One Winter Day in Therms, Pounds and Degrees, Hour by Hour

5:58 a.m. A thermostat clicks, a gas valve opens, and the furnace starts turning therms into warm air. Before the house goes quiet again tonight, its rooms will run through three pieces of physics that everyone meets every day and almost nobody puts numbers on: how much of the fuel becomes heat, what a bathroom scale is really measuring, and why a window that was clear at noon is full of reflections after dark.

Nobody in particular lives here. The house is a set of readings, and this is one winter day of them, hour by hour, worked through the Thermal Efficiency Calculator, the Gravitational Force Calculator and the Snell's Law Calculator. Every figure below came out of those calculators first, and only then got a sentence around it.

5:58 a.m. The furnace fires

Say the house needs 5 therms of heat to get through a cold day. That is a round example, not an average. At the end of the day the gas meter shows 6.25 therms burned. Divide one by the other and the furnace ran at 80.0 %. The other 1.25 therms went up the vent as hot exhaust and water vapor, a fifth of what was paid for.

For a non-condensing furnace that is a perfectly normal number. It only looks poor next to a condensing unit: at 90 % the same 5 therms of heat need 5.56 therms of gas, which is 0.69 therms less on this one day.

So what does the number on the furnace label actually change? The table holds the heat fixed at those 5 therms and runs the gas through every AFUE rating people look up. Each rating links to a page with the calculator already set to it.

Two things jump out. The early steps are the big ones: 80 to 90 saves 0.69 therms today, while 97 to 98 saves 0.05. And an old furnace with a standing pilot at 56 % would have burned 8.93 therms for the same warmth. That is 261.67 kWh of fuel energy, of which a little over half ever reached a room.

7:10 a.m. The bathroom scale

The scale says 180 lb. It is not lying, but it is not measuring what it claims to either. A spring or a load cell cannot feel mass, only the force pressing down on it, and it turns that force back into pounds using the gravity it was set up for. The force itself is 800.68 N, or 180.00 lbf, which is exactly what a pound-force is defined to be.

Work the same pull out from scratch, with the Earth's mass of 5.9722 × 10²⁴ kg against 81.65 kg at 6,371 km from the center, and Newton's law gives 801.79 N. The 0.14 % gap is not a mistake in either figure. Standard gravity is a defined reference that already allows for the spinning planet. The bare law does not.

Two more bodies are pulling at the same moment. The Sun, 1.496 × 10¹¹ m away, pulls on those 180 lb with 0.484 N. The Moon manages 0.00271 N. Neither shows up on the scale, because the person, the scale and the whole planet are falling around the Sun together, and a scale only notices what stops a fall.

10:32 a.m. The space station goes over

Somewhere overhead the International Space Station completes another lap. It goes round about every 90 minutes, 16 times a day, roughly 400 km up. At 6,771 km from the Earth's center the same 81.65 kg would be pulled with 709.86 N, or 159.58 lbf. That is 88.5 % of the pull back in the bathroom.

Why, then, does nobody up there stand on a scale? Because the scale falls exactly as fast as they do. The station moves sideways quickly enough that its fall keeps missing the ground. Gravity at that height is barely weaker than at home. Weightlessness is what falling feels like.

1:15 p.m. The deep end of the pool

The tile says 8 ft. From the edge, looking straight down, the bottom seems to sit at 6.00 ft, just 75.0 % of the real depth. Light coming up out of the water speeds up and bends away from the normal, and the eye follows it back along a straight line that ends too high. At a slant the bottom looks shallower still.

The pool has a second trick, and it only shows from underneath. Light in water can get out into the air only if it meets the surface within 48.63° of the normal. Beyond that critical angle every bit of it is reflected back down. From the bottom of the pool the whole sky is squeezed into a circle 97.25° wide, and outside that circle the underside of the water is a mirror. At 60°, for instance, 100 % of the light stays in.

