The Physics
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Opus in profectus

# Pressure

## Problems

### practice

1. Two related problems…
1. Determine the mass of Earth's atmosphere using the value of the standard atmosphere.
2. Determine the scale height of Earth's atmosphere — the height the atmosphere would have if its density stayed constant instead of decreasing with altitude. (The scale height is a useful approximation for some calculations in atmospheric sciences.)
2. Determine the maximum height that a lift pump can raise water from a well.
3. The first measurements of blood pressure were made in 1726 by the English botanist, physiologist, and clergyman, Stephen Hales. Hales performed several experments on horses deemed "unfit for service". You must recall that at the time horses were primarily used as working animals. Those that were seriously injured, chronically ill, or otherwize unable to perform their duties were routinely slaughtered and eaten. Read Hales' description of one such experiment, then determine the blood pressure of his poor, unfortunate horse.

In December I laid a common field gate on the ground, with some straw upon it, on which a white mare was cast on her right side, and in that posture bound fast to the Gate; she was fourteen hands and three inches high [150 cm], lean, tho' not to a great degree, and about ten or twelve years old. This and the above-mentioned horse and mare were to have been killed, as being unfit for service….

Then laying bare the left carotid artery, I fixed to it towards the heart the brass pipe, and to that the wind-pipe of a goose; to the other end of which a glass tube was fixed, which was twelve feet nine inches long [388 cm]. The design of using the wind-pipe was by its pliancy to prevent the inconveniencies that might happen when the mare struggled; if the tube had been immediately fixed to the artery, without the intervention of this pliant pipe.

There had been lost before the tube was fixed to the artery, about seventy cubic inches of blood [1.15 L]. The blood rose in the tube in the same manner as in the case of the two former horses, till it reached to nine feet six inches height [290 cm]. I then took away the tube from the artery, and let out by measure sixty cubick inches of blood [0.98 L], and then immediately replaced the tube to see how high the blood would rise in it after each evacuation; this was repeated several times, till the mare expired….

Stephen Hales, 1733

4. When the human body is accelerated vertically, blood pressure in the brain will drop. Determine the maximum vertical acceleration that a human can withstand before losing consciousness; that is, determine the acceleration that would reduce the blood pressure in the brain to zero. Assume a typical systolic pressure of 16 kPa and that the base of the brain is 20 cm above the top of the heart.

### conceptual

1. A question about American football. How is it that a weekend carpenter can apply a relatively small amount of force on a 5 pound hammer and drive a nail though a board, but a 300 pound tackle running at top speed will never be able to pierce the helmet of a quarterback?

### numerical

1. Determine the pressure under the thickest part of the Antarctic ice cap (4776 m) in kPa and atm.
2. Use the values of standard atmospheric pressure in torr and pascals to determine the density of mercury — the liquid metal used in old fashioned barometers, not the planet closest to the Sun.
3. A hydraulic brake system on a bicycle consists of a master cylinder with a diameter of 2 mm connected to two slave cylinders with a diameter of 1 cm each. A rider grips the brake levers and applies a force of 240 N to the master cylinder. What total frictional force do the two brake pads apply to the opposite sides of the brake disks? (Brake pads on steel have a coefficient of friction of 0.4.)
4. A car can be weighed by measuring the "footprint" of each tire and multiplying by that tire's gauge pressure. Determine the weight of this car (in English units) using the data in the following table. (Adapted from: Beakman's World, Episode 401, Sweat, Beakmania, and Weighing a Car, 1996.)
Determine the weight of this car
tire length (in.) width (in.) pressure (p.s.i.) weight (lbs)
front right 6 4 24
front left 6 4 24
rear right 4 22
rear left 4 22
total
5. Determine the mass of the atmospheres of Venus and Mars. Follow the example of the practice problem in this section. Use a table like the one below to organize your thoughts.
P (atm) mplanet (kg) rplanet (km) g (m/s2) matm (kg)
Venus 90.000 4.8685 × 1024 6051.8
Earth 01.000 5.9736 × 1024 6371.0 9.8 5.27 × 1018 kg
Mars 00.007 6.4185 × 1023 3390.0
6. Breakthrough Starshot is a \$100 million research program founded in 2015 whose aim is to test proof of concept technologies for a proposed unmanned flyby mission to Alpha Centauri (the star nearest the Sun) within a generation. Read the following excerpt from the key technical paper describing the program. The author envisions a swarm of ultralight, sailed space probes propelled from low earth orbit by the light pressure generated by a battery of high powered lasers stationed on Earth.

