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Chapter 1: Problem 9

1.9 If \(P\) is a force and \(x\) a length, what are the dimensions (in the \(F L T \text { system })\) of (a) \(d P / d x\) (b) \(d^{3} P / d x^{3},\) and (c) \(\int P d x ?\)

Short Answer

Expert verified
The dimensions of \(d P / d x\), \(d^{3} P / d x^{3}\), and \(\int P d x\) are \(F/L\), \(F/L^{3}\), and \(FL\) respectively in the FTL system.

Step by step solution

01

Dimension of \(d P / d x\)

The dimension of \(d P / d x\) is obtained by subtracting the dimension of \(x\) (which is L) from the dimension of \(P\) (which is F). This leads to \(F/L\).
02

Dimension of \(d^{3} P / d x^{3}\)

The dimension of \(d^{3} P / d x^{3}\) is obtained by subtracting three times the dimension of \(x\) (which is L) from the dimension of \(P\) (which is F). This results in \(F/L^{3}\).
03

Dimension of \(\int P d x\)

The dimension of \(\int P d x\) is obtained by adding the dimension of \(x\) (which is L) to the dimension of \(P\) (which is F). This gives us \(FL\).

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Key Concepts

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

Force
In physics, force is a fundamental concept that describes the interaction that causes an object to change its motion. It is often measured in Newtons (N) and defined by Isaac Newton's second law of motion:
  • Force is the result of mass multiplied by acceleration, represented as \( F = ma \).
  • It is a vector quantity, which means it has both magnitude and direction.
In dimensional analysis specifically, force is represented by the symbol \( F \) in the \( F L T \) (Force, Length, Time) system. Understanding force helps us relate measurable quantities and predict how objects behave when acted upon by various forces. This plays a crucial role in solving physics problems involving motion and equilibrium.
Length
Length is a measure of distance between two points, and it is one of the fundamental quantities in physics. It is typically measured in meters (m) in the metric system.
  • It plays a crucial role in many physical equations and laws, including calculations involving speed, acceleration, and area.
  • In dimensional analysis, length is denoted by \( L \) in the \( F L T \) system, forming a foundation for expressing other physical dimensions in relation to one another.
Understanding the role of length helps grasp the spatial aspects of movement, growth, and physical configuration in various scientific contexts.
Differential Calculus
Differential calculus is a branch of mathematics focused on the concept of the derivative, which represents the rate of change or slope of a function. It is critically used to analyze how quantities change over time or with respect to another quantity.
  • For instance, the derivative \( \frac{dP}{dx} \) calculates how force (\( P \)) varies with respect to length (\( x \)).
  • In a physical context, it helps determine how one variable influences another, providing insights into motion and change.
This makes differential calculus an essential tool in physics, allowing us to describe physical laws in terms that can be easily manipulated and understood.
Integration
Integration is another fundamental concept in calculus. It is essentially the inverse operation of differentiation and involves calculating the area under curves or the accumulation of quantities over a given interval.
  • In a physical sense, integration can determine the total, accumulated effect of a quantity, such as force over a distance.
  • The expression \( \int P \, dx \) represents the total work done when a force \( P \) is applied over a length \( x \).
Understanding integration is crucial in solving problems where the summation of quantities is important, such as determining quantities like displacement, area, volume, and total energy.

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Most popular questions from this chapter

Often the assumption is made that the flow of a certain fluid can be considered as incompressible flow if the density of the fluid changes by less than \(2 \%\). If air is flowing through a tube such that the air pressure at one section is 9.0 psi and at a downstream section it is 8,6 psi at the same temperature, do you think that this flow could be considered an incompressible flow? Support your answer with the necessary calculations. Assume standard atmospheric pressire.

If \(p\) is a pressure, \(V\) a velocity, and \(\rho\) a fluid density, what are the dimensions (in the \(M L T\) system) of (a) \(p / \rho\) (b) \(p V \rho,\) and (c) \(p / \rho V^{2} ?\)

The momentum flux (discussed in Chapter 5) is given by the product \(\dot{m} V\), where \(\dot{m}\) is mass flow rate and \(V\) is velocity. If mass flow rate is given in units of mass per unit time, show that the momentum fux can be expressed in units of force.

The information on a can of pop indicates that the can contains \(355 \mathrm{mL}\). The mass of a full can of pop is \(0.369 \mathrm{kg}\), while an empty can weighs 0.153 N. Determine the specific weight, density, and specific gravity of the pop and compare your results with the corresponding values for water at \(20^{\circ} \mathrm{C}\). Express your results in SI units.

1.17 A formula to estimate the volume rate of flow, \(Q\), flowing over a dam of length, \(B\), is given by the equation \\[ Q=3.09 B H^{3 / 2} \\] where \(H\) is the depth of the water above the top of the dam (called the head). This formula gives \(Q\) in \(\mathrm{ft}^{3} / \mathrm{s}\) when \(B\) and \(H\) are in feet. Is the constant, \(3.09,\) dimensionless? Would this equation be valid if units other than feet and seconds were used?
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