Wonder needs good foundations.

At the same starting radius, how does speed change an orbit?

Predict the path below and above circular speed.

1. Pause and set speed to 0.8 times circular speed.
2. Run, then reset and compare ratios 1.0 and 1.2.
3. Record closest and farthest distances.

What to measure

Compare radius r, speed v, elapsed time and energy error.

The idea underneath

Circular motion requires gravity to provide exactly the inward acceleration needed. Other launch speeds produce eccentric paths, collisions or escape, depending on energy and the planet’s radius.

Build the argument

1. Start from F = GMm/r² and divide by satellite mass: a = μ/r².
2. For circular motion set v²/r = μ/r². Then v = √(μ/r) and T = 2π√(r³/μ).
3. Combine kinetic and potential energy per mass: ε = v²/2 − μ/r. Its sign separates bound and unbound trajectories.

Worked example 1

With μ=3.986×10¹⁴ m³/s² and r=7×10⁶ m, v_c=7546 m/s.

Worked example 2

At that radius T=5829 s≈97.1 min. Doubling r multiplies T by 2√2.

Applications

Orbital mechanics connects satellite navigation, communications and interplanetary transfers. A launch is a choice of position and velocity, not a command to follow a circle.

Common misconception

Zero weight is not zero gravity. Orbital radius is measured from the centre. Numerical energy drift must not be mistaken for real energy production.

A further investigation

What radius factor doubles a circular orbital period?

Check your understanding

Where does acceleration point?

• Toward the centre
• Along velocity
• It is zero

Model limits

Newtonian two-body model with fixed central mass. No drag, third bodies, oblateness or relativity. Visual sizes are illustrative; initial conditions are fictional, not live ephemerides.

University Physics Volume 1, 13.4 Satellite Orbits and Energy

How can a satellite keep falling without reaching the ground?

Without gravity, would it move radially outward or along the tangent?

1. Start a circular orbit and show vectors.
2. Pause and advance one step at a time.
3. Compare how velocity and acceleration turn.

What to measure

Compare x, y, v_x, v_y and |a| at two positions.

The idea underneath

Velocity carries the body along a tangent while gravity bends its path inward. Apparent weightlessness is shared free fall, not the absence of gravity. In circular motion acceleration changes direction without changing speed.

Build the argument

1. Start from F = GMm/r² and divide by satellite mass: a = μ/r².
2. For circular motion set v²/r = μ/r². Then v = √(μ/r) and T = 2π√(r³/μ).
3. Combine kinetic and potential energy per mass: ε = v²/2 − μ/r. Its sign separates bound and unbound trajectories.

Worked example 1

At r=7×10⁶ m, acceleration is 8.13 m/s², far from zero.

Worked example 2

In 1 s the small inward fall is approximately aΔt²/2=4.07 m while tangential travel is about 7.55 km.

Applications

Orbital mechanics connects satellite navigation, communications and interplanetary transfers. A launch is a choice of position and velocity, not a command to follow a circle.

Common misconception

Zero weight is not zero gravity. Orbital radius is measured from the centre. Numerical energy drift must not be mistaken for real energy production.

A further investigation

Use a dot product to check perpendicular velocity and acceleration.

Check your understanding

Why do orbiting occupants feel weightless?

• There is no gravity
• Spacecraft and contents fall together

Model limits

Newtonian two-body model with fixed central mass. No drag, third bodies, oblateness or relativity. Visual sizes are illustrative; initial conditions are fictional, not live ephemerides.

University Physics Volume 1, 13.2 Gravitation Near Earth’s Surface

University Physics Volume 1, 13.4 Satellite Orbits and Energy

How much weaker is gravity at twice the centre distance?

Half as strong or one quarter? Record a prediction.

1. Set the initial centre radius to 7000 km and record acceleration.
2. Double radius to 14000 km, reset, and compare.
3. Hold radius fixed and double central mass.

What to measure

Calculate a₂/a₁ and a·r²/μ; the second should equal one.

The idea underneath

Acceleration scales with central mass and inversely with squared centre distance. Radius is measured from the centre, not the surface. Logarithmic plots provide a useful data-analysis check: log(g) against log(r) has slope −2.

Build the argument

1. Start from F = GMm/r² and divide by satellite mass: a = μ/r².
2. For circular motion set v²/r = μ/r². Then v = √(μ/r) and T = 2π√(r³/μ).
3. Combine kinetic and potential energy per mass: ε = v²/2 − μ/r. Its sign separates bound and unbound trajectories.

Worked example 1

At r=7×10⁶ m, g=8.13 m/s²; at 14×10⁶ m, g=2.03 m/s².

Worked example 2

Doubling both mass and radius gives g₂/g₁=2/4=0.5.

Applications

Orbital mechanics connects satellite navigation, communications and interplanetary transfers. A launch is a choice of position and velocity, not a command to follow a circle.

Common misconception

Zero weight is not zero gravity. Orbital radius is measured from the centre. Numerical energy drift must not be mistaken for real energy production.

A further investigation

If radius doubles, what mass factor preserves acceleration?

Check your understanding

If satellite mass doubles, what happens to its acceleration?

• It doubles
• It stays the same

Model limits

Newtonian two-body model with fixed central mass. No drag, third bodies, oblateness or relativity. Visual sizes are illustrative; initial conditions are fictional, not live ephemerides.

University Physics Volume 1, 13.2 Gravitation Near Earth’s Surface

What separates a bound orbit from an escape trajectory?

Predict the energy sign at 1.4 and 1.5 times circular speed.

1. Increase launch speed gradually.
2. Read K, U and E; find the zero-energy threshold.
3. Monitor numerical energy error over a bound orbit.

What to measure

Specific energy ε=E/m in J/kg and numerical energy error.

The idea underneath

With potential zero at infinity, negative energy is bound, zero is marginal escape, and positive energy is unbound. Numerical drift is not physical energy production. Near zero total energy, compare error against the initial binding-energy scale to avoid a misleading relative error.

Build the argument

1. Start from F = GMm/r² and divide by satellite mass: a = μ/r².
2. For circular motion set v²/r = μ/r². Then v = √(μ/r) and T = 2π√(r³/μ).
3. Combine kinetic and potential energy per mass: ε = v²/2 − μ/r. Its sign separates bound and unbound trajectories.

Worked example 1

At r=7×10⁶ m, v_esc≈10672 m/s, 41.4% above circular speed.

Worked example 2

Circular specific energy is −28.47 MJ/kg. A 1000 kg body has E=−28.47 GJ.

Applications

Orbital mechanics connects satellite navigation, communications and interplanetary transfers. A launch is a choice of position and velocity, not a command to follow a circle.

Common misconception

Zero weight is not zero gravity. Orbital radius is measured from the centre. Numerical energy drift must not be mistaken for real energy production.

