lost with Q = mcΔT.…
Heat Transfer — Engine Block (Worked Example)
Heat 5 kg of aluminium 20°C → 120°C. Find energy (c = 900).…
Kinetic Energy — A Sprinter at Top Speed (Worked Example)
A 70 kg sprinter runs at 8.0 m/s. Calculate the kinetic energy with a full unit check.…
Lens Magnification — Microscope Objective (Worked Example)
Object 2 cm from lens, image 30 cm away. Find magnification.…
Lens Magnification — Cinema Projector (Worked Example)
Film 10 cm from lens, screen 300 cm away. Find magnification.…
Magnetic Force — CRT Electron Beam (Worked Example)
An electron at 2×10⁷ m/s in a 1 mT steering field. Find the force.…
Newton's Second Law — SUV Accelerating from Rest (Worked Example)
A 1500 kg SUV reaches 20 m/s in 8 s. Find acceleration, then the net force — full step-by-step solution.…
Ohm's Law — Current Through a Light Bulb (Worked Example)
A 240 Ω bulb on 120 V mains. Find the current and power — full step-by-step solution.…
Pendulum Period — Grandfather Clock (Worked Example)
A 1.0 m pendulum. Find its period with T = 2π√(L/g) and see why clocks use this length.…
Pendulum Period — Playground Swing (Worked Example)
A 2.5 m swing. Find its period with the pendulum formula.…
Photoelectric Effect — Potassium (Worked Example)
450 nm light on potassium (Φ = 2.3 eV). Find max KE.…
Electrical Power — Space Heater Current (Worked Example)
A 1500 W heater on a 120 V outlet. Find the current draw and element resistance — worked solution.…
Pressure at Depth — Dam Base (Worked Example)
Find water pressure at the 50 m deep base of a dam.…
Pressure at Depth — Swimming Pool (Worked Example)
Find absolute pressure 3 m down in a pool with P = P₀ + ρgh.…
Fluid Pressure — Scuba Diver at 15 m (Worked Example)
Find the gauge and absolute pressure on a diver 15 m down in seawater — full solution.…
RMS Speed — Hydrogen, the Fastest Gas (Worked Example)
Hydrogen molecules at 300 K — find their rms speed with v = √(3RT/M) and see why Earth lost its hydrogen.…
RMS Speed — Oxygen (Worked Example)
Oxygen molecules at 300 K — find their rms speed with v = √(3RT/M), step by step.…
Significant Figures — Lab Measurement Math (Worked Example)
Multiply 12.5 cm × 8.24 cm and add 15.2 g + 3.45 g with correct significant figures — worked rules.…
Sound Intensity — Home Speaker (Worked Example)
A 2 W speaker at 3 m distance. Find the intensity.…
Sound Intensity — Workshop Machine (Worked Example)
A machine emits 0.5 W; worker stands 2 m away. Find the intensity.…
Specific Heat — Heating Water for Tea (Worked Example)
Heat 2.0 kg of water from 20 °C to 80 °C. Calculate the energy required — step-by-step.…
Elastic Potential Energy — Archery Bow (Worked Example)
A bow (k = 400 N/m) drawn 0.5 m. Find stored elastic potential energy.…
Elastic Potential Energy — Mousetrap (Worked Example)
A mousetrap spring (k = 60 N/m) flexes 0.05 m. Find stored energy.…
Hooke's Law — Mattress Spring (Worked Example)
A mattress spring (k = 800 N/m) compresses 0.12 m. Find the restoring force.…
Hooke's Law — Kitchen Scale (Worked Example)
A scale spring (k = 250 N/m) stretches 0.20 m. Find the weight measured.…
Spring Period — Oscillating Scale Pan (Worked Example)
A 2 kg mass on a 250 N/m spring. Find the oscillation period.…
Standing Waves — Guitar Fundamental (Worked Example)
A 0.65 m string, wave speed 400 m/s. Find the fundamental frequency.…
Standing Waves — Organ Pipe (Worked Example)
A 2 m open pipe, sound speed 343 m/s. Find the fundamental.…
Standing Waves — Violin String (Worked Example)
A 0.33 m string, wave speed 280 m/s. Find the fundamental frequency.…
SUVAT Displacement — Elevator Rising (Worked Example)
An elevator accelerates upward from rest at 1.2 m/s² for 3 s. How far does it rise?…
