Video Transcript
A mercury barometer is shown in the
figure, where the tube of the barometer makes an angle of 45 degrees with the
horizontal. If the length of mercury in the
tube is 150 centimeters, calculate the atmospheric pressure in this situation. Note that the density of mercury is
13600 kilograms per meter cubed and 𝑔 is 10 meters per second squared.
Here, we are shown a mercury
barometer that has its tube of mercury at an angle of 45 degrees with the
horizontal. And we want to calculate the
atmospheric pressure in this situation. Let’s begin by recalling an
equation used to calculate the pressure in a column of fluid. Atmospheric pressure, 𝑃, is equal
to the density of the fluid, 𝜌, multiplied by the acceleration due to gravity, 𝑔,
multiplied by the height of the object, ℎ.
Taking a look at our question, we
are given the density of the mercury in our barometer as 13600 kilograms per meter
cubed and the acceleration due to gravity, 𝑔, as 10 meters per second squared. We are also given the length of the
tube and the angle it is at. So we already have two of the
values that we need to solve our equation: the density and acceleration due to
gravity. We just need to figure out what the
height of the mercury is. We can figure this out using
trigonometry.
Notice that we have a right
triangle here, where the hypotenuse is the length of our tube. We will convert 150 centimeters
into 1.5 meters so that we are working with SI units. We are trying to find the height of
the mercury, which is the opposite side to the given angle. Therefore, we are going to use the
sine function to find this height. Recall that the sine is equal to
the opposite side over the hypotenuse. So we have the sin of 45 degrees is
equal to the height of the opposite side, ℎ, over 1.5 meters, the length of the
hypotenuse. Because we are looking for the
unknown height, ℎ, let’s make ℎ the subject by multiplying both sides by the length
of the hypotenuse, 1.5 meters. We now see that the height of the
mercury is equal to the sin of 45 degrees multiplied by 1.5 meters.
We now have all the values that we
need to solve the equation for atmospheric pressure, 𝑃. Substituting our values into the
equation, we see that the atmospheric pressure is equal to 13600 kilograms per meter
cubed multiplied by 10 meters per second squared multiplied by the sin of 45 degrees
multiplied by 1.5 meters. Completing this calculation, we see
that the atmospheric pressure is equal to 1.44 times 10 to the power of five newtons
per meter squared. And we have arrived at our final
answer. The atmospheric pressure in this
situation is equal to 1.44 times 10 to the power of five newtons per meter
squared.