A further investigation

Why does a smaller timestep reduce numerical error without changing launch conditions?

Check your understanding

Escape speed is what multiple of circular speed?

• √2
• 2
• 1/2

Model limits

Newtonian two-body model with fixed central mass. No drag, third bodies, oblateness or relativity. Visual sizes are illustrative; initial conditions are fictional, not live ephemerides.

University Physics Volume 1, 13.4 Satellite Orbits and Energy

How does charge sign affect field and force direction?

Will a negative test charge accelerate with or against the field?

1. Place one positive source and inspect field arrows.
2. Reverse its sign and release a test particle.
3. Change the test charge and compare acceleration.

What to measure

E_x and E_y in N/C; F=qE in N.

The idea underneath

Field direction is defined using a positive test charge. A negative test charge feels the opposite force. The field itself depends on the sources, not the test charge.

Build the argument

1. Evaluate each source separately and add its vector contribution. Potential adds as a signed scalar.
2. Use F = qE or F = q(v × B), then divide by mass to obtain acceleration.
3. The electric model uses consistent softened potential and field with ε = 0.2 m, so values very near a source differ from ideal point charges.

Worked example 1

For Q=1 nC at r=1 m, E=8.99 N/C outward.

Worked example 2

In that field q=−2 nC gives F=−17.98 nN, toward the source. These unsoftened calculations apply away from the core.

Applications

Electrostatic lenses steer electrons; magnetic fields separate particles by momentum and charge. Both depend on vector direction as much as field strength.

Common misconception

A field line is not generally a particle path. A negative charge accelerates opposite E. A purely magnetic force does no work.

A further investigation

What happens to field strength when distance triples?

Check your understanding

Where does a negative source’s field point?

• Toward the charge
• Away from the charge

Model limits

Static sources and Coulomb superposition. A softened core regularises singularities. Field lines are visual constructions whose density depends on seeding. This is not a time-dependent Maxwell solver.

University Physics Volume 2, 5.4 Electric Field

Does the field cancel between equal opposite charges?

At a dipole midpoint, will potential and field both be zero?

1. Place equal +Q and −Q on opposite sides.
2. Inspect midpoint field and potential.
3. Change −Q to +Q and compare.

What to measure

Vector sum E=E₁+E₂ and scalar sum V=V₁+V₂.

The idea underneath

Between opposite sources the field vectors align and add, while signed scalar potentials cancel at the midpoint. For equal like charges the midpoint field cancels instead. This is a practical linear-algebra distinction between scalar and vector addition.

Build the argument

1. Evaluate each source separately and add its vector contribution. Potential adds as a signed scalar.
2. Use F = qE or F = q(v × B), then divide by mass to obtain acceleration.
3. The electric model uses consistent softened potential and field with ε = 0.2 m, so values very near a source differ from ideal point charges.

Worked example 1

For ±1 nC at x=±1 m, midpoint |E|=17.98 N/C and V=0.

Worked example 2

For two +1 nC sources at the same locations, midpoint E=0 and V=17.98 V, before core softening.

Applications

Electrostatic lenses steer electrons; magnetic fields separate particles by momentum and charge. Both depend on vector direction as much as field strength.

Common misconception

A field line is not generally a particle path. A negative charge accelerates opposite E. A purely magnetic force does no work.

A further investigation

Double one like charge and locate the new zero-field point.

Check your understanding

Does V=0 imply E=0?

• Always
• No

Model limits

Static sources and Coulomb superposition. A softened core regularises singularities. Field lines are visual constructions whose density depends on seeding. This is not a time-dependent Maxwell solver.

University Physics Volume 2, 5.4 Electric Field

University Physics Volume 2, 7.2 Electric Potential and Potential Difference

Is a field line a particle trajectory?

Will a particle with sideways velocity exactly follow a field line?

1. Set up a dipole and show potential.
2. Release a particle and compare its trajectory with the lines.
3. Change initial velocity and compare paths in the same field.

What to measure

V in volts, U=qV in joules, particle position and speed.

The idea underneath

A field line represents local direction. A particle carries momentum: acceleration follows qE/m, but velocity need not align with E. For a positive charge descending in potential, potential energy becomes kinetic energy. This distinction matters in electron optics.

Build the argument

1. Evaluate each source separately and add its vector contribution. Potential adds as a signed scalar.
2. Use F = qE or F = q(v × B), then divide by mass to obtain acceleration.
3. The electric model uses consistent softened potential and field with ε = 0.2 m, so values very near a source differ from ideal point charges.

Worked example 1

A +2 nC charge falling through 10 V gains 20 nJ of kinetic energy.

Worked example 2

For m=10⁻⁹ kg, starting from rest gives v=√(2ΔK/m)=6.32 m/s.

Applications

Electrostatic lenses steer electrons; magnetic fields separate particles by momentum and charge. Both depend on vector direction as much as field strength.

Common misconception

A field line is not generally a particle path. A negative charge accelerates opposite E. A purely magnetic force does no work.

A further investigation

Why can a negative charge gain kinetic energy while moving to higher voltage?

Check your understanding

What does a field arrow directly determine?

• Positive-charge acceleration direction
• Every possible initial velocity

Model limits

Static sources and Coulomb superposition. A softened core regularises singularities. Field lines are visual constructions whose density depends on seeding. This is not a time-dependent Maxwell solver.

University Physics Volume 2, 7.2 Electric Potential and Potential Difference

University Physics Volume 2, 5.4 Electric Field

Does a magnetic field increase particle speed?

Will direction, speed, or both change?

1. Select the magnetic experiment.
2. Follow the circular trajectory in a uniform field.
3. Explain how reversing charge sign would reverse the turning direction.

What to measure

Radius r, rotation period T, and constant speed.

The idea underneath

Magnetic force is perpendicular to velocity and B. It does no work, so it bends motion without increasing speed. A known B and a measured radius can reveal m/|q|, the principle behind mass spectrometry.

Build the argument

1. Evaluate each source separately and add its vector contribution. Potential adds as a signed scalar.
2. Use F = qE or F = q(v × B), then divide by mass to obtain acceleration.
3. The electric model uses consistent softened potential and field with ε = 0.2 m, so values very near a source differ from ideal point charges.

Worked example 1

For illustrative q/m=1 C/kg, v=2 m/s and B=1.5 T, r=1.333 m.

Worked example 2

With these values T=2π/1.5=4.189 s. Doubling B halves both r and T.

Applications

Electrostatic lenses steer electrons; magnetic fields separate particles by momentum and charge. Both depend on vector direction as much as field strength.

Common misconception

A field line is not generally a particle path. A negative charge accelerates opposite E. A purely magnetic force does no work.

A further investigation

What is the force if v is parallel to B?

