Pressure

Problems

practice

  1. Determine the mass of the Earth's atmosphere.
  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

  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.
  5. 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

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" ca. 1995-1996.)
    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  
    Determine the weight of this car
  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) matmosphere (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

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 a Tire Pressure Gauge Works.)
    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.

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.