SUVAT Displacement — Speed Skater (Worked Example)
A skater accelerates from rest at 1.5 m/s² for 6 s. Find the distance covered with s = ut + ½at².…
SUVAT v² Equation — Dropped Object (Worked Example)
An object is dropped from 20 m. Find impact speed with v² = u² + 2as.…
SUVAT Velocity — Motorcycle Merging (Worked Example)
A motorcycle at 5 m/s accelerates at 4.2 m/s² for 5 s. Find its merging speed.…
SUVAT Velocity — Sprinter Accelerating from Rest (Worked Example)
A sprinter accelerates at 2.5 m/s² for 4.0 s from rest. Find final velocity with v = u + at — full steps.…
SUVAT Velocity — Accelerating Train (Worked Example)
A train at 15 m/s accelerates at 0.8 m/s² for 20 s. Find its final speed step by step.…
Thermal Expansion — Steel Bridge in Summer (Worked Example)
A 200 m steel bridge warms by 40 °C. Calculate how much longer it gets — worked solution.…
Torque — Pushing a Door Open (Worked Example)
Push a 0.8 m-wide door with 55 N at the handle: find the torque and compare pushing near the hinge.…
Torque — Pushing a Door (Worked Example)
Push a door 0.9 m from hinges with 30 N. Find the torque.…
Torque — Loosening a Bolt with a Wrench (Worked Example)
A 0.35 m wrench with 180 N of force: compute the torque step by step and see why longer wrenches win.…
Torque — Tightening a Bolt (Worked Example)
A 0.25 m wrench pulled with 80 N. Find the torque on the bolt.…
Trigonometry — Height of a Loading Ramp (Worked Example)
A 5.0 m ramp rises at 30°. Find its vertical height and horizontal run — SOH CAH TOA in action.…
Vector Addition — Hiker's Displacement (Worked Example)
Hike 3.0 km east then 4.0 km north. Find the resultant displacement — magnitude and direction.…
Wave Period — Guitar String (Worked Example)
A guitar string vibrates at 110 Hz. Find its period.…
Wave Period — Ocean Swell Frequency (Worked Example)
Ocean swells arrive every 8 s. Find their frequency.…
Wave Period — Tuning Fork (Worked Example)
A tuning fork vibrates at 440 Hz. Find the period of one vibration.…
Wave Speed — Vibration on a Guitar String (Worked Example)
A string vibrates at 120 Hz with wavelength 2.5 m. Find the wave speed — quick worked solution.…
Work — Lifting a Box Against Gravity (Worked Example)
A 25 kg box is lifted 2.0 m straight up. Calculate the work done with a unit check.…
Medium
Angular Velocity — Earth's Rotation (Worked Example)
Earth turns once per day. Find its angular velocity in rad/s.…
Buoyant Force — Hot-Air Balloon Lift (Worked Example)
2000 m³ balloon; outside air 1.2 kg/m³, inside 0.9 kg/m³. Find net lift.…
Buoyant Force — Iceberg (Worked Example)
A 1000 m³ iceberg, 90% submerged in seawater. Find the buoyant force.…
Capacitance — Defibrillator Capacitor (Worked Example)
A 32 μF capacitor at 5000 V. Find the charge.…
Capacitance — Keyboard Key (Worked Example)
A 10 pF key switch at 5 V. Find the charge per press.…
Carnot Efficiency — Diesel Engine (Worked Example)
Th = 2200 K, Tc = 300 K. Find the Carnot limit.…
Carnot Efficiency — Nuclear Plant (Worked Example)
Th = 590 K, Tc = 300 K. Find the Carnot limit.…
Carnot Efficiency — Steam Power Plant (Worked Example)
Th = 800 K, Tc = 300 K. Find the maximum theoretical efficiency.…
Centripetal Acceleration — Car on a Curve (Worked Example)
A car takes a 50 m radius turn at 20 m/s. Find centripetal acceleration.…
Centripetal Acceleration — Roller Coaster Hill (Worked Example)
A coaster crosses a 12 m radius hilltop at 18 m/s. Find centripetal acceleration.…
Coulomb's Law — Force Between Two Charges (Worked Example)
Two 2.0 µC charges 0.50 m apart. Calculate the electrostatic force — full solution with unit check.…