Check your understanding

Does a pure magnetic force do work on a point charge?

• Yes
• No

Model limits

Analytic circle in uniform static B. No electric field, radiation, collisions or relativity. Schematic educational values, not a manufacturing design.

University Physics Volume 2, 11.3 Motion of a Charged Particle in a Magnetic Field

At fixed speed, what happens to wavelength as frequency rises?

Predict crest spacing when frequency doubles.

1. Select the travelling wave.
2. Hold propagation speed fixed and change frequency.
3. Follow the distance a crest travels in one complete cycle.

What to measure

f in Hz; λ in m; v=fλ in m/s; T=1/f in s.

The idea underneath

Frequency counts cycles per second; wavelength is the distance a crest travels in one cycle. Amplitude is maximum displacement from equilibrium. In this nondispersive model, changing frequency changes wavelength without changing propagation speed.

Build the argument

1. Write a travelling disturbance as y = A sin(kx − ωt + φ), with k = 2π/λ and ω = 2πf.
2. Keeping phase constant gives v = ω/k = fλ. Add displacements, not amplitudes, when phases differ.
3. Two waves travelling in opposite directions form nodes when their sum is zero for all times.

Worked example 1

For v=4 m/s and f=2 Hz, λ=2 m and T=0.5 s.

Worked example 2

At the same v=4 m/s, f=4 Hz gives λ=1 m.

Applications

The same language describes musical instruments, communications, seismic waves and interference-based measurement. Frequency content reveals structure hidden in a time trace.

Common misconception

The moving pattern is not the motion of a piece of the medium. Interference redistributes energy. A DFT has finite resolution and a Nyquist limit.

A further investigation

How can a wave move right while a material point moves up and down?

Check your understanding

At fixed v, doubling f makes λ…

• Half as large
• Twice as large

Model limits

Analytic linear educational model, not a numerical fluid solver. No damping, dispersion or nonlinear coupling.

University Physics Volume 1, 16.2 Mathematics of Waves

Can two nonzero waves add to zero?

Predict the sum of equal waves with phase difference π.

1. Select the interference experiment.
2. Change relative phase from 0 to π ≈ 3.14 rad (180°).
3. Compare both component waves and their sum.

What to measure

At a point inspect y₁, y₂ and y₁+y₂; measure resultant amplitude.

The idea underneath

Linear displacements add. Equal contributions reinforce in phase and cancel in antiphase. Cancellation at a point does not destroy energy; spatial interference redistributes it. This principle connects to noise cancellation and antenna arrays. A phase-offset trace is a local slice, not a complete spatial radiation model.

Build the argument

1. Write a travelling disturbance as y = A sin(kx − ωt + φ), with k = 2π/λ and ω = 2πf.
2. Keeping phase constant gives v = ω/k = fλ. Add displacements, not amplitudes, when phases differ.
3. Two waves travelling in opposite directions form nodes when their sum is zero for all times.

Worked example 1

For A=0.2 m and Δφ=0, resultant amplitude is 0.4 m.

Worked example 2

At Δφ=π equal contributions cancel. At Δφ=π/2 the resultant is 0.283 m.

Applications

The same language describes musical instruments, communications, seismic waves and interference-based measurement. Frequency content reveals structure hidden in a time trace.

Common misconception

The moving pattern is not the motion of a piece of the medium. Interference redistributes energy. A DFT has finite resolution and a Nyquist limit.

A further investigation

Why do 0.2 m and 0.1 m antiphase waves not cancel completely?

Check your understanding

Equal in-phase waves produce amplitude…

• 2A
• 0
• A/2

Model limits

Analytic linear educational model, not a numerical fluid solver. No damping, dispersion or nonlinear coupling.

University Physics Volume 1, 16.5 Interference of Waves

Why do some points never move in a standing wave?

How many internal nodes will the second harmonic have?

1. Select standing waves and inspect the fundamental.
2. Increase harmonic number from 1 to 2 and 3.
3. Count nodes and compare their spacing with λ/2.

What to measure

Harmonic n, node positions, f_n and λ_n.

The idea underneath

Equal counterpropagating waves make a stationary amplitude envelope. Fixed ends must remain nodes, allowing only integer numbers of half-wavelengths. These modes explain string-instrument pitches and help engineers avoid damaging structural resonances.

Build the argument

1. Write a travelling disturbance as y = A sin(kx − ωt + φ), with k = 2π/λ and ω = 2πf.
2. Keeping phase constant gives v = ω/k = fλ. Add displacements, not amplitudes, when phases differ.
3. Two waves travelling in opposite directions form nodes when their sum is zero for all times.

Worked example 1

For L=2 m and v=4 m/s, f₁=1 Hz and λ₁=4 m.

Worked example 2

For n=3, f₃=3 Hz and λ₃=4/3 m, with two internal nodes.

Applications

The same language describes musical instruments, communications, seismic waves and interference-based measurement. Frequency content reveals structure hidden in a time trace.

Common misconception

The moving pattern is not the motion of a piece of the medium. Interference redistributes energy. A DFT has finite resolution and a Nyquist limit.

A further investigation

At fixed v, what happens to the fundamental if length doubles?

Check your understanding

How many internal nodes occur in the second harmonic?

• One
• Two
• None

Model limits

Analytic linear educational model, not a numerical fluid solver. No damping, dispersion or nonlinear coupling.

University Physics Volume 1, 16.6 Standing Waves and Resonance

How can simple waves build a complex signal?

Where will a third harmonic create a spectral peak?

1. Select signal and spectrum.
2. Set the fundamental to 2 Hz.
3. Change the third-harmonic amplitude; compare the time trace and DFT.

What to measure

Sample rate f_s, N, bin spacing Δf=f_s/N, Nyquist limit f_s/2, and peak frequencies.

The idea underneath

The DFT projects samples onto sinusoidal basis functions. The spectrum is calculated from the synthetic signal, not decoration. A one-second observation resolves bins 1 Hz apart. Noninteger cycles cause spectral leakage. Frequencies beyond half the sample rate can alias into lower ones; practical electronics use antialias filters.

Build the argument

1. Write a travelling disturbance as y = A sin(kx − ωt + φ), with k = 2π/λ and ω = 2πf.
2. Keeping phase constant gives v = ω/k = fλ. Add displacements, not amplitudes, when phases differ.
3. Two waves travelling in opposite directions form nodes when their sum is zero for all times.

Worked example 1

For f₀=2 Hz and A₃=0.3, peaks occur at 2 and 6 Hz with amplitudes 1 and 0.3.

Worked example 2

For f_s=256 Hz and N=256, Δf=1 Hz and Nyquist is 128 Hz. A 150 Hz component aliases to 106 Hz.

