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Chapter 1: Problem 10

Use SI unit analysis to show that the equation \(A=4 \pi r^{2},\) where \(A\) is the area and \(r\) is the radius of a sphere, is dimensionally correct.

Short Answer

Expert verified
The equation is dimensionally correct; both sides have dimensions \([L^2]\).

Step by step solution

01

Identify the SI Units for Each Variable

In the equation \(A = 4 \pi r^{2}\), we need to find the SI units for each variable. The area \(A\) of a sphere is typically measured in square meters \((\text{m}^2)\). The radius \(r\) is a length, so its SI unit is meters \((\text{m})\).
02

Examine the Dimensional Formula for Each Side

The equation \(A = 4 \pi r^{2}\) states that the left side, \(A\), should have dimensions of area \([L^2]\), where \(L\) is length. For the right side, \(4\pi r^{2}\), the constant \(4\pi\) is dimensionless. We need to examine the dimensions of \(r^{2}\). Given \(r\) has dimensions \(L\), \(r^{2}\) therefore has dimensions \(L^2\).
03

Compare and Confirm the Dimensions for Both Sides

Now compare the dimensions of both sides of the equation. The left side has dimensions \([L^2]\) as it is an area. The right side, after including the dimensionless constant \(4\pi\) and \(r^{2}\), also has dimensions \([L^2]\). Thus, both sides of the equation \(A = 4 \pi r^{2}\) have the same dimensions.

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Key Concepts

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

SI Units
The International System of Units (SI) is a systematic set of units that is widely accepted and used globally for measurement, making scientific communication clear and standardized.
  • The foundation of SI units is based on seven base units, which include meters, kilograms, and seconds, among others.
  • These base units are then used to derive other units, such as meters squared (\(m^2\), area).
  • In the context of our problem, understanding SI units helps verify the correctness of physical equations by ensuring each side shares the same dimensional form.
      • For measuring the surface area of a sphere, we use square meters (\(m^2\)), since area is derived from length. It is through these standardized units that scientists and engineers can communicate with precision and universality.
Area of a Sphere
The area of a sphere is a geometric property that can be visualized as the amount of surface needed to cover the sphere entirely.
The formula for calculating the area of a sphere is \(A = 4 \pi r^2\). Here, the constant \(4\pi\) is used, which is common in spherical geometry because a sphere's surface area is related to its radius squared.
  • The constant \(4\pi\) is derived from mathematical proofs involving integrating a circle's circumference over a hemisphere.
  • It can help to visualize a sphere as an infinite number of small circles stacked together, where the radius affects how much surface is covered.
Hence, this formula relates directly to the concept that all surfaces curved in three dimensions can be described using their radii.
Radius
The radius of a sphere is the straight-line distance from its center to any point on its surface. The radius (\(r\)) plays a crucial role in calculating not only the area but also the volume of a sphere.
  • The SI unit for radius is meters (\(m\)), as it is a measure of length.
  • In the formula \(A = 4 \pi r^2\), squaring the radius helps determine the total surface area as the radius influences every point equally.
Understanding the radius provides insight into how surface areas and volumes scale with changes in size and is fundamental in numerous fields such as physics and engineering.
Dimensionless Constant
A dimensionless constant is a numerical value that does not alter the dimensions of a physical quantity. In the equation \(A = 4 \pi r^2\), \(4\pi\) is a dimensionless constant.
  • Dimensionless constants often arise in mathematical and physical formulas where a number is needed to precisely define a relationship, but it doesn't have any units itself.
  • They are crucial for maintaining the dimensional consistency of an equation, ensuring that both sides remain balanced without affecting the unit measure.
In our equation for the area of a sphere, it helps assert that the squared term of the radius maintains the correct unit \(m^2\), thereby verifying the equation's dimensional correctness.

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Most popular questions from this chapter

The general equation for a parabola is \(y=a x^{2}+b x+c,\) where \(a, b,\) and \(c\) are constants. What are the units of each constant if \(y\) and \(x\) are in meters?

How many minutes of arc does the Earth rotate in 1 min of time?

The Roman Coliseum used to be flooded with water to re-create ancient naval battles. Assuming the circular floor be \(250 \mathrm{~m}\) in diameter and the water to have a depth of \(10 \mathrm{ft},\) (a) how many cubic meters of water are required? (b) How much mass would this water have in kilograms? (c) How much would the water weigh in pounds?

(a) If you wanted to express your height with the largest number, which units would you use: (1) meters, (2) feet, (3) inches, or (4) centimeters? Why? (b) If you are \(6.00 \mathrm{ft}\) tall, what is your height in centimeters?

Newton's second law of motion (Section 4.3 ) is expressed by the equation \(F=m a,\) where \(F\) represents force, \(m\) is mass, and \(a\) is acceleration. (a) The SI unit of force is, appropriately, called the newton (N). What are the units of the newton in terms of base quantities? (b) An equation for force associated with uniform circular motion (Section 7.3) is \(F=m v^{2} / r,\) where \(v\) is speed and \(r\) is the radius of the circular path. Does this equation give the same units for the newton?
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