Doppler Effect — Approaching Ambulance Siren (Worked Example)
An ambulance at 30 m/s sounds a 700 Hz siren. Find the heard frequency — step-by-step Doppler shift.…
Doppler Effect — Receding Ambulance (Worked Example)
A 700 Hz siren recedes at 25 m/s. Find the observed frequency.…
Doppler Effect — Race Car (Worked Example)
A 300 Hz engine note approaches at 60 m/s. Find observed frequency.…
Doppler Effect — Approaching Train Horn (Worked Example)
A 400 Hz horn approaches at 30 m/s. Find the observed frequency.…
Electric Field — Point Charge (Worked Example)
A 2 μC charge; find the field 0.3 m away with E = kQ/r².…
Electric Potential — Defibrillator Charge (Worked Example)
A defibrillator delivers 200 J at 1500 V. Find the charge moved.…
Electric Potential — Van de Graaff Generator (Worked Example)
A 5 μC sphere of radius 0.2 m. Find its potential with V = kQ/r.…
Flow Rate — Aorta Blood Flow (Worked Example)
Aorta diameter 2.5 cm, blood speed 0.15 m/s. Find cardiac output.…
Newton's Law of Gravitation — Satellite in Orbit (Worked Example)
A 1200 kg satellite orbits 400 km up. Find Earth's gravitational pull on it.…
Newton's Law of Gravitation — Two Asteroids (Worked Example)
Two 10¹² kg asteroids 500 m apart. Find their mutual gravitational attraction.…
Half-Life — Cobalt-60 Source (Worked Example)
8 g of Co-60 (t½ = 5.27 y) after 15.81 years. Find remaining mass.…
Half-Life — Uranium-238 Over Geologic Time (Worked Example)
1 kg of U-238 (t½ = 4.5×10⁹ y) after 9×10⁹ years. Find remaining mass.…
Heat Engine Efficiency — Car Engine Energy Audit (Worked Example)
An engine takes in 800 J and expels 500 J per cycle. Find work output and thermal efficiency.…
Heat Transfer — Solar Water Heater (Worked Example)
Heat 100 kg of water 15°C → 45°C. Find energy required.…
Heat Transfer — Heating a Pool (Worked Example)
Heat 50,000 kg of water by 2°C. Find the energy needed.…
Ideal Gas Law — Moles in a Party Balloon (Worked Example)
A 10 L balloon at 300 K and 1 atm. Find the moles of gas inside — full ideal-gas solution.…
Thin Lens Equation — Camera Focus (Worked Example)
A 50 mm camera lens focuses on a subject 2.0 m away. Find the image distance and magnification.…
Lens Magnification — Magnifying Glass (Worked Example)
f = 10 cm lens, object at 8 cm. Find image distance and magnification.…
Lens Magnification — Telescope Objective (Worked Example)
Distant mountain 10 m... lens forms image 0.5 m away. Find magnification.…
Magnetic Force — Aurora Electron (Worked Example)
An electron at 5×10⁶ m/s in Earth's 5×10⁻⁵ T field. Find the force.…
Magnetic Force — Cyclotron Proton (Worked Example)
A proton at 10⁷ m/s in a 1.5 T field. Find the magnetic force.…
Magnetic Force — Mass Spectrometer Ion (Worked Example)
An ion (q = 3.2×10⁻¹⁹ C) at 2×10⁵ m/s in 0.8 T. Find the force.…
Lens Magnification — Camera Lens (Worked Example)
A camera lens (f = 5.0 cm) photographs a subject 50 cm away. Find where the sensor sits and the image scale.…
Lens Magnification — Classroom Projector (Worked Example)
A projector lens (f = 10 cm) sits 12 cm from the slide. Find the image distance and magnification.…
Mass-Energy Equivalence — One Gram of Mass (Worked Example)
Convert 1.0 g of mass fully to energy via E = mc². Compare the result to a TNT equivalent.…
Convex Mirror — Car Side Mirror Image (Worked Example)
A convex mirror (f = −0.40 m) shows a car 6.0 m behind. Find image distance and magnification.…
Conservation of Momentum — Rear-End Collision (Worked Example)
A 1200 kg car at 15 m/s hits a stationary 1500 kg car and they stick. Find the post-crash speed.…
Parallel Circuits — Branch Currents (Worked Example)
30 Ω and 60 Ω resistors in parallel on 12 V. Find equivalent resistance and each branch current.…