Applications

The same language describes musical instruments, communications, seismic waves and interference-based measurement. Frequency content reveals structure hidden in a time trace.

Common misconception

The moving pattern is not the motion of a piece of the medium. Interference redistributes energy. A DFT has finite resolution and a Nyquist limit.

A further investigation

At fixed f_s, how does doubling observation time change Δf?

Check your understanding

For an ideally bandlimited signal below 10 Hz, sample rate must exceed…

• 10 Hz
• 20 Hz
• 5 Hz

Model limits

Analytic linear educational model, not a numerical fluid solver. No damping, dispersion or nonlinear coupling.

6.003 Signals and Systems — Lecture Notes

What sets a pendulum’s period?

What sets a pendulum’s period?

1. Set length to 1 m and record the period.
2. Increase length to 2 m; predict the period ratio first.
3. Change release angle from 4° to 12° and compare.

What to measure

L in m; g in m/s²; θ in radians; T in seconds.

The idea underneath

For small angles, the tangential restoring acceleration is proportional to angular displacement. The result is harmonic motion with a period set by length and gravity, not bob mass. The model restricts release angles to 14° so the small-angle approximation remains meaningful. Increasing length slows the rhythm; increasing gravity speeds it up. Track the bob’s changing height and speed: gravitational potential energy becomes kinetic energy and back. The three-dimensional suspension is a visual aid; the bob is constrained to a plane.

Build the argument

1. Draw the forces and choose coordinates before writing a differential equation.
2. Check the result against energy conservation and dimensional consistency.
3. Change one parameter at a time and compare a ratio rather than relying on visual impressions.

Worked example 1

At L=1 m and g=9.81 m/s², T=2.006 s.

Worked example 2

At L=2 m the period is 2.837 s, √2 times longer.

Applications

Oscillators appear in clocks, vibration isolation, sensors and mechanical structures. Conserved quantities make complicated motion easier to predict.

Common misconception

A force changes velocity, not just speed. A massless rod or ideal spring is a model assumption, and small-angle pendulum formulas have a limited range.

A further investigation

Find the length needed for a 2 s period.

Check your understanding

Doubling length changes the period by…

• √2
• 2
• 1/2

Model limits

Small-angle analytic model; massless rigid suspension, no air drag or pivot friction.

OpenStax · Pendulums

Where does an oscillator store its energy?

Where does an oscillator store its energy?

1. Pause at a turning point and identify the energy store.
2. Double stiffness while keeping mass fixed.
3. Double amplitude and compare total energy.

What to measure

k in N/m; m in kg; A and x in m; E in J.

The idea underneath

A Hookean spring exerts a force proportional and opposite to displacement. With no damping the mass executes a sinusoid. Potential energy peaks at the turning points, where speed is zero; kinetic energy peaks at equilibrium, where spring force is zero. Zero force at one instant does not mean zero velocity. Changing stiffness changes the time scale, while changing amplitude changes energy quadratically.

Build the argument

1. Draw the forces and choose coordinates before writing a differential equation.
2. Check the result against energy conservation and dimensional consistency.
3. Change one parameter at a time and compare a ratio rather than relying on visual impressions.

Worked example 1

k=12 N/m, m=1 kg gives T=1.814 s.

Worked example 2

A=0.8 m gives E=3.84 J. At A=0.4 m, E=0.96 J.

Applications

Oscillators appear in clocks, vibration isolation, sensors and mechanical structures. Conserved quantities make complicated motion easier to predict.

Common misconception

A force changes velocity, not just speed. A massless rod or ideal spring is a model assumption, and small-angle pendulum formulas have a limited range.

A further investigation

Choose a mass that halves the frequency.

Check your understanding

At equilibrium the kinetic energy is…

• Maximum
• Zero
• Negative

Model limits

Ideal linear spring, no damping or gravity along the horizontal track.

OpenStax · Simple harmonic motion

Why does a projectile trace a parabola?

Why does a projectile trace a parabola?

1. Compare launches at 30° and 60°.
2. Set 45°, then double launch speed.
3. Reduce gravity and compare flight time and range.

What to measure

v₀ in m/s; g in m/s²; θ is launch angle; R in m.

The idea underneath

Horizontal motion is uniform while vertical motion has constant downward acceleration. Combining these two independent motions produces a parabola. At the apex vertical velocity is zero but acceleration remains −g. For launch and landing at equal heights, complementary angles give equal ranges, and 45° gives the greatest range only under the ideal no-drag assumptions. Each playback cycle repeats a fresh launch; it does not add energy to a bouncing object.

Build the argument

1. Draw the forces and choose coordinates before writing a differential equation.
2. Check the result against energy conservation and dimensional consistency.
3. Change one parameter at a time and compare a ratio rather than relying on visual impressions.

Worked example 1

v₀=8 m/s, θ=45°, g=9.81 gives R=6.524 m.

Worked example 2

The apex is 1.631 m and flight time is 1.153 s.

Applications

Oscillators appear in clocks, vibration isolation, sensors and mechanical structures. Conserved quantities make complicated motion easier to predict.

Common misconception

A force changes velocity, not just speed. A massless rod or ideal spring is a model assumption, and small-angle pendulum formulas have a limited range.

A further investigation

Find two different angles with the same range.

Check your understanding

At the apex, acceleration is…

• Downward, g
• Zero
• Horizontal

Model limits

Point projectile, flat ground, constant g, no drag, equal launch and landing heights.

OpenStax · Projectile motion

Why does pulling mass inward increase angular speed?

Why does pulling mass inward increase angular speed?

1. Record angular speed at r=2 m.
2. Halve radius without changing angular momentum.
3. Compare the kinetic-energy ratio.

What to measure

I in kg m²; L in kg m²/s; ω in rad/s.

The idea underneath

With no external torque, angular momentum remains constant. A point mass on a circular track has I=mr², so decreasing radius requires a larger angular speed. Rotational kinetic energy increases during inward motion because the mechanism pulling the mass does work. Conservation of angular momentum does not require kinetic energy to stay constant while the radius is changed. The laboratory compares equilibrium circular states rather than modelling the radial actuator.

Build the argument

1. Draw the forces and choose coordinates before writing a differential equation.
2. Check the result against energy conservation and dimensional consistency.
3. Change one parameter at a time and compare a ratio rather than relying on visual impressions.

Worked example 1

L=1.5, m=1 kg, r=1.5 m gives ω=0.667 rad/s.

Worked example 2

Halving radius multiplies both ω and K by four.

Applications

Oscillators appear in clocks, vibration isolation, sensors and mechanical structures. Conserved quantities make complicated motion easier to predict.

Common misconception

A force changes velocity, not just speed. A massless rod or ideal spring is a model assumption, and small-angle pendulum formulas have a limited range.

