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Chapter 15: Problem 16

In a classroom demonstration, the pressure inside a soft drink can is suddenly reduced to essentially zero. Assuming the can to be a cylinder with a height of \(12 \mathrm{cm}\) and a diameter of \(6.5 \mathrm{cm}\), find the net inward force exerted on the vertical sides of the can due to atmospheric pressure.

Short Answer

Expert verified
The net inward force is approximately 24,855,804 N.

Step by step solution

01

Calculate the radius of the cylinder

To find the radius, divide the diameter by 2. Given that the diameter is 6.5 cm, the radius \( r \) is obtained as follows: \[ r = \frac{6.5}{2} = 3.25 \text{ cm} \]
02

Calculate the lateral surface area of the cylinder

The lateral surface area \( A \) of a cylinder is determined by the formula \( A = 2\pi r h \). We know \( r = 3.25 \text{ cm} \) and the height \( h = 12 \text{ cm} \). Plug these into the formula to get: \[ A = 2 \pi (3.25)(12) = 78 \pi \text{ cm}^2 \]
03

Calculate the atmospheric pressure in Pascals

The standard atmospheric pressure is approximately \( 101,325 \text{ Pa} \) (Pascals), which will be the pressure acting on the outside surface of the can.
04

Calculate the net inward force

Once we have the surface area and the atmospheric pressure, we can calculate the force \( F \) exerted on the can by using the equation \( F = P \times A \). Substituting the values, we have: \[ F = 101,325 \times 78 \pi \approx 24,855,804 \text{ N} \]

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Key Concepts

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

Cylinder Surface Area
The surface area of a cylinder is an essential concept when determining forces acting on it. In this case, we focus on the lateral surface area, which is the area of the curved "side" part of the can. It does not include the top and bottom surfaces. Calculating the lateral surface area involves understanding its geometry. A cylinder's lateral surface unfolds like a rectangle. The length of this rectangle corresponds to the cylinder's circumference, while its height is the same as the cylinder's height.
The formula for lateral surface area is given by:
  • \[ A = 2\pi r h \]
where:
  • \( r \) is the radius of the cylinder's base, and
  • \( h \) is its height.
In our exercise, with \( r = 3.25 \, \text{cm} \) and \( h = 12 \, \text{cm} \), we use the formula to calculate the surface area of the can. This process gives us a surface area of \( 78\pi \, \text{cm}^2 \). The lateral area is crucial as it dictates the total area on which atmospheric pressure acts.
Force Calculation
Once we know the area, figuring out the force involves using the concept from physics where force is expressed as the product of pressure and area. This relationship is defined by the equation:
  • \[ F = P \times A \]
Here, \( F \) denotes the force, \( P \) represents the pressure exerted on a surface, and \( A \) stands for the area over which the pressure acts.
For our scenario, atmospheric pressure pushes on the vertical sides of the can, and the calculation leverages the known atmospheric pressure of roughly \( 101,325 \, \text{Pa} \) (Pascals). Multiply this pressure by the calculated lateral surface area (\( 78\pi \, \text{cm}^2 \)) to derive the net inward force. Simplifying this gives approximately \( 24,855,804 \, \text{N} \). Recognize that this substantial force results from the pressure distributed over the entire vertical surface of the can.
Pressure Units
Understanding pressure units is crucial in physics and engineering problems. Pressure refers to the amount of force applied over an area and is quantified in units like Pascals (Pa), atmospheres (atm), or bars. In the metric system, 1 Pascal equals a force of 1 Newton per square meter (\[ 1 \, ext{Pa} = 1 \, ext{N/m}^2 \]). The standard atmospheric pressure, typically used here, is approximately \( 101,325 \, \text{Pa} \).
When working on exercises involving atmospheric or any type of pressure, it's important to consistently use correct units. This ensures accurate calculations, particularly when converting between different measurement systems. In most scientific contexts, especially involving gases or liquids' behavior under pressure, Pascals serve as the standard unit. Familiarity with this allows easier interpretation of physical scenarios, like the force enacted by atmospheric pressure, purely from a numerical and conceptual standpoint.

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Most popular questions from this chapter

A circular wine barrel \(75 \mathrm{cm}\) in diameter will burst if the net upward force exerted on the top of the barrel is \(643 \mathrm{N}\). A tube \(1.0 \mathrm{cm}\) in diameter extends into the barrel through a hole in the top, as indicated in Figure \(15-26 .\) Initially, the barrel is filled to the top and the tube is empty above that level. What weight of water must be poured into the tube in order to burst the barrel?

IP A pan half-filled with water is placed in the back of an SUV. (a) When the SUV is driving on the freeway with a constant velocity, is the surface of the water in the pan level, tilted forward, or tilted backward? Explain. (b) Suppose the SUV accelerates in the forward direction with a constant acceleration \(a\). Is the surface of the water tilted forward, or tilted backward? Explain. (c) Show that the angle of tilt, \(\theta,\) in part (b) has a magnitude given by tan \(\theta=a / g,\) where \(g\) is the acceleration of gravity.

Water flows in a cylindrical, horizontal pipe. As the pipe narrows to half its initial diameter, the pressure in the pipe changes. (a) Is the pressure in the narrow region greater than, less than, or the same as the initial pressure? Explain. (b) \(\mathrm{Cal}-\) culate the change in pressure between the wide and narrow regions of the pipe. Give your answer symbolically in terms of the density of the water, \(\rho,\) and its initial speed \(v\).

A horizontal pipe carries oil whose coefficient of viscosity is \(0.00012 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\). The diameter of the pipe is \(5.2 \mathrm{cm},\) and its length is \(55 \mathrm{m}\) (a) What pressure difference is required between the ends of this pipe if the oil is to flow with an average speed of \(1.2 \mathrm{m} / \mathrm{s}\) ? (b) What is the volume flow rate in this case?

Suppose that when you ride on your \(7.70-\mathrm{kg}\) bike the weight of you and the bike is supported equally by the two tires. If the gauge pressure in the tires is \(70.5 \mathrm{Ib} / \mathrm{in}^{2}\) and the area of contact between each tire and the road is \(7.13 \mathrm{cm}^{2},\) what is your weight?
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