Every clear material has its own critical angle, and the higher its refractive index, the smaller the angle gets. The escape cone is simply twice as wide. Each material in the table opens the calculator already set to it.

The last row is the entire reason a diamond sparkles. Light that gets in has a cone under 49° wide to get out through, so it bounces around the facets first and leaves in bright flashes.

4:40 p.m. Low sun on the lake

The sun now sits 80° from straight overhead, so its light arrives almost skimming the water, and 34.78 % of it bounces straight off. That is the glare. Earlier, with the sun 45° from overhead, the same surface reflected only 2.78 % and let the rest in, bending it down to 32.05° under the surface.

One angle in between is special. At about 53°, the Brewster angle for air into water, the reflected light is completely polarized and 3.91 % of it comes off the surface. That glare is exactly what polarized sunglasses are cut to block, which is why a lake looks darker and clearer through them when the sun sits around that height.

6:30 p.m. Moonrise

The Earth and the Moon pull on each other with 1.982 × 10²⁰ N. The force is the same in both directions and the response is not: the Moon accelerates toward us at 0.0027 m/s², the Earth toward the Moon at 3.318 × 10^-5 m/s².

Carry this morning's scale up there and it shows 29.73 lb, which is 16.5 % of Earth's pull on the same 180 lb of mass. A dropped key takes 1.11 s to fall a meter instead of 0.45 s. A 1 ft hop turns into 6.05 ft, same push off the ground, no suit.

9:05 p.m. The window turns into a mirror

Nothing happened to the glass since lunch. A pane of crown glass reflects 4.21 % of the light that hits it head-on, at each of its two surfaces. In daylight that sliver is drowned out by the view outside. After dark the brightest thing beyond the pane is the reflected living room, so the window quietly stops being a window. Switch off the lamp and the street comes back.

10:45 p.m. The space heater

A 1.5 kW heater runs in the bedroom for eight hours. That is 12 kWh of electricity, and all of it becomes heat: 12 kWh, or 40,945.70 BTU. Nothing is lost at all, which is the ceiling for anything that makes heat out of what it consumes. It is also why "100 percent efficient" and "cheap to run" are two different claims.

A heat pump delivering the same 12 kWh of heat at 250 % takes only 4.80 kWh from the wall and moves the other 7.20 kWh in from the outdoor air. On a colder night, at 175 %, it needs 6.86 kWh. Above 100 % is not a bookkeeping trick. The heat was carried in, not made.

The whole day in one table

What if the house had made two different choices?

Most of the day's numbers are not choices at all. The pull on 180 lb of mass, the critical angle of water and the glare off a lake at a low sun come out the same in every house on the street. Two of them are choices, though, and they are the two that show up on a bill.

Of the two, the heater swap moves the bigger share of its energy, because a heat pump is not burning anything better, it is refusing to make heat it can move instead. The furnace swap is smaller per day and adds up over a season. Neither changes a single thing about the scale, the pool or the window.

Tools discussed in this article

• Thermal Efficiency Calculator: efficiency from what goes in and what comes out, or the fuel needed for a heat demand, in therms, kWh, BTU or MJ
• Gravitational Force Calculator: Newton's law between any two masses, and what a scale reading becomes on eleven worlds from Pluto to the Sun
• Snell's Law Calculator: where light bends between two materials, how much bounces, and the critical angle on its own when no angle is typed

More physics tools

The rest of the physics shelf, in the order it was built:

• Momentum Calculator, Force Calculator and Kinetic Energy Calculator
• Potential Energy Calculator, Work Calculator and Mechanical Power Calculator
• Torque Calculator, Kinetic Friction Calculator and Pendulum Period Calculator
• Thermal Expansion Calculator, Specific Heat Calculator and Thermal Conductivity Calculator, the other half of the story behind this morning's therms
• Hooke's Law Calculator, Simple Harmonic Motion Calculator and Buoyancy Calculator
• Acceleration Formula Calculator, Density Calculator and Temperature Converter