A further investigation

Explain where the extra kinetic energy comes from.

Check your understanding

At fixed L, halving radius multiplies ω by…

• 4
• 2
• 1/2

Model limits

A point mass on a prescribed circular track; radius changes select new ideal states.

OpenStax · Angular momentum

How does tilting a mirror steer a beam?

How does tilting a mirror steer a beam?

1. Choose 30° incidence and identify both angles.
2. Rotate the mirror by 10° with the incoming beam fixed.
3. Compare the outgoing directions before and after.

What to measure

All angles are measured from the local normal, in degrees in the controls.

The idea underneath

Specular reflection preserves the angle to the local surface normal. Rotating a mirror rotates that normal and changes the outgoing beam direction by twice the mirror rotation when the incoming direction is held fixed. The scene displays a family of parallel rays on an optical bench and the normal at their intersection. Rays are geometric paths, not the transverse oscillation of the electromagnetic field.

Build the argument

1. Identify the optical axis and the normal to each interface.
2. Apply reflection, Snell’s law, or the paraxial lens equation according to the model.
3. Check limiting cases: equal indices, object at the focal plane, or a slit much wider than the wavelength.

Worked example 1

An incident ray at 35° reflects at 35° to the normal.

Worked example 2

A 5° mirror rotation steers a fixed incoming beam by 10°.

Applications

Cameras, microscopes and fibre links combine geometric paths with wave limitations. An image can be sharp geometrically and still be limited by diffraction.

Common misconception

Angles are measured from the normal, not the surface. A virtual image is located by backward ray extensions; no rays physically converge there.

A further investigation

Choose a mirror tilt that steers the beam by 24°.

Check your understanding

Reflection angles are measured from…

• The normal
• The surface
• The camera

Model limits

Perfect plane mirror and geometric rays; no roughness, absorption or polarisation.

OpenStax · Refraction

When does light bend, and when can it not escape?

When does light bend, and when can it not escape?

1. Set n₁=1 and n₂=1.5, then increase incidence.
2. Swap the indices and look for the critical angle.
3. Compare with equal-index media.

What to measure

n is refractive index; c/n is phase speed; angles are to the normal.

The idea underneath

A wave changes speed when it crosses media with different refractive indices. Continuity of phase along the interface gives Snell’s law. Entering a higher-index medium bends a ray toward the normal. From higher to lower index, an angle can be reached beyond which no propagating transmitted ray exists: total internal reflection. The laboratory explicitly reports this state rather than drawing an impossible refracted ray.

Build the argument

1. Identify the optical axis and the normal to each interface.
2. Apply reflection, Snell’s law, or the paraxial lens equation according to the model.
3. Check limiting cases: equal indices, object at the focal plane, or a slit much wider than the wavelength.

Worked example 1

At 30° from air to n=1.5, θ₂=19.47°.

Worked example 2

For n₁=1.5 to n₂=1, θc=41.81°.

Applications

Cameras, microscopes and fibre links combine geometric paths with wave limitations. An image can be sharp geometrically and still be limited by diffraction.

Common misconception

Angles are measured from the normal, not the surface. A virtual image is located by backward ray extensions; no rays physically converge there.

A further investigation

Find a pair of indices with a critical angle near 30°.

Check your understanding

Total internal reflection requires…

• Higher to lower index
• Lower to higher index
• Equal indices

Model limits

Homogeneous, lossless media; no dispersion or Fresnel intensity calculation.

OpenStax · Refraction

How can a lens make both real and virtual images?

How can a lens make both real and virtual images?

1. Place the object at three focal lengths.
2. Move toward the focal plane and watch the image distance.
3. Move inside the focus and inspect backward extensions.

What to measure

f, dₒ and dᵢ in m; negative dᵢ denotes a virtual image.

The idea underneath

A converging thin lens redirects paraxial rays so parallel light reaches the focal point. An object outside the focal length makes a real inverted image on the other side. Inside the focal length, outgoing rays diverge and their backward extensions locate an upright virtual image. At exactly one focal length the image is at infinity; the readout handles this singular case explicitly. The glass shape is illustrative; the calculation uses the thin-lens equation.

Build the argument

1. Identify the optical axis and the normal to each interface.
2. Apply reflection, Snell’s law, or the paraxial lens equation according to the model.
3. Check limiting cases: equal indices, object at the focal plane, or a slit much wider than the wavelength.

Worked example 1

f=1 m and dₒ=3 m give dᵢ=1.5 m and M=−0.5.

Worked example 2

f=1 m and dₒ=0.5 m give dᵢ=−1 m and M=2.

Applications

Cameras, microscopes and fibre links combine geometric paths with wave limitations. An image can be sharp geometrically and still be limited by diffraction.

Common misconception

Angles are measured from the normal, not the surface. A virtual image is located by backward ray extensions; no rays physically converge there.

A further investigation

Locate the object that makes a real image of equal size.

Check your understanding

A negative image distance means…

• Virtual image
• No light
• Negative energy

Model limits

Paraxial thin converging lens. No aberrations, thickness, dispersion or diffraction.

OpenStax · Thin lenses

Why does a narrow slit make a wide pattern?

Why does a narrow slit make a wide pattern?

1. Set λ=550 nm and width a=3 µm.
2. Halve the width and compare the first minimum.
3. Increase screen distance without changing the angular pattern.

What to measure

a is slit width; λ is wavelength; β is dimensionless.

The idea underneath

Different parts of an illuminated slit act as coherent contributors. At certain angles their contributions cancel, leaving minima. Narrowing the slit spreads the central maximum; increasing wavelength does the same. This is a far-field intensity model, not a ray shadow. The bench visualises the slit and screen in three dimensions and samples the exact sinc-squared angular pattern.

Build the argument

1. Identify the optical axis and the normal to each interface.
2. Apply reflection, Snell’s law, or the paraxial lens equation according to the model.
3. Check limiting cases: equal indices, object at the focal plane, or a slit much wider than the wavelength.

Worked example 1

λ=500 nm, a=2 µm gives sinθ₁=0.25, θ₁=14.48°.

Worked example 2

At small angles the central width is approximately 2Lλ/a.

Applications

Cameras, microscopes and fibre links combine geometric paths with wave limitations. An image can be sharp geometrically and still be limited by diffraction.

Common misconception

Angles are measured from the normal, not the surface. A virtual image is located by backward ray extensions; no rays physically converge there.

A further investigation

Double slit width and wavelength together. What stays fixed?

Check your understanding

A narrower slit produces…

• A wider central maximum
• A narrower maximum
• No diffraction

Model limits

Monochromatic coherent illumination, single ideal slit, Fraunhofer far field; no near-field diffraction.

OpenStax · Single-slit diffraction

How do invisible molecular impacts become pressure?