Pendulum Period — Finding Length from Period (Worked Example)
A pendulum's period is measured as 1.42 s. Work backwards to find its length.…
Pendulum Period — Pendulum on the Moon (Worked Example)
A 1.0 m pendulum on the Moon (g = 1.62 m/s²). Find its period.…
Photoelectric Effect — Cesium under Green Light (Worked Example)
500 nm light on cesium (Φ = 2.1 eV). Find max electron KE.…
Photoelectric Effect — Zinc under UV (Worked Example)
250 nm UV on zinc (Φ = 4.3 eV). Find the emitted electron's max KE.…
Photoelectric Effect — Potassium under Violet Light (Worked Example)
Violet light (400 nm) hits potassium (work function 2.29 eV). Find the ejected electrons' maximum kinetic ener…
Photoelectric Effect — Threshold Wavelength of Cesium (Worked Example)
Work backwards: given cesium's work function (2.14 eV), find the longest wavelength that can still eject elect…
Photoelectric Effect — Zinc under Ultraviolet (Worked Example)
Zinc needs 4.3 eV to release electrons. Shine 200 nm UV on it and compute the electrons' kinetic energy.…
Pressure at Depth — Mariana Trench (Worked Example)
Find pressure at 10,900 m in the Mariana Trench.…
Pressure at Depth — Submarine (Worked Example)
A research submarine dives to 300 m in seawater. Find the crushing absolute pressure with P = P₀ + ρgh — full …
Projectile Motion — Soccer Ball Kick (Worked Example)
A ball is kicked at 20 m/s at 30°. Find range, max height and flight time — full solution.…
RMS Speed — Helium Atoms at Room Temperature (Worked Example)
How fast do helium atoms zip around at 300 K? Apply the kinetic-theory RMS formula step by step.…
RMS Speed — Hydrogen, the Fastest Molecule (Worked Example)
Hydrogen is the lightest molecule. Find its RMS speed at 300 K and see why it escapes Earth's atmosphere.…
RMS Speed — Oxygen Molecules at 300 K (Worked Example)
Oxygen is 8× heavier than helium. Compute its RMS speed and see the mass dependence directly.…
RMS Speed — Helium Escape (Worked Example)
Find helium's rms speed at 300 K and explain why balloons deflate.…
RMS Speed — Nitrogen at Room Temperature (Worked Example)
Find the rms speed of N₂ molecules at 300 K with v = √(3RT/M).…
Series Circuits — Two Bulbs on One Battery (Worked Example)
60 Ω and 120 Ω bulbs in series on 12 V. Find total resistance, current and each voltage drop.…
Snell's Law — Air to Glass (Worked Example)
Light enters glass (n = 1.5) at 60°. Find the refracted angle.…
Snell's Law — Water to Air (Worked Example)
Light leaves water (n = 1.33) at 40°. Find the angle in air.…
Snell's Law — Light Leaving Water (Worked Example)
Light at 30° in water (n = 1.33) enters air. Find the refraction angle — step-by-step.…
Sound Intensity — Concert Speaker (Worked Example)
A 100 W speaker at 10 m. Find the sound intensity using the inverse-square law.…
Sound Intensity — Thunder (Worked Example)
Lightning radiates 10⁷ W of sound; observer 2 km away. Find intensity.…
Elastic Potential Energy — Bungee Cord (Worked Example)
A bungee cord (k = 100 N/m) stretches 15 m. Find stored elastic energy.…
Elastic Potential Energy — Pogo Stick (Worked Example)
A pogo spring (k = 8000 N/m) compresses 0.15 m. Find stored energy.…
Hooke's Law — Car Suspension (Worked Example)
A suspension spring (k = 30000 N/m) compresses 0.08 m. Find the force per spring.…
Hooke's Law — Trampoline (Worked Example)
A trampoline mat (k = 5000 N/m) deflects 0.35 m. Find the upward force.…
Spring Period — Car Suspension Bounce (Worked Example)
400 kg per wheel on 30000 N/m springs. Find the bounce period.…
Spring Period — Diving Board (Worked Example)
A 70 kg diver on a board with k = 4000 N/m. Find the bounce period.…
