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Chapter 11: Problem 22

A meat baster consists of a squeeze bulb attached to a plastic tube. When the bulb is squeezed and released, with the open end of the tube under the surface of the basting sauce, the sauce rises in the tube to a distance h, as the drawing shows. Using \(1.013 \times 10^{5}\) Pa for the atmospheric pressure and 1200 \(\mathrm{kg} / \mathrm{m}^{3}\) for the density of the sauce, find the absolute pressure in the bulb when the distance \(h\) is \(\quad\) (a) 0.15 \(\mathrm{m}\) and (b) 0.10 \(\mathrm{m} .\)

Short Answer

Expert verified
The pressure is approximately 9.835x10^4 Pa for h=0.15 m and 9.895x10^4 Pa for h=0.10 m.

Step by step solution

01

Understanding the Situation

A meat baster is used to suck sauce from a container by creating a pressure difference. Atmospheric pressure pushes the sauce up the tube when the internal pressure in the bulb is reduced by squeezing and releasing it.
02

Identify Given Values

- Atmospheric Pressure \ \(P_0 = 1.013 \times 10^5 \, \text{Pa}\)\ - Density of Sauce \ \(\rho = 1200 \, \text{kg/m}^3\)\- Gravitational Acceleration \ \(g = 9.81 \, \text{m/s}^2\)\- Height difference \ \(h = 0.15 \, \text{m}\) for part (a) and \(h = 0.10 \, \text{m}\) for part (b).
03

Apply Pressure Formula

The pressure in the bulb is determined by balancing the atmospheric pressure with the pressure exerted by the liquid column. The formula is: \[ P_{bulb} = P_0 - \rho g h \]
04

Calculate Pressure in the Bulb for h = 0.15 m

Plug in the values to calculate the pressure when \(h = 0.15 \, \text{m}\): \[ P_{bulb} = 1.013 \times 10^5 \text{ Pa} - 1200 \times 9.81 \times 0.15 \text{ m} \] Calculate the result to find \(P_{bulb}\).
05

Result for h = 0.15 m

The calculated pressure in the bulb is approximately \(P_{bulb} \approx 9.835 \times 10^4 \, \text{Pa}\).
06

Calculate Pressure in the Bulb for h = 0.10 m

Use the same formula with \(h = 0.10 \, \text{m}\): \[ P_{bulb} = 1.013 \times 10^5 \text{ Pa} - 1200 \times 9.81 \times 0.10 \text{ m} \] Compute this to find \(P_{bulb}\).
07

Result for h = 0.10 m

The calculated pressure in the bulb is approximately \(P_{bulb} \approx 9.895 \times 10^4 \, \text{Pa}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pressure Difference
Pressure difference is the concept that enables the meat baster to function. When you squeeze the bulb, the air inside it is partially expelled, creating a lower pressure within the bulb. As you release the bulb, this pressure becomes even less than the atmospheric pressure. This pressure difference between the inside of the bulb and the surrounding atmosphere is what causes the fluid, in this case, the sauce, to rise up into the tube.
Here鈥檚 why the pressure difference is essential:
  • The atmospheric pressure outside the bulb is higher than the pressure inside the bulb once it is released.
  • High external pressure pushes the sauce into the tube, filling the void created by the low internal pressure.
Understanding pressure differences is crucial, as it explains numerous everyday phenomena, such as drinking through a straw or the functioning of a vacuum cleaner.
Atmospheric Pressure
Atmospheric pressure is the force exerted by the weight of the air above us. It is the pressure we experience at any point in the Earth's atmosphere. In the case of the meat baster, atmospheric pressure is an essential factor that assists in raising the sauce up the tube.
For this exercise, we use a specific value for atmospheric pressure, given as \(P_0 = 1.013 \times 10^5 \text{ Pa}\). This is a general standard value used for practical calculations close to sea level.
When using a meat baster:
  • Atmospheric pressure acts on the surface of the sauce as well as within the tube.
  • As the pressure inside the bulb decreases due to squeezing, atmospheric pressure remains constant, which aids in moving the fluid against gravity.
This difference allows us to perform tasks like basting, where external atmospheric pressure does the work of moving the sauce into the tube.
Density of Fluid
Density of a fluid, noted as \(\rho\), plays a vital role in determining how much pressure is exerted by the fluid column in the meat baster tube. The density is defined as the mass per unit volume of a fluid, with units often expressed in \(\text{kg/m}^3\).
For the basting sauce in our exercise, the density is given as 1200 \(\text{kg/m}^3\). This density directly affects how the sauce behaves in response to the pressure differences:
  • Higher density sauces will exert more pressure per unit height than thinner ones.
  • The formula \(P_{bulb} = P_0 - \rho g h\) is used to relate the atmospheric pressure and the additional pressure exerted by the column of sauce.
The density of the fluid is crucial in understanding how much the pressure needs to be reduced inside the bulb to raise the sauce to a certain height \(h\) in the tube.

