Math Solutions

White light enters a glass prism (apex angle 60°). Show how different wavelengths (colors) refract at different angles because the refractive index depends on wavelength. Calculate the deviation for red (n = 1.514) and violet (n = 1.532) light, and show the resulting rainbow spectrum.

Light travels from water (n = 1.33) toward the water-air boundary. Find the critical angle at which total internal reflection occurs, and show what happens to the reflected and refracted rays as the angle of incidence increases past the critical angle.

Light travels from air (n = 1.00) into glass (n = 1.50) at an angle of incidence of 45°. Find the angle of refraction using Snell's law, and explore how the refracted angle depends on the incidence angle and the refractive indices of the two media.

An object is placed 15 cm in front of a convex (diverging) mirror with focal length f = −10 cm. Draw the principal rays, find the image, and show that a convex mirror always produces a virtual, upright, diminished image regardless of object position.

An object is placed 25 cm in front of a concave (converging) mirror with focal length f = 10 cm. Draw the three principal rays, locate the image using the mirror equation, and determine whether the image is real or virtual, upright or inverted, magnified or diminished.

An object is 20 cm from a concave (diverging) lens with focal length f = −15 cm. Draw the three principal rays, locate the image using the thin-lens equation, and verify that the image is always virtual, upright, and diminished regardless of where the object is placed.

Move an object from far beyond 2F all the way to inside the focal point of a converging lens. Show how the image changes — from real, diminished, and inverted when the object is beyond 2F, through the same-size crossover at 2F, to magnified and inverted between F and 2F, to image-at-infinity at F, and finally to a virtual, upright, magnified image inside F. Use f = 10 cm and verify each case with the thin-lens equation.

An object is placed 30 cm from a converging (convex) lens with focal length f = 10 cm. Draw the ray diagram and locate the image. Find its position, size (magnification), and describe whether it is real or virtual, upright or inverted.

A 10-centimeter-thick steel wall has 500°C on one side and 20°C on the other. Compute the heat flow through it by conduction, and compare conduction, convection, and radiation as the three fundamental mechanisms of heat transfer.

A 1-meter copper rod at 20°C is heated to 100°C. Compute its change in length. Compare linear, areal, and volumetric expansion for different materials, and explain why railway tracks have gaps between segments.

Two gases of different colors are separated by a barrier in a box. When the barrier is removed, the gases mix. Compute the entropy change using Boltzmann's formula S = k_B ln Ω, and show why the mixing is statistically irreversible.

Plot the distribution of molecular speeds in an ideal gas. Show how the distribution changes with temperature and with the mass of the gas molecules, and mark the three characteristic speeds: most probable, mean, and root-mean-square.

For each of the four canonical processes (isochoric, isobaric, isothermal, adiabatic), show how heat Q, work W, and internal energy change ΔU distribute via bar charts, and verify the first law ΔU = Q − W in each case.

A heat engine absorbs 1000 J from a hot reservoir at 600 K and exhausts heat to a cold reservoir at 300 K. What is its maximum theoretical (Carnot) efficiency, and how does actual efficiency compare?

A Carnot engine operates between a hot reservoir at 600 K and a cold reservoir at 300 K. Trace the four steps on a PV diagram and compute the maximum theoretical efficiency.

A gas starts at P₁ = 100 kPa, V₁ = 30 L, T₁ = 300 K, and expands adiabatically (no heat exchanged, Q = 0). Compare the adiabatic curve to an isothermal curve from the same starting state.

A gas at 300 K expands from 20 L to 40 L while in thermal contact with a reservoir that keeps T constant. How much work does the gas do, and what does the process look like on a PV diagram?

A gas is held in a rigid piston at 300 K with volume 25 L and 1 mole of gas. Show how the pressure changes when you vary temperature, volume, or amount of gas.

Solve x² − 5x + 6 = 0 by finding where the parabola y = x² − 5x + 6 crosses the x-axis.

Show how y = a(x − h)² + k shifts and scales a parabola. Adjust h and k with sliders.