Spring Period — Trampoline Bounce (Worked Example)
A 50 kg jumper on a 5000 N/m trampoline. Find the bounce period.…
Standing Waves — Flute Second Harmonic (Worked Example)
A 0.6 m flute, sound speed 343 m/s. Find the 2nd harmonic frequency.…
Standing Waves — Fundamental on a String (Worked Example)
A 1.2 m string with wave speed 240 m/s. Find the fundamental wavelength and frequency — worked solution.…
SUVAT Displacement — Cyclist Sprint (Worked Example)
A cyclist at 8 m/s accelerates at 0.6 m/s² for 10 s. Find total distance with both SUVAT terms.…
SUVAT Displacement — Aircraft Takeoff Roll (Worked Example)
A jet accelerates from rest at 2.5 m/s² for 30 s. Find the runway distance used.…
SUVAT v² Equation — Braking Distance (Worked Example)
A car at 25 m/s brakes at −5 m/s² to rest. Find stopping distance with v² = u² + 2as.…
SUVAT v² Equation — Runway Length Needed (Worked Example)
A jet needs 80 m/s liftoff speed at 2.5 m/s² acceleration. Find minimum runway length.…
SUVAT Velocity — Rocket Second-Stage Burn (Worked Example)
A rocket at 200 m/s accelerates at 12 m/s² for 45 s. Compute burnout velocity with full working.…
Torque — Bicycle Pedal (Worked Example)
A 0.17 m crank pushed with 200 N. Find torque on the chainring.…
Torque — Balancing a Seesaw (Worked Example)
A 30 kg child sits 2.0 m from the pivot. Where must a 45 kg child sit to balance? Torque balance, step by step…
Torque — Seesaw Balance (Worked Example)
A 300 N child sits 1.5 m from the pivot. Find the torque about the pivot.…
Wave Period — Microwave Oven (Worked Example)
A microwave oven operates at 2.45 GHz. Find the period of the microwaves.…
Hard
Radioactive Activity — Banana Equivalent Dose (Worked Example)
K-40 in a banana (t½ = 1.25×10⁹ y). Find the activity.…
Radioactive Activity — Technetium-99m Tracer (Worked Example)
10¹² Tc-99m atoms (t½ = 6 h). Find the activity.…
Radioactive Activity — Radon in a Basement (Worked Example)
10¹⁰ Rn-222 atoms (t½ = 3.8 days). Find the activity in becquerels.…
Radioactive Activity — Smoke Detector (Worked Example)
0.3 μg of Am-241 (t½ = 432 y). Find the activity.…
Bernoulli's Equation — Lift on an Airplane Wing (Worked Example)
Air flows at 85 m/s over a wing and 75 m/s beneath. Estimate the pressure difference and lift.…
Centripetal Acceleration — ISS Orbit (Worked Example)
The ISS moves at 7660 m/s at radius 6.77×10⁶ m. Find its centripetal acceleration.…
Centripetal Acceleration — Washing Machine Spin (Worked Example)
A washer drum (r = 0.25 m) spins at 1200 rpm. Find the drum wall's centripetal acceleration.…
Doppler Effect — Bat Echolocation (Worked Example)
A bat emits 80 kHz flying 8 m/s toward a moth. Find the frequency at the moth.…
Electric Field — Inside a Hydrogen Atom (Worked Example)
Find the field the proton creates at the electron's orbit (Bohr radius).…
Newton's Law of Gravitation — Earth–Moon Force (Worked Example)
Compute the gravitational force between Earth and the Moon from masses and distance.…
Photoelectric Effect — Threshold Wavelength (Worked Example)
Find the longest wavelength that can eject electrons from zinc (Φ = 4.3 eV).…
Photoelectric Effect — Sodium Under Violet Light (Worked Example)
400 nm light hits sodium (work function 2.75 eV). Find the max kinetic energy of ejected electrons.…
Snell's Law — Light Escaping Diamond (Worked Example)
Light in diamond (n = 2.42) hits the surface at 30°. Does it escape?…
Snell's Law — Fiber Optic Cladding (Worked Example)
Light in glass (n = 1.5) hits the cladding at 50°. Find the critical angle and outcome.…
SUVAT v² Equation — Ski Jump Inrun (Worked Example)
A skier at 12 m/s accelerates at 4 m/s² over 50 m. Find takeoff speed.…