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Chapter 11: Problem 14

United States currency is printed using intaglio presses that generate a printing pressure of \(8.0 \times 10^{4} \mathrm{lb} / \mathrm{in} .^{2} \mathrm{A} \$ 20 \mathrm{bill}\) is 6.1 in. by \(2.6 \mathrm{in} .\) Calculate the magnitude of the force that the printing press applies to one side of the bill.

Short Answer

Expert verified
The force applied is approximately \(1.3 \times 10^6 \text{ lb}\).

Step by step solution

01

Determine the Area of the Bill

The area of the bill is calculated by using the formula for area, which is length times width. Given that the dimensions of the bill are 6.1 inches by 2.6 inches, the area can be found as follows:\[\text{Area} = \text{Length} \times \text{Width} = 6.1 \, \text{in.} \times 2.6 \, \text{in.}\]
02

Calculate the Area of the Bill

Substitute the given values into the formula for area to get:\[\text{Area} = 6.1 \, \text{in.} \times 2.6 \, \text{in.} = 15.86 \, \text{in.}^2\]
03

Use Pressure to Find the Force

The force exerted by the printing press can be determined using the relationship between pressure and force, where \( \text{Force} = \text{Pressure} \times \text{Area} \). The given printing pressure is \( 8.0 \times 10^4 \, \text{lb/in.}^2 \).
04

Calculate the Force Applied

Substitute the values for pressure and the calculated area into the force formula:\[\text{Force} = \left( 8.0 \times 10^4 \, \text{lb/in.}^2 \right) \times 15.86 \, \text{in.}^2 = 1.2688 \times 10^6 \, \text{lb}\]
05

Finalize the Calculation

Round the result to an appropriate number of significant figures. For this calculation, we can round the force to two significant figures since the given pressure has two significant figures:\[\text{Force} \approx 1.3 \times 10^6 \, \text{lb}\]

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Key Concepts

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

Pressure
Pressure is the amount of force applied per unit area. It describes how force is distributed over a surface. In physics, pressure is an important concept because it helps us understand how forces affect objects.

When you apply pressure, you concentrate force over an area. This relationship is defined by the formula:
  • Pressure = \(\frac{\text{Force}}{\text{Area}}\)
For example, when a printing press applies pressure on paper, it determines how much force per inch is exerted on the surface of the paper. This is often measured in pounds per square inch (lb/in虏).

In problems like the one given, knowing the pressure helps us figure out the force applied on an object as long as we know the area affected.
Force
Force is a push or pull acting upon an object as a result of its interaction with another object. In physical problems, knowing the amount of force is crucial to understanding how objects are moved or altered.

The link between force and pressure can be expressed by:
  • Force = Pressure \(\times\) Area
This formula tells us that the force exerted depends on how much pressure is applied and over how large an area. In our exercise, the force generated by the printing press is calculated using this relation. We multiply the pressure (in lb/in虏) by the area (in in虏) of the $20 bill's surface.

This application ensures uniform force distribution over the paper during printing. Understanding this helps in not only physics lessons but also in practical applications, like how machines operate to apply precise amounts of pressure.
Area Calculation
Calculating area is fundamental in determining how much surface is available for applying force. The area can be calculated by multiplying the length by the width of an object.

For a rectangle like a bill, the formula is:
  • Area = Length \(\times\) Width
In our exercise, the $20 bill is given as 6.1 inches by 2.6 inches. To find the area of this bill:
  • Area = 6.1 in \(\times\) 2.6 in = 15.86 in虏
Understanding how to compute an area is crucial because this number is used to calculate the resulting force from applied pressure in physics problems. The concept of area extends beyond physics and is widely used in everyday situations like calculating space for furniture or determining land sizes.
Significant Figures
Significant figures are digits in a number that contribute to its accuracy. They reflect precision in measurements and calculations. When rounding numbers, it is important to consider these figures to not overstate or understate the precision of the value.

According to the rules of significant figures, when multiplying or dividing, the number of significant figures in the result should match the smallest number of significant figures in any of the numbers used in the calculation.

In our exercise:
  • The pressure is given as 8.0 \(\times\) 10鈦� lb/in虏, which has two significant figures.
  • The calculated area is 15.86 in虏.
  • The final force was initially calculated as 1.2688 \(\times\) 10鈦� pounds.
  • We round to 1.3 \(\times\) 10鈦� pounds, using two significant figures, reflecting the precision from the pressure value.
This ensures that the final result is not more precise than the data given and maintains scientific reliability.

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Most popular questions from this chapter

A liquid is flowing through a horizontal pipe whose radius is \(0.0200 \mathrm{m}\). The pipe bends straight upward through a height of \(10.0 \mathrm{m}\) and joins another horizontal pipe whose radius is \(0.0400 \mathrm{m} .\) What volume flow rate will keep the pressures in the two horizontal pipes the same?

An antifreeze solution is made by mixing ethylene glycol \(\rho=1116\) \(\mathrm{kg} / \mathrm{m}^{3}\) ) 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.

Two hoses are connected to the same outlet using a Y-connector, as the drawing shows. The hoses \(A\) and \(B\) have the same length, but hose \(B\) has the larger radius. Each is open to the atmosphere at the end where the water exits. Water flows through both hoses as a viscous fluid, and Poiseuille's \(\operatorname{law}\left[Q=\pi R^{4}\left(P_{2}-P_{1}\right) /(8 \eta L)\right]\) applies to each. In this law, \(P_{2}\) is the pressure upstream, \(P_{1}\) is the pressure downstream, and \(Q\) is the volume flow rate. The ratio of the radius of hose \(\mathrm{B}\) to the radius of hose \(\mathrm{A}\) is \(R_{\mathrm{B}} / R_{\mathrm{A}}=1.50 .\) Find the ratio of the speed of the water in hose \(B\) to the speed in hose \(A\).

A solid concrete block weighs 169 N and is resting on the ground. Its dimensions are \(0.400 \mathrm{m} \times 0.200 \mathrm{m} \times 0.100 \mathrm{m} .\) A number of identical blocks are stacked on top of this one. What is the smallest number of whole blocks (including the one on the ground) that can be stacked so that their weight creates a pressure of at least two atmospheres on the ground beneath the first block?

An object is solid throughout. When the object is completely submerged in ethyl alcohol, its apparent weight is \(15.2 \mathrm{N}\). When completely submerged in water, its apparent weight is \(13.7 \mathrm{N}\). What is the volume of the object?
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