We propose a roadmap to a program that will lead to sending relativistic probes to the nearest stars and will open up a vast array of possibilities of flight both within our solar system and far beyond. Spacecraft from gram level complete spacecraft on a wafer (“wafersats”) that reach more than ¼ c and reach the nearest star in 20 years to spacecraft with masses more than 105 kg (100 tons) that can reach speeds of greater than 1,000 km/s. These systems can be propelled to speeds currently unimaginable with existing propulsion technologies. To do so requires a fundamental change in our thinking of both propulsion and in many cases what a spacecraft is….

The laser array has been described in a series of papers we have published and is called DE‑STAR (Directed Energy System for Targeting of Asteroids and exploRation)…. As an example, on the eventual upper end, a full scale DE‑STAR 4 (50–70 GW) will propel a wafer scale spacecraft with a 1 m laser sail to about 26% the speed of light in about 10 minutes (20 kgo accel), reach Mars (1 AU) in 30 minutes, pass Voyager 1 in less than 3 days, pass 1,000 AU in 12 days and reach Alpha Centauri in about 20 years. The same directed energy driver (DE‑STAR 4) can also propel a 100 kg payload to about 1% c and a 10,000 kg payload to more than 1,000 km/s. While such missions would be truly remarkable, the system is scalable to any level of power and array size where the tradeoff is between the desired mass and speed of the spacecraft.

Philip Lubin, 2015

Determine the following quantities for a wafersat propelled by a DE‑STAR 4 laser battery….

1. its average acceleration
2. the force accelerating it
3. the light pressure on the sail

Assume an acceleration that's 0.1% of the value calculated in part a. and a light pressure equal to the value calculated in part c. and determine the following quantities for a 100 kg payload.

1. it's average acceleration
2. the force acting on it
3. the area of the sail
4. the time needed to accelerate to the desired speed
7. A hydraulic automotive lift has a capacity of 3,600 kg at 1.0 MPa.
1. What is the inner diameter of the lifting cylinder?
2. What force does the supply pump need to apply to the hydraulic fluid if the piston is 2.5 cm in diameter?

### investigative

1. Obtain a pen-sized tire pressure gauge for a car or bicycle — the kind with a cylindrical body and sliding, calibrated rod. When the gauge is connected to the tire valve, the air inside the tire pushes a piston attached to a spring and the calibrated rod. The whole apparatus moves until the force of the air escaping from the tire equals the force of the spring pushing back. (For an in-depth description with illustrations see How tire pressure gauges work.)
1. Measure the diameter of the bore.
2. Choose a convenient pressure value and measure the length of the stem from that value to the zero.
3. Calculate the spring coefficient from these two measurements and the pressure value you chose.

### statistical

1. Astronomical pressures
1. Derive an expression for the pressure in a spherical, astronomical body with uniform density.
2. Use this formula to estimate the pressure at the center of…
1. the Earth
2. the Sun

### calculus

1. earth.txt
The data in this text file gives the density and gravitational field strength of the Earth at various depths below the surface. Using data analysis software (preferably something that can do numerical integration) generate a data column for the the pressure at various depths below the surface. The value in the center of the core will be on the order of 360 GPa, so you can ignore the contribution of atmospheric pressure in your calculations.