How do invisible molecular impacts become pressure?

1. Record pressure at 300 K and a 5 m box side.
2. Double temperature at fixed side length.
3. Increase side length; remember volume scales cubically.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

An ideal gas is a statistical model of many independent particles. Elastic wall collisions reverse the normal velocity component while preserving kinetic energy. The sum of many momentum transfers creates pressure. The displayed ensemble samples a Maxwell distribution; molecules do not all move at the RMS speed. The readout is for one mole of ideal gas in a cubic container, while rendered particles are representative samples. Their playback is slowed and their sizes exaggerated. Increasing temperature broadens the velocity distribution, while increasing volume at fixed temperature reduces pressure.

Build the argument

1. Specify the ensemble, particle mass, temperature and boundary conditions.
2. Average over many particles before comparing microscopic and macroscopic quantities.
3. For independent isotropic diffusion, each coordinate has variance 2Dt, giving mean squared displacement 6Dt in three dimensions.

Worked example 1

One mole at 300 K in 125 m³ gives P=19.95 Pa.

Worked example 2

Doubling box side divides pressure by eight at fixed T.

Applications

Microscopic statistics connect to pressure, diffusion and transport. The visible particles represent an ensemble; their rendered size and playback speed are illustrative.

Common misconception

Temperature is a distribution of molecular kinetic energies, not the speed of every molecule. A stochastic trajectory cannot predict a particular future collision.

A further investigation

Doubling temperature at fixed volume makes pressure…

Check your understanding

Doubling temperature at fixed volume makes pressure…

• Twice as large
• Half as large
• Unchanged

Model limits

Monatomic ideal-gas thermodynamics; independent point particles and elastic walls; no intermolecular collisions or real-gas corrections.

OpenStax · Molecular model of an ideal gas

What does thermal speed actually mean?

What does thermal speed actually mean?

1. Compare 4 g/mol and 28 g/mol at 300 K.
2. Increase temperature and watch the histogram broaden.
3. Compare speed ratios with the square-root law.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

Each Cartesian molecular velocity component follows a Gaussian distribution at equilibrium. Speed is the length of the three-component vector, so its distribution is not Gaussian. The most probable speed, mean speed and RMS speed are three different numbers. The RMS value connects directly to kinetic energy. A heavier gas at the same temperature has lower characteristic speeds because the mean kinetic energy per particle is the same. The three-dimensional ensemble uses the same seeded normal samples as the visible speed histogram, making changes reproducible.

Build the argument

1. Specify the ensemble, particle mass, temperature and boundary conditions.
2. Average over many particles before comparing microscopic and macroscopic quantities.
3. For independent isotropic diffusion, each coordinate has variance 2Dt, giving mean squared displacement 6Dt in three dimensions.

Worked example 1

At 300 K and 28 g/mol, vᵣₘₛ≈517 m/s.

Worked example 2

Quadrupling temperature doubles RMS speed.

Applications

Microscopic statistics connect to pressure, diffusion and transport. The visible particles represent an ensemble; their rendered size and playback speed are illustrative.

Common misconception

Temperature is a distribution of molecular kinetic energies, not the speed of every molecule. A stochastic trajectory cannot predict a particular future collision.

A further investigation

At equal temperature, a heavier molecule has…

Check your understanding

At equal temperature, a heavier molecule has…

• Lower RMS speed
• Higher mean kinetic energy
• Zero motion

Model limits

Classical dilute equilibrium gas; finite representative ensemble, not a many-body collision solver.

OpenStax · Molecular model of an ideal gas

Can random motion have a predictable law?

Can random motion have a predictable law?

1. Run the tracer and inspect the trail from two camera angles.
2. Double radius while holding temperature fixed.
3. Compare the theoretical RMS displacement at equal time.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

A suspended tracer receives many uneven microscopic impulses. Its individual path is irregular, but the mean squared displacement of an ensemble follows a simple law. The laboratory integrates independent Gaussian increments with variance 2DΔt in each direction. This is an overdamped diffusion model, not a resolved simulation of individual solvent impacts. The effective coefficient increases with temperature and decreases with tracer radius, following the qualitative Stokes–Einstein scaling for fixed viscosity. Compare the ensemble spread, not the endpoint of a single random run.

Build the argument

1. Specify the ensemble, particle mass, temperature and boundary conditions.
2. Average over many particles before comparing microscopic and macroscopic quantities.
3. For independent isotropic diffusion, each coordinate has variance 2Dt, giving mean squared displacement 6Dt in three dimensions.

Worked example 1

D=0.06 m²/s over 10 s gives RMS=√3.6=1.897 m.

Worked example 2

Four times the time gives twice the RMS distance.

Applications

Microscopic statistics connect to pressure, diffusion and transport. The visible particles represent an ensemble; their rendered size and playback speed are illustrative.

Common misconception

Temperature is a distribution of molecular kinetic energies, not the speed of every molecule. A stochastic trajectory cannot predict a particular future collision.

A further investigation

RMS displacement grows with time as…

Check your understanding

RMS displacement grows with time as…

• √t
• t²
• A constant

Model limits

Gaussian overdamped diffusion at fixed viscosity; no hydrodynamic interactions. Distances are model coordinates.

MIT OpenCourseWare · Random walks and diffusion

Why does a concentrated cloud spread?

Why does a concentrated cloud spread?

1. Start with a narrow cloud and pause after five seconds.
2. Double D and compare at equal elapsed time.
3. Increase sample count to reduce visual fluctuations.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

Independent random walks turn a narrow initial concentration into a broader distribution. There is no outward force pushing particles away from the centre: more particles simply leave dense regions than enter them. For an initially Gaussian cloud, variance grows as σ²=σ₀²+2Dt on each axis. The cloud centre fluctuates for a finite sample, so increasing sample count improves statistical stability without changing the diffusion coefficient. The scene has open boundaries; the container grid is a scale reference, not a confining wall.

Build the argument

1. Specify the ensemble, particle mass, temperature and boundary conditions.
2. Average over many particles before comparing microscopic and macroscopic quantities.
3. For independent isotropic diffusion, each coordinate has variance 2Dt, giving mean squared displacement 6Dt in three dimensions.

Worked example 1

σ₀=0.25 m, D=0.08 m²/s, t=5 s gives σ=0.929 m per axis.

Worked example 2

The three-dimensional RMS radius is √(3σ₀²+6Dt).

Applications

Microscopic statistics connect to pressure, diffusion and transport. The visible particles represent an ensemble; their rendered size and playback speed are illustrative.

Common misconception

Temperature is a distribution of molecular kinetic energies, not the speed of every molecule. A stochastic trajectory cannot predict a particular future collision.

A further investigation

Increasing particle count changes the diffusion coefficient?

