The density of mercury changes approximately linearly with...

The density of mercury changes approximately linearly with temperature as $\rho_{\mathrm{Hg}}=13595-2.5 \mathrm{~T}$ $\mathrm{kg} / \mathrm{m}^{3}(T$ in Celsius), so the same pressure difference will result in a manometer reading that is influenced by temperature. If a pressure difference of $100 \mathrm{kPa}$ is measured in the summer at $35^{\circ} \mathrm{C}$ and in the winter at $-15^{\circ} \mathrm{C}$, what is the difference in column height between the two measurements?

Key Concepts

Manometer Operation and Pressure Measurement

Manometers are instruments that measure pressure differences by balancing the weight of a fluid column against the pressure being measured. Understanding how fluid properties like density change with temperature is essential in interpreting manometer readings accurately and compensating for thermal variations in pressure measurements.

Fluid Density and Temperature Dependence

The density of fluids can vary with temperature, often in a nearly linear fashion for certain liquids like mercury. This temperature dependence is critical because it affects how the fluid responds in various applications, such as manometers, where a change in density directly influences the height of the fluid column for a given pressure difference.

Hydrostatic Pressure

Hydrostatic pressure refers to the pressure exerted by a fluid at rest due to the force of gravity. It is calculated using the relationship ?P = ?gh, where ? is the fluid density, g is gravitational acceleration, and h is the height of the fluid column. This concept is fundamental when analyzing manometers and other devices that measure pressure via fluid columns.

Transcript

00:01 So let's start with the statement right now.

00:03 It says that the density of mercury, right? the density of mercury, it changes approximately linearly with the temperature, right? and the density of mercury is given by this equation that is already in the statement.

00:23 That is 13 ,595.

00:25 Negative 2 .5 d kilogram per meter cube, right? now the t is in degrees celsius.

00:35 Now, it says that the same pressure difference will result in a nanometer reading that is influenced by the temperature.

00:42 It says that if the pressure difference, if a pressure difference of 100 kilopascals, right? now, the pressure difference is equal to 100 kilopascals at a temperature is made in the summer at a temperature of 35 degrees centigrade and this is for the summer right then we have the temperature for the winter that is negative 15 degree centigrade right so we are required to answer that what is the difference in column height between the two measurements right so for that um let this temperature for summer let it be t s and the temperature for winter let it be t is and the temperature for winter let it be t sub w right now we have been given the density of the mercury right and they have also told us that the density of the mercury it changes approximately nearly with the temperature right so first of all we would calculate the summer density right so the summer density it would be row sub s and we would use the given equation right that is 13 ,595 negative 2 .5 times the summer temperature right that is equal to 35 degree centigrade right so when we calculated we get that's equal to 13 ,507 .5 kilogram per meter right now this is the summer density and i'm moving forward to the winter density right that is a row sub w right and again you're going to use the given expression for the density that is 13 ,595 negative 2 .5 into the winter temperature that is negative 15.

02:36 So when we calculate, we get that's equal to 13 ,600 and 32 .5 kilogram per meter.

02:49 Right.

02:51 So now that we have the density for the summer and winter, right, we can calculate the column height as the summer...

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