Video Transcript
A diver swims in water of density
1015 kilograms per meters cubed, as shown in the diagram. What is the difference between the
water pressure at the diver’s head and at his feet? Answer to the nearest pascal.
In this question, we’ve been given
a diagram of a diver, which shows his head and feet at different depths. Since the diver’s feet are at a
greater depth than his head, we know that the diver’s feet will experience a greater
value of water pressure. It’s our job to find the difference
in pressure between these two points.
Recall that the pressure produced
by a fluid, such as water, can be calculated using the equation 𝑃 equals 𝜌 times
𝑔 times ℎ, where 𝜌 is the density of the fluid, 𝑔 is the acceleration due to
gravity, and ℎ is the depth of the fluid where the pressure is to be calculated.
The water pressure at the diver’s
feet, which we’ll call 𝑃 sub feet, is equal to the water density 𝜌 times the
acceleration due to gravity 𝑔 times the depth of his feet in the water, which we’ll
call ℎ sub feet. Similarly, the water pressure at
the diver’s head, 𝑃 sub head, is equal to 𝜌 times 𝑔 times the depth of the
diver’s head in the water, ℎ sub head. We’ll call the difference in water
pressure between the head and the feet Δ𝑃. Recall that Δ is a Greek symbol
often used to denote a change in a quantity.
To find the difference in pressure
between these two points, we simply need to subtract 𝑃 sub head from 𝑃 sub
feet. Now, substituting in our
expressions for the two pressures, we find that Δ𝑃 is equal to 𝜌𝑔ℎ sub feet minus
𝜌𝑔ℎ sub head. Since 𝜌 and 𝑔 are the same at
both the head and the feet, we can factorize out these common terms and get Δ𝑃
equals 𝜌𝑔 times ℎ sub feet minus ℎ sub head.
Now, we’re ready to substitute in
numerical values. We’re told that the density of the
water here, 𝜌, is equal to 1015 kilograms per cubic meter. And we can recall that near the
surface of the Earth, acceleration due to gravity equals 9.8 meters per second
squared. From the diagram, we know that the
diver’s feet are at a depth of 1.8 meters and the diver’s head is at a depth of 1.2
meters. So the difference in these depths,
ℎ sub feet minus ℎ sub head, is equal to 1.8 meters minus 1.2 meters, or 0.6
meters.
Now, substituting all these values
in, we find that the pressure difference, Δ𝑃, is equal to 1015 kilograms per cubic
meter times 9.8 meters per second squared times 0.6 meters.
Before we calculate, let’s check
out the units here and notice that two powers of meters cancel out of the numerator
and denominator. Thus, the units associated with
this expression are kilograms per meter second squared. We can recall that this combination
of units is equivalent to pascals, which is a good sign because we are asked to
solve for a pressure value in pascals.
Finally, plugging this into a
calculator, we find that Δ𝑃 is equal to 5968.2 pascals. The question asks us to give our
answer to the nearest pascal. So we simply round this value down
to reach our final answer. Thus, we’ve found that the
difference between the water pressure at the diver’s head and at his feet is equal
to 5968 pascals.