Check your understanding

Increasing particle count changes the diffusion coefficient?

• No
• It doubles it
• It makes it zero

Model limits

Independent Gaussian walkers in an unbounded medium; no reactions, advection or concentration-dependent D.

MIT OpenCourseWare · Random walks and diffusion

Why are only certain energies allowed?

Why are only certain energies allowed?

1. Increase n from 1 to 3 and count internal nodes.
2. Double well width and inspect energy.
3. Pause and compare signed amplitude with probability.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

An electron confined by infinitely high walls must have zero wavefunction at both boundaries. Only standing sine waves with an integer number of half-wavelengths fit. Their allowed kinetic energies scale as n²/L². The signed ribbon shows the real part of ψ; the translucent envelope shows |ψ|². A stationary energy state has time-independent probability density even though its complex phase evolves. The display exaggerates transverse size for readability; the confinement is one-dimensional.

Build the argument

1. State the potential and boundary conditions, then solve or approximate the Schrödinger equation.
2. Normalise the wavefunction so the integral of |ψ|² is one.
3. Interpret measurable distributions through |ψ|²; colour indicates signed amplitude or phase only where labelled.

Worked example 1

An electron in a 1 nm well has E₁≈0.376 eV.

Worked example 2

For n=2, E₂=4E₁; doubling width divides all energies by four.

Applications

Confinement, tunnelling and orbital structure underpin semiconductor devices, scanning tunnelling microscopes and atomic spectroscopy. The scene shows probability or wave amplitude, never a hidden classical orbit.

Common misconception

A probability cloud is not a swarm of electrons. Wavefunction sign or phase is different from probability density. These are ideal single-particle models.

A further investigation

The nth state has how many internal nodes?

Check your understanding

The nth state has how many internal nodes?

• n−1
• n+1
• Always zero

Model limits

Single nonrelativistic electron in a one-dimensional infinite well; transverse thickness is illustrative.

OpenStax · Particle in a box

How can transmission occur below the barrier energy?

How can transmission occur below the barrier energy?

1. Set E=1 eV, barrier 4 eV and width 0.2 nm.
2. Increase width in 0.05 nm steps.
3. Raise particle energy while remaining below the barrier.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

For E below a finite rectangular barrier, the stationary wavefunction decays exponentially inside it rather than vanishing. Matching the wavefunction and its derivative at both boundaries produces a nonzero transmission probability. The readout uses the exact one-dimensional rectangular-barrier formula, while the displayed envelope illustrates evanescence. A small width increase can suppress transmission by orders of magnitude. This is not a classical particle borrowing energy: transmitted particles retain the incident energy.

Build the argument

1. State the potential and boundary conditions, then solve or approximate the Schrödinger equation.
2. Normalise the wavefunction so the integral of |ψ|² is one.
3. Interpret measurable distributions through |ψ|²; colour indicates signed amplitude or phase only where labelled.

Worked example 1

At E=1 eV, V₀=4 eV and a=0.2 nm, T≈8.45%.

Worked example 2

The decay constant is approximately 8.87 nm⁻¹ for V₀−E=3 eV.

Applications

Confinement, tunnelling and orbital structure underpin semiconductor devices, scanning tunnelling microscopes and atomic spectroscopy. The scene shows probability or wave amplitude, never a hidden classical orbit.

Common misconception

A probability cloud is not a swarm of electrons. Wavefunction sign or phase is different from probability density. These are ideal single-particle models.

A further investigation

Increasing barrier width usually makes transmission…

Check your understanding

Increasing barrier width usually makes transmission…

• Smaller
• Larger
• Exactly one

Model limits

Stationary nonrelativistic electron; rectangular barrier and E<V₀; envelope illustration is not a time-dependent scattering solution.

OpenStax · Quantum tunnelling

What does a narrow wave packet cost in momentum?

What does a narrow wave packet cost in momentum?

1. Compare broad and narrow initial packets.
2. Record Δk for each Δx.
3. Increase spreading rate and follow the envelope.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

A localised wave packet requires a superposition of wave numbers. A Gaussian packet is a minimum-uncertainty state with ΔxΔk=1/2, so narrowing position broadens its wave-number distribution. Free evolution spreads the envelope because its component waves have different group velocities. The display uses an analytic Gaussian spreading envelope with an illustrative phase clock. It wraps the packet centre at the display edge for continuous inspection; the wrap is not a physical periodic boundary calculation.

Build the argument

1. State the potential and boundary conditions, then solve or approximate the Schrödinger equation.
2. Normalise the wavefunction so the integral of |ψ|² is one.
3. Interpret measurable distributions through |ψ|²; colour indicates signed amplitude or phase only where labelled.

Worked example 1

Δx=0.5 nm gives Δk=1 nm⁻¹.

Worked example 2

Δx=1 nm gives Δk=0.5 nm⁻¹.

Applications

Confinement, tunnelling and orbital structure underpin semiconductor devices, scanning tunnelling microscopes and atomic spectroscopy. The scene shows probability or wave amplitude, never a hidden classical orbit.

Common misconception

A probability cloud is not a swarm of electrons. Wavefunction sign or phase is different from probability density. These are ideal single-particle models.

A further investigation

Halving Δx changes the minimum Δk to…

Check your understanding

Halving Δx changes the minimum Δk to…

• Twice its value
• Half its value
• Zero

Model limits

Analytic Gaussian envelope; the phase/playback rates are illustrative and are not a calibrated electron propagation time.

OpenStax · The Schrödinger equation

What does an atomic orbital represent?

What does an atomic orbital represent?

1. Switch from 1s to 2p and rotate the cloud.
2. Identify the nodal plane in the 2p state.
3. Increase Z and compare size and energy.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

A hydrogenic orbital is a stationary wavefunction in the Coulomb potential of a nucleus. The point cloud samples its probability density, not the positions of many electrons travelling on hidden paths. A 1s state is spherically symmetric. A real 2p state has two lobes of opposite amplitude sign and a nodal plane through the nucleus. Increasing nuclear charge contracts the cloud and lowers its energy by Z². The colour of each lobe encodes wavefunction sign, not positive or negative electric charge.

Build the argument

1. State the potential and boundary conditions, then solve or approximate the Schrödinger equation.
2. Normalise the wavefunction so the integral of |ψ|² is one.
3. Interpret measurable distributions through |ψ|²; colour indicates signed amplitude or phase only where labelled.

Worked example 1

Hydrogen 1s has E≈−13.606 eV; 2p has E≈−3.401 eV.

Worked example 2

For Z=2, all hydrogenic energy magnitudes multiply by four.

Applications

Confinement, tunnelling and orbital structure underpin semiconductor devices, scanning tunnelling microscopes and atomic spectroscopy. The scene shows probability or wave amplitude, never a hidden classical orbit.

