How To Measure The Height Of A Building With A Barometer
(And other problems in elementary physics)
Now with 15 answers!
1. Measure the barometric pressure at the top and at the bottom
Measure the barometric pressure (in mm of mercury) at the base of the
building. Take the barometer to the roof of the building, and measure the
pressure again. The difference tells you the weight of a column of air the
same height as the building. Multiply the pressure difference (in mm of
mercury) by the relative density of mercury compared to the air around the
building. This gives you the height of the building (in mm). Use standard
tables to convert to the required unit of measurement.
2. Drop the barometer and time how long it takes to fall
Take the barometer to the roof of the building. Walk to the edge. Drop the
barometer over the edge and time how long it takes to hit the ground. You
may either watch the barometer fall, or listen for it hitting the ground,
depending on the height of the building and the accuracy required. Don't
forget to correct for the speed of sound if listening for the crunch. Use the
fact that height is gravity times the square of the time, divided by two to
calculate height from the (known) gravity and the (measured) time.
3. Use the barometer as a measuring stick
Place the barometer upright against the wall. Mark the top of the barometer.
Label the mark '1'. Move the barometer vertically so that the bottom of the
barometer is at the mark. Mark the top of the barometer, labelling this mark
'2'. Continue like this until you reach the top of the building. Multiply
the number on the last mark by the height of the barometer. This will be
accurate to within one barometer height. For greater accuracy, add onehalf
the height of the barometer to account for the portion above the last mark.
Note: For tall buildings, some extra equipment may be required to assist in
climbing the wall.
4. Offer the barometer to the superintendent
Find the superintendent of the building. Offer him a deluxe display
barometer if he will tell you the height of the building. If the
superintendent is not available, or doesn't like barometers, try other
parties such as the local survey officer, the original architect, or a member
of the construction crew.
5. Measure the shadow of the barometer and the building
On a sunny day, measure the length of the shadow cast by the barometer, and
the length of the shadow cast by the building. Multiply this ratio by the
height of the barometer to get the height of the building.
6. Measure the shadow of the building, calibrated by the barometer
On a sunny day, place the barometer on the ground. Mark both ends. Stand the
barometer upright on the mark closer to the sun, so the shadow will be
approaching the other mark. Note the exact time when the shadow reaches the
other mark.
On the following sunny day, mark the end of the shadow cast by the building
at exactly that time. Measure the distance from there to the building. This
is the height of the building.
Note: The accuracy of this technique depends on the number of days between
consecutive sunny days.
7. Find a barometer with heights of local buildings on it
Go to all the local gift shops. Look for a fancy souvenir barometer, the
kind which shows important local landmarks. Find one which shows the heights
of local buildings and considers this building important enough to be
listed. Use this barometer.
8. Compare the barometer height to the building height
Hold the barometer one foot in front of yourself and find a position where
the building appears to be the same size as the barometer. Now measure the
distance to the building (in feet) and multiply by the height of the
barometer.
9. Trade the barometer for a long measuring tape
Go to a local shop and trade the barometer for the longest measuring tape
they have. Take the tape onto the roof of the building. Holding one end,
drop the other end over the edge of the building. Raise the measuring tape
until the far end is just touching, not resting on, the ground. Read the
height of the building from the measuring tape.
Note: For particularly tall buildings, this may require a particularly good
hardware store.
10. Drop the barometer on the roof and on the ground
Hold the barometer straight in front of you and drop it. Measure, very
carefully, how long it takes to hit the ground. Go up on the roof and hold
the barometer in the same position. Drop it and measure, again very
carefully, how long it takes to hit the roof. Since gravity falls off as the
square of the distance from the centre of the planet, you can use the
difference in times to calculate the height of the building relative to the
distance from the base of the building to the centre of the planet. The
local library can provide you with the distance to the centre of the planet
in the required units.
Note: The ratio of the times is the same as the ratio of the distances from
the drop points to the centre of the planet.
11. Use the barometer as a pendulum
Buy a rope long enough to reach the top of the building. Tie the barometer
to one end and go up on the roof. Lower the barometer until it is exactly
one inch above the ground. Holding the top of the rope at the top of the
building, swing the barometer, and measure the period of the pendulum. From
this you can calculate the length of the rope. Add one inch and you have the
height of the building.
12. Drop the barometer on a windless day
Drop (and shatter) the mercury barometer at the base of the buidling on a
windless day. Measure the increase in the mercury vapour concentration at
the top of the building. Solve the diffusion equation to determine the
distance from the shattered barometer to the top of the building.
13. Seal the building and fill with water
Place the barometer on the ground floor of the building. Seal all the
building's doors and windows. Fill the building with water. Read the
pressure measurement from the barometer. This gives the weight of a column
of water the same height as the building. Use this and the ratio of the
density of mercury to the density of water to calculate the height of the
building.
Note: It is common courtesy to evacuate the building before using this
technique.
14. Take the barometer to an airless world
If you have access to an airless world, take the building there. Throw the
barometer horizontally off the building. If the barometer hits the ground,
retrieve it and try again, throwing harder. The objective is to throw it
hard enough to achieve a circular orbit. Once the barometer is in orbit
around the planet, you can measure the period of the orbit. Compare this
with the period of the orbit when you throw the barometer from the base of
the building. Use this ratio and Kepler's laws to determine the height of
the building (relative to the radius of the planet).
15. Clap the barometer and listen for the echo
(This one was contributed by one of our readers in October 2001  thanks,
Chandra!) Clap the barometer against the top of the building. Measure the
time taken to hear the echo from the ground. Find the height of the
building by multiplying half the echo delay by the velocity of sound.
Where did the barometer question and answers come from?
This list of answers was compiled originally by a group of university
students and extended by answers from the net. The question has been around
for a very long time. You may want to
read the original
story.
Beauty is the first test; there is no permanent place in the world for
ugly mathematics
Godfrey Harold Hardy, 1940

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