Zak's Lab

Inelastic collision then maximum compression of a spring: find the initial velocity of the dart.

Definite integral sqrt(1+cos(x)) trig identity and absolute value of cosine integral.

Find the magnetic field on the axis of a current loop using the Biot-Savart law for magnetic field.

Shockwaves produced by supersonic motion in a circular path.

Stop using this u-substitution! Integrate cos(2x) by using the chain rule backwards.

Oscillation of an electric dipole in a uniform electric field.

Test for convergence (n!)^3/(3n)! ratio test limit expand factorials and cancel to compute the limit

Integral arctan(1/x). Integration by parts: integrate inverse tangent.

Spring shoots a block up a ramp: find the speed at the equilibrium length, maximum sliding distance

Verify that the general solution f(x)=c_1sin(ix)+c_2cos(ix) satisfies the ODE f''(x)=f(x).

Find the moment of inertia of a thick cylindrical shell, moment of inertia integral.

The art of physics: Doppler effect visualization using new Doppler code for two objects.

Energy of a radio wavelength photon from an FM radio station: energy of the photon using E=hf.

Animation of Doppler effect: moving source and stationary listener.

Disk dropped on a rotating disk angular momentum conservation problem (rotational collision).

Derivation of the final velocity formulas for elastic collisions + physical checks on the solution.

Introduction to capacitance, parallel plate capacitor derivation of C=epsilon_0*A/d, and example.

Wave function for a standing wave, boundary conditions on a string of length L, normal modes, etc.

Trig substitution integral letting x=tan(theta): integrate 1/(1+x^2)^(3/2).

Derivation of the one dimensional wave equation using partial derivatives of the wave function.

Particles in a surface wave: a combination of longitudinal and transverse wave motion.

Converting RPM to rad/s and finding tangential velocity and centripetal acceleration.

Unit analysis and conversion factors: number of seconds in one year unit conversion problem.

Derivative of 2^x, exponential function derivative with non-standard base.

Is gravity positive or negative? Sign of acceleration of gravity with free fall example.

The most destructive seismic wave: animation of Rayleigh waves in a solid.

Ampere's law with non-uniform current density: magnetic field of a thick cylindrical shell conductor

Animation Doppler effect for a stationary source and moving listener.

Integral (cos(x))^4 use trig identity (cos(x))^2=(1/2)*(1+cos(2x)) to reduce powers twice.

This integral breaks Symbolab! Integrate sqrt(1+sin(x)) on zero to pi.

Animation of polar cosine roses: a quick survey of polar rose curves.

Finding the slope of a tangent line as a limit of secant lines.

Find the moment of inertia of a disk about a diameter by using thin rods, moment of inertia integral

Buoyant force on a hot air balloon problem: find the volume of hot air required to lift a balloon.

Electric field in the cylindrical capacitor using Gauss' Law [Cylindrical Shell Capacitor Pt. 1]

A short survey of polar sine roses, animations of rose curves using linear interpolation.

Electric potential as an integral over the charge distribution + uniformly charged ring potential.

Photoelectric effect example: work function in eV and stopping potential for a given wavelength.

Calculate the work function in eV and calculate the stopping potential in the photoelectric effect.

Electron deflected by charged plates projectile motion electrostatics problem.

Rayleigh criterion for diffraction limited resolution through a circular aperture: Rayleigh equation

Derivation: moment of inertia of a thin rod rotating about its center, moment of inertia integral.

Integral sqrt(1-(cos(x))^2) on 0 to 2pi, reduces to integral abs(sin(x)) - split the integral!

Definite integral sqrt(1-cos(t)) on 0 to 2pi. Two ways to integrate sqrt(1-cos(t)) and sign issue!

Find the Laplace transform of the sine function (Laplace transform of sin(t)).

The physics of how a siphon works: pressure as a function of depth at the moment the siphon starts.

Wave interference using sound: beat frequency for two sine waves with similar frequencies.

Motion perpendicular to the direction of wave velocity: time derivatives of the wave function.

Improper integral of 1/(1+x^2) on 1 to infinity.

Arc length of cosh(x) on -1 to 1: finding the arc length of the hyperbolic cosine.