Common misconception

A probability cloud is not a swarm of electrons. Wavefunction sign or phase is different from probability density. These are ideal single-particle models.

A further investigation

The two lobe colours represent…

Check your understanding

The two lobe colours represent…

• Wavefunction sign
• Opposite charges
• Two electrons

Model limits

One-electron Coulomb atom, real 1s and 2p_y orbitals; no electron interactions, spin or fine structure.

OpenStax · Hydrogen atom

Can a dimensionless number reveal an entire family of orbits?

Can a dimensionless number reveal an entire family of orbits?

1. Keep launch ratio at 1 and change mass.
2. Compare characteristic orbital speeds.
3. Predict a time-scale ratio before changing radius.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

The ratio of launch speed to circular speed removes the units from the launch problem. At fixed relative planet size, equal ratios produce similar classes of trajectory even when mass and radius change. Dimensional analysis also predicts the characteristic time √(r³/μ). Scaling radius without considering the physical planet surface can change collision behaviour, so similarity requires matching every relevant dimensionless ratio.

Build the argument

1. Write a prediction before running the experiment.
2. Hold all but one variable constant, collect repeated measurements, and calculate dimensionless ratios.
3. Compare the observations to an analytic limit and document the remaining discrepancy.

Worked example 1

Doubling μ increases circular speed by √2.

Worked example 2

Doubling r increases orbital time by 2√2.

Applications

A useful model makes a testable prediction and states where it fails. Reproducible settings and exported measurements turn a visual experiment into evidence.

Common misconception

More decimal places do not guarantee greater accuracy. Distinguish discretisation error, uncertainty in inputs, and limits of the physical model.

A further investigation

Which launch ratio marks marginal escape?

Check your understanding

Which launch ratio marks marginal escape?

• √2
• 1
• 2π

Model limits

Zero weight is not zero gravity. Orbital radius is measured from the centre. Numerical energy drift must not be mistaken for real energy production.

University Physics Volume 1, 13.4 Satellite Orbits and Energy

How can a conserved quantity test a model?

How can a conserved quantity test a model?

1. Record total energy and the initial amplitude.
2. Pause near equilibrium and inspect displacement.
3. Change mass and explain why total energy stays fixed at fixed k,A.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

A conservation law is an independent check on a simulated trajectory. In a harmonic oscillator, evaluate kinetic and potential energy at different phases rather than checking only the visual amplitude. Their sum should remain fixed. This model has an analytic solution, so any observed discrepancy beyond floating-point rounding would indicate an implementation or unit error, not a physical dissipation mechanism. Comparing two parameter settings requires accounting for the work done while changing them.

Build the argument

1. Write a prediction before running the experiment.
2. Hold all but one variable constant, collect repeated measurements, and calculate dimensionless ratios.
3. Compare the observations to an analytic limit and document the remaining discrepancy.

Worked example 1

k=12 N/m and A=0.8 m store 3.84 J.

Worked example 2

At x=A/2, one quarter of E is potential and three quarters kinetic.

Applications

A useful model makes a testable prediction and states where it fails. Reproducible settings and exported measurements turn a visual experiment into evidence.

Common misconception

More decimal places do not guarantee greater accuracy. Distinguish discretisation error, uncertainty in inputs, and limits of the physical model.

A further investigation

At a turning point, kinetic energy is…

Check your understanding

At a turning point, kinetic energy is…

• Zero
• All the energy
• Negative

Model limits

A force changes velocity, not just speed. A massless rod or ideal spring is a model assumption, and small-angle pendulum formulas have a limited range.

OpenStax · Simple harmonic motion

Is an elegant approximation always accurate?

Is an elegant approximation always accurate?

1. Convert 10° to radians before estimating error.
2. Compare 4° and 14° release angles.
3. Estimate whether a 0.1% timing target is plausible.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

The small-angle pendulum replaces sinθ with θ. The first neglected term is −θ³/6, so relative force error grows roughly as θ²/6. Even when an approximation looks visually convincing, that error may matter for precision timing. This lab deliberately stays below 15° and reports the small-angle period. It does not pretend to show nonlinear large-angle motion. Compare the predicted correction with the accuracy required by a proposed measurement.

Build the argument

1. Write a prediction before running the experiment.
2. Hold all but one variable constant, collect repeated measurements, and calculate dimensionless ratios.
3. Compare the observations to an analytic limit and document the remaining discrepancy.

Worked example 1

At 10°, θ=0.1745 rad and θ²/16≈0.19%.

Worked example 2

At 4°, the leading period correction is about 0.030%.

Applications

A useful model makes a testable prediction and states where it fails. Reproducible settings and exported measurements turn a visual experiment into evidence.

Common misconception

More decimal places do not guarantee greater accuracy. Distinguish discretisation error, uncertainty in inputs, and limits of the physical model.

A further investigation

The argument of sin in a power series must be in…

Check your understanding

The argument of sin in a power series must be in…

• Radians
• Degrees
• Metres

Model limits

A force changes velocity, not just speed. A massless rod or ideal spring is a model assumption, and small-angle pendulum formulas have a limited range.

OpenStax · Pendulums

What does a finite observation hide?

What does a finite observation hide?

1. Set the fundamental to 3 Hz with harmonics disabled.
2. Change to 3.5 Hz and inspect neighbouring bars.
3. Add the third harmonic and locate its actual frequency.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

A discrete Fourier transform measures projection onto a finite set of sinusoidal basis functions. In this instrument 256 samples cover one second, giving 1 Hz bin spacing and a 128 Hz Nyquist limit. A frequency exactly aligned to a bin has a clean peak. A fractional-bin tone spreads over neighbouring bins because the finite rectangular window cuts it at unequal phases. More interpolation would make a smoother picture without increasing the underlying resolution.

Build the argument

1. Write a prediction before running the experiment.
2. Hold all but one variable constant, collect repeated measurements, and calculate dimensionless ratios.
3. Compare the observations to an analytic limit and document the remaining discrepancy.

Worked example 1

At 3 Hz the third harmonic occurs at 9 Hz.

Worked example 2

Doubling the observation window would halve bin spacing, at fixed sample rate.

Applications

A useful model makes a testable prediction and states where it fails. Reproducible settings and exported measurements turn a visual experiment into evidence.

Common misconception

More decimal places do not guarantee greater accuracy. Distinguish discretisation error, uncertainty in inputs, and limits of the physical model.

A further investigation

Smoothing the plotted curve increases physical resolution?

Check your understanding

Smoothing the plotted curve increases physical resolution?

• No
• Always
• Only in 3D

Model limits

Direct DFT, 256 Hz sampling, 256 samples, rectangular window; only the first 40 Hz are displayed.

6.003 Signals and Systems — Lecture Notes