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Most popular questions from this chapter

Multiple-Concept Example 8 presents an approach to problems of this kind. The hydraulic oil in a car lift has a density of \(8.30 \times 10^{2} \mathrm{kg} / \mathrm{m}^{3}\) . The weight of the input piston is negligible. The radii of the input piston and output plunger are \(7.70 \times 10^{-3} \mathrm{m}\) and \(0.125 \mathrm{m},\) respectively. What input force \(F\) is needed to support the 24500 \(\mathrm{-N}\) combined weight of a car and the output plunger, when (a) the bottom surfaces of the piston and plunger are at the same level, and (b) the bottom surface of the output plunger is 1.30 m above that of the input piston?

One way to administer an inoculation is with a 鈥済un鈥� that shoots the vaccine through a narrow opening. No needle is necessary, for the vaccine emerges with sufficient speed to pass directly into the tissue beneath the skin. The speed is high, because the vaccine \(\left(\rho=1100 \mathrm{kg} / \mathrm{m}^{3}\right)\) is held in a reservoir where a high pressure pushes it out. The pressure on the surface of the vaccine in one gun is 4.1 106 Pa above the atmospheric pressure outside the narrow opening. The dosage is small enough that the vaccine鈥檚 surface in the reservoir is nearly stationary during an inoculation. The vertical height between the vaccine鈥檚 surface in the reservoir and the opening can be ignored. Find the speed at which the vaccine emerges.

A water bed for sale has dimensions of 1.83 \(\mathrm{m} \times 2.13 \mathrm{m} \times 0.229 \mathrm{m} .\) The floor of the bedrroom will tolerate an additional weight of no more than 66660 \(\mathrm{N}\) . Find the weight of the water in the bed and determine whether the bed should be purchased.

An antifreeze solution is made by mixing ethylene glycol \(\left(\rho=1116 \mathrm{kg} / \mathrm{m}^{3}\right)\) with water. Suppose that the specific gravity of such a solution is \(1.0730 .\) Assuming that the total volume of the solution is the sum of its parts, determine the volume percentage of ethylene glycol in the solution.

The drawing shows a hydraulic system used with disc brakes. The force \(\overrightarrow{\mathbf{F}}\) is applied perpendicularly to the brake pedal. The pedal rotates about the axis shown in the drawing and causes a force to be applied perpendicularly to the input piston (radius \(=9.50 \times 10^{-3} \mathrm{m} )\) in the master cylinder. The resulting pressure is transmitted by the brake fluid to the output plungers (radii \(=1.90 \times 10^{-2} \mathrm{m}\) ), which are covered with the brake linings. The linings are pressed against both sides of a disc attached to the rotating wheel. Suppose that the magnitude of \(\overrightarrow{\mathbf{F}}\) is 9.00 \(\mathrm{N}\) . Assume that the input piston and the output plungers are at the same vertical level, and find the force applied to each side of the rotating disc.
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