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Chapter 1: Problem 54

The variables \(x, v,\) and \(a\) have the dimensions of \([\mathrm{L}],[\mathrm{L}] /[\mathrm{T}],\) and \([\mathrm{L}] /[\mathrm{T}]^{2},\) respectively. These variables are related by an equation that has the form \(v^{n}=2 a x,\) where \(n\) is an integer constant \((1,2,3,\) etc. \()\) without dimensions. What must be the value of \(n,\) so that both sides of the equation have the same dimensions? Explain your reasoning.

Short Answer

Expert verified
The value of \(n\) must be 2 for both sides of the equation to have the same dimensions.

Step by step solution

01

Identify dimensions of each variable

The variable dimensions are given as follows: \[x\] is \([L]\), \[v\] is \([L]/[T]\), and \[a\] is \([L]/[T]^2\). These dimensions are essential to analyzing the equation.
02

Analyze the dimensions of the right side of the equation

The right side of the equation is \(2ax\). The constant 2 is dimensionless, so we focus on \(ax\): \[ [L]/[T]^2 \times [L] = [L]^2/[T]^2 \]. Therefore, the dimension of the right side of the equation is \([L]^2/[T]^2\).
03

Determine the dimensions of the left side of the equation

The left side of the equation is \(v^n\). The dimension of \(v\) is \([L]/[T]\), so raising \(v\) to the power of \(n\), we get: \[[L]^n/[T]^n\].
04

Equate dimensions from both sides and solve for n

To have dimensional consistency between both sides of the equation, their dimensions must be equal: \[[L]^n/[T]^n = [L]^2/[T]^2\]. This implies that \(n = 2\) since both lengths \([L]\) and times \([T]\) are squared on the right side.

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Key Concepts

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

Physics Equations
Physics equations serve as the language of the physical world, allowing us to describe relationships between different quantities and their behaviors. Equations in physics express connections through mathematical symbols and constants. By using dimensional analysis, we can ensure that these equations are not only quantitatively correct but also dimensionally consistent. In the given problem, the equation is \(v^{n}=2ax\), which relates the variables velocity \(v\), acceleration \(a\), and displacement \(x\). An integer constant \(n\) is used to balance their dimensions.

For equations to make physical sense, the dimensions on both sides must match. This property is critical because it ensures the equation's independence of the units used in measurement. If both sides of an equation have the same physical dimensions, the equation is said to be dimensionally consistent. The solution involves finding the correct exponent \(n\) that achieves this balance.
Physical Dimensions
Physical dimensions represent the intrinsic characteristics of a physical quantity, such as length, time, mass, etc. These dimensions are commonly expressed using symbols like \([L]\) for length, \([T]\) for time, and \([M]\) for mass. In our exercise:

  • \(x\) is displacement with dimension \([L]\)
  • \(v\) is velocity with dimension \([L]/[T]\)
  • \(a\) is acceleration with dimension \([L]/[T]^2\)

By identifying the physical dimensions of each variable, we can perform dimensional analysis to ensure equations are balanced. The approach involves breaking down each term of an equation into its constituent dimensions. This allows us to verify that each part of the equation has the correct dimensionality and infer more information about how the variables interact.
Dimensional Consistency
Dimensional consistency, sometimes referred to as dimensional homogeneity, is a fundamental principle ensuring that both sides of an equation express the same type of physical quantity. It implies that units on each side of an equation must match, making it universally valid regardless of the unit system employed.

In the problem, we explored the equation \(v^n = 2ax\). The right side of the equation \(2ax\) has the dimension \([L]^2/[T]^2\) because:
  • \([L]/[T]^2\) from \(a\)
  • \([L]\) from \(x\)
  • Combine to \([L]^2/[T]^2\)

The left side involves \(v\) raised to the power of \(n\):
  • \(v^n\) becomes \([L]^n/[T]^n\)
  • For dimensional consistency \([L]^n/[T]^n = [L]^2/[T]^2\)
  • Thus, \(n = 2\)

This process highlights how dimensional consistency is crucial for verifying the plausibility and correctness of a physics equation.

