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Chapter 1: Problem 13

The position of a particle moving under uniform acceleration is some function of time and the acceleration. Suppose we write this position \(s=k a^{m} t^{n},\) where \(k\) is a dimensionless constant. Show by dimensional analysis that this expression is satisfied if \(m=1\) and \(n=2 .\) Can this analysis give the value of \(k ?\)

Short Answer

Expert verified
For the given expression of position, it can be confirmed through dimensional analysis that \(m = 1\) and \(n = 2\). Dimensional analysis cannot provide the value of 'k'. The value of 'k' would need to be determined through other means specific to the physical scenario under consideration.

Step by step solution

01

Identify the dimensions

The first step is to identify the dimensions of the quantities involved in the expression. The dimensions of position 's' are [L] (where [L] represents length), the dimensions of acceleration 'a' are \([LT^{-2}]\) (length per time squared), and the dimensions of time 't' are [T]. The constant 'k' is dimensionless and therefore its dimensions are [].
02

Substitute the dimensions into the expression

Substitute the dimensions of the quantities into the expression for position. The expression becomes: \([L] = k ([LT^{-2}])^{m} [T]^{n}\)
03

Simplify the expression

Simplify the expression, which leads to: \([L] = k [L^{m} T^{-2m} T^{n}]\). This can be further simplified by combining like dimensions to get \([L] = k [L^{m} T^{n-2m}]\).
04

Compare dimensions on each side

In order to satisfy the principle of homogeneity, the dimensions on each side of the equation should be the same. Hence \(m = 1\) (since the power of L on left side of equation = power of L on right side of equation) and \(n - 2m = 0\) (since the dimension T does not appear on the left side of the equation). Solving the second equation gives \(n = 2m\). Given that \(m = 1\), hence \(n = 2\).
05

Answering the second part of the question

Dimensional analysis only checks the homogeneity of the physical quantities in the equation. The constant 'k' being dimensionless, does not contribute to the dimension of the equation. Therefore, dimensional analysis cannot be used to find the value of 'k'. The value of 'k' depends on the specific nature of the physical phenomena which is described by the equation.

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Key Concepts

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

Uniform Acceleration
Uniform acceleration refers to a constant acceleration, meaning the rate of change of velocity does not vary over time. This is a fundamental concept in physics and greatly simplifies calculations for moving objects.
In real-world scenarios, examples of uniform acceleration include free-falling objects under gravity (neglecting air resistance) or a car moving at a constant rate of increase in speed.
  • Constant Rate: The velocity of an object increases or decreases at a consistent rate.
  • Simplification: Uniform acceleration helps in predicting the future motion of objects easily using equations of motion.
By understanding uniform acceleration, students can apply it to solve various motion-related problems with greater ease.
Position Function
The position function describes the location of a particle in relation to time and other changes, such as acceleration.
In physics, particularly with constant acceleration, the position function provides a mathematical way to capture an object's movement precisely.
  • Equation Form: The position can be given by the formula \(s = k a^{m} t^{n}\), where \(s\) is the position, \(a\) is acceleration, and \(t\) is time.
  • Dependence: This formula shows dependability on time \(t\) and acceleration \(a\), which are key factors in an object's motion.
The accurate expression assists in predicting where an object will be at any point in time, given its starting conditions and motion characteristics.
Homogeneity Principle
The homogeneity principle is an essential principle in dimensional analysis. It states that for an equation to be physically meaningful, both sides must have the same dimensions.
In practice, this means that you can only equate measures that are similarly dimensioned.
  • Dimensional Consistency: A valid equation ensures both sides mirror the same dimensions, like length, mass, and time.
  • Checks and Balances: It helps verify whether expressions derived from physical laws are correct in terms of their dimensional aspects.
By employing this principle, physicists and students can intuitively verify mathematical models of real-world phenomena, ensuring reliability and correctness.

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Most popular questions from this chapter

A hydrogen atom has a diameter of approximately \(1.06 \times 10^{-10} \mathrm{m},\) as defined by the diameter of the spherical electron cloud around the nucleus. The hydrogen nucleus has a diameter of approximately \(2.40 \times 10^{-15} \mathrm{m}\) (a) For a scale model, represent the diameter of the hydrogen atom by the length of an American football field \((100 \text { yd }=300 \mathrm{ft}),\) and determine the diameter of the nucleus in millimeters. (b) The atom is how many times larger in volume than its nucleus?

The nearest stars to the Sun are in the Alpha Centauri multiple-star system, about \(4.0 \times 10^{13} \mathrm{km}\) away. If the Sun, with a diameter of \(1.4 \times 10^{9} \mathrm{m},\) and Alpha Centauri A are both represented by cherry pits \(7.0 \mathrm{mm}\) in diameter, how far apart should the pits be placed to represent the Sun and its neighbor to scale?

In physics it is important to use mathematical approximations. Demonstrate that for small angles \(\left(<20^{\circ}\right)\) $$\tan \alpha \approx \sin \alpha \approx \alpha=\pi \alpha^{\prime} / 180^{\circ}$$ where \(\alpha\) is in radians and \(\alpha^{\prime}\) is in degrees. Use a calculator to find the largest angle for which tan \(\alpha\) may be approximated by \(\sin \alpha\) if the error is to be less than \(10.0 \%\)

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The distance from the Sun to the nearest star is about \(4 \times 10^{16} \mathrm{m} .\) The Milky Way galaxy is roughly a disk of diameter \(\sim 10^{21} \mathrm{m}\) and thickness \(\sim 10^{19} \mathrm{m} .\) Find the order of magnitude of the number of stars in the Milky Way. Assume the distance between the Sun and our nearest neighbor is typical.
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