Here is a puzzle that was set by the famous mathematician, Wittgenstein. See if you can solve it before reading the answer!
Assume the Earth is a perfect sphere.
Stretch a string round it tightly.
Now remove the string and add 1 metre to it’s length.
Wrap the string round the Earth again, such that it is equidistant from the surface all the way round.
Q:- What is the height, h, of the string above the Earth’s surface?
Can you slip a hair under it? A credit card? could you trip over it?
Answer below
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The answer is….
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You could trip over it!
By adding just one metre to the string’s length, means it will have enough slack to circle the Earth about a foot off the ground!
Remember, the circle equation from school?
The circumference, C=PI*d, where d is the diameter of earth.
So C2 (after adding one metre) = PI * d2, where d2 is the new diameter of the string circle.
So d2-d1 = the height of the string above the land, = C2/PI minus C/PI
Therefore, d2-d1 = 1 metre / PI = about 30cm, or one foot imperial!
STOP PRESS!!!!! This answer is a little wrong. The true answer is half the difference in diameters (ie the difference in the radii), about half a foot, or approx 15cm. See the comments below.
I’m sorry but the answer is wrong.
d2-d1 is TWICE the height of the string above the land. If you look at it from the raidus’ point of view (which is half the diameter):
r2-r1 is the real height of the string above the land:
r2-r1=h
if 2*r2=d2 and 2r1=d1, by substitution we can get:
d2/2 – d1/2 = h
to simplify:
(d2-d1)/2 = h
therefore: d2-d1=2h
going back to the original problem:
Let C1 = Circumference 1
C2 = Circumference 2
D1 = Diameter 1
D2 = Diameter 2
R1 = Radius 1
R2 = Radius 2
We know that:
1. pi*D1 = C1
2. pi*D2 = C2
or
1. pi*2*R1 = C1
2. pi*2*R2 = C2
from the problem above, we can formulate that:
C1 + 1meter = C2
D2 – D1 = 2*h (based from the explanation above)
or
R2 – R1 = h
Substituting C2, D2 and R2 in terms of C1, D1 and R1 respectively:
1. pi*D1 = C1
2. pi*(D1+2*h) = C1 + 1meter
or
1. pi*2*R1 = C1
2. pi*2*(R1+h) = C1 + 1meter
Now substitute equation 1 to equation 2:
3. pi*(D1+2*h) = pi*D1 + 1meter
pi*D1 + pi*2*h = pi*D1 + 1meter
h = 1/(2*pi)
h = 0.15915 meters or 15.915 cm
or
3. pi*2*(R1+h) = pi*2*R1 + 1meter
pi*2*R1 + pi*2*h = pi*2*R1 + 1meter
pi*2*h = 1meter
h = 1/(2*pi)
h = 0.15915 meters or 15.915 cm
kasoge, you are right!
The calculation should be based on the radius rather than the diameter.
The total diameter of the string circle minus the diameter of the earth is basically 2* the height of the string above the earth’s surface. So the answer is approx half a foot!
thanks!