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Most popular questions from this chapter

The following are dimensions of various physical parameters that will be discussed later on in the text. Here [L], [T], and [M] denote, respectively, dimensions of length, time, and mass. $$ \begin{array}{|l|l|l|l|} \hline & \text { Dimension } & & \text { Dimension } \\ \hline \text { Distance }(x) & {[\mathrm{L}]} & \text { Acceleration }(a) & {[\mathrm{L}] /[\mathrm{T}]^{2}} \\ \hline \text { Time }(t) & {[\mathrm{T}]} & \text { Force }(F) & {[\mathrm{M}][\mathrm{L}] /[\mathrm{T}]^{2}} \\ \hline \text { Mass }(m) & {[\mathrm{M}]} & \text { Energy }(E) & {[\mathrm{M}][\mathrm{L}]^{2} /[\mathrm{T}]^{2}} \\ \hline \text { Speed }(v) & {[\mathrm{L}] /[\mathrm{T}]} & & \\ \hline \end{array} $$ Which of the following equations are dimensionally correct? a. \(F=m a\) b. \(x=\frac{1}{2} a t^{3}\) c. \(E=\frac{1}{2} m v\) d. \(E=\max\) e. \(v=\sqrt{F x / m}\)

Two bicyclists, starting at the same place, are riding toward the same campground by two different routes. One cyclist rides \(1080 \mathrm{~m}\) due east and then turns due north and travels another \(1430 \mathrm{~m}\) before reaching the campground. The second cyclist starts out by heading due north for \(1950 \mathrm{~m}\) and then turns and heads directly toward the campground. (a) At the turning point, how far is the second cyclist from the campground? (b) What direction (measured relative to due east) must the second cyclist head during the last part of the trip?

Consider the following four force vectors: $$ \begin{array}{l} \overrightarrow{\mathbf{F}}_{1}=50.0 \text { newtons, due east } \\\ \overrightarrow{\mathbf{F}_{2}}=10.0 \text { newtons, due east } \\\ \overrightarrow{\mathbf{F}_{3}}=40.0 \text { newtons, due west } \\\ \overrightarrow{\mathbf{F}}_{4}=30.0 \text { newtons, due west } \end{array} $$ Which two vectors add together to give a resultant with the smallest magnitude, and which two vectors add to give a resultant with the largest magnitude? In each case specify the magnitude and direction of the resultant.

Vector \(\overrightarrow{\mathrm{A}}\) has a magnitude of 6.00 units and points due east. Vector \(\overrightarrow{\mathrm{B}}\) points due north. (a) What is the magnitude of \(\overrightarrow{\mathbf{B}},\) if the vector \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\) points \(60.0^{\circ}\) north of east? (b) Find the magnitude of \(\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}\).

A pilot flies her route in two straight-line segments. The displacement vector \(\overrightarrow{\mathrm{A}}\) for the first segment has a magnitude of \(244 \mathrm{~km}\) and a direction \(30.0^{\circ}\) north of east. The displacement vector \(\overrightarrow{\mathbf{B}}\) for the second segment has a magnitude of \(175 \mathrm{~km}\) and a direction due west. The resultant displacement vector is \(\overrightarrow{\mathbf{R}}=\overrightarrow{\mathrm{A}}+\overrightarrow{\mathbf{B}}\) and makes an angle \(\theta\) with the direction due east. Make a drawing to scale showing the vectors \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathrm{B}}\) placed tail to head and the resultant vector \(\overrightarrow{\mathbf{R}}\). Without doing any calculations decide whether (a) the magnitude of \(\overrightarrow{\mathbf{R}}\) is greater or smaller than the magnitude of \(\overrightarrow{\mathrm{A}},(\mathrm{b})\) the magnitude of \(\overrightarrow{\mathbf{R}}\) is greater or smaller than the magnitude of \(\overrightarrow{\mathbf{B}}\), and (c) the angle \(\theta\) is greater than, smaller than, or equal to \(30.0^{\circ}\). Using the component method, find the magnitude of \(\overrightarrow{\mathbf{R}}\) and the directional angle \(\theta\). Check to see that your answers are consistent with your answers to the Concept Questions.
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