Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 1: Problem 37

An equation for the period \(T\) of a planet (the time to make one orbit about the Sun) is \(4 \pi^{2} r^{3} /(G M),\) where \(T\) is in \(\mathrm{s}, r\) is in \(\mathrm{m}, G\) is in $\mathrm{m}^{3} /\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right),\( and \)M$ is in kg. Show that the equation is dimensionally correct.

Short Answer

Expert verified
Answer: No, the equation is not dimensionally correct.

Step by step solution

01

Write the given equation

The equation for the period \(T\) of a planet is $$ T = \frac{4 \pi^{2} r^{3}}{G M} $$ where \(T\) is in seconds (s), \(r\) is in meters (m), \(G\) is in \(\mathrm{m}^{3}/\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right)\), and \(M\) is in kilograms (kg).
02

Determine the dimensions of all terms

We'll find the dimensions of the terms in the equation. The dimensions are represented by [T] for time, [L] for length, and [M] for mass. - For \(T\), the dimension is [T]. - For \(r\), the dimension is [L]. - For \(G\), the dimension is [L\(^3\)/M\(\cdot\)T\(^2\)]. (Given) - For \(M\), the dimension is [M].
03

Substitute the dimensions in the given equation

Now, we will substitute the dimensions of each term in the given equation: $$ [T] = \frac{[L^3]}{[L^3M^{-1}T^{-2}] \cdot [M]} $$
04

Simplify the equation

After substituting the dimensions, we need to simplify the equation by cancelling out the common terms and performing the required operations. $$ [T] = [\frac{L^3}{L^3 M^{-1} T^{-2}}\cdot M] $$ $$ [T] = [M^1 L^3\cdot T^2] $$
05

Check if the dimensions are the same

We can now compare the dimensions of both sides of the equation: $$ [T] = [T^2] $$ Since the dimensions of both sides of the equation are not the same, the given equation for the period of a planet is not dimensionally correct.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In a laboratory you measure the decay rate of a sample of radioactive carbon. You write down the following measurements: $$\begin{array}{lrrrrrrr} \hline \text { Time (min) } & 0 & 15 & 30 & 45 & 60 & 75 & 90 \\\ \text { Decays/s } & 405 & 237 & 140 & 90 & 55 & 32 & 19 \\\ \hline \end{array}$$ (a) Plot the decays per second versus time. (b) Plot the natural logarithm of the decays per second versus the time. Why might the presentation of the data in this form be useful?
The weight \(W\) of an object is given by \(W=m g,\) where \(m\) is the object's mass and \(g\) is the gravitational field strength. The SI unit of field strength \(g\), expressed in SI base units, is \(\mathrm{m} / \mathrm{s}^{2} .\) What is the SI unit for weight, expressed in base units?
It is useful to know when a small number is negligible. Perform the following computations. (a) \(186.300+\) 0.0030 (b) \(186.300-0.0030\) (c) $186.300 \times 0.0030\( (d) \)186.300 / 0.0030$ (e) For cases (a) and (b), what percent error will result if you ignore the \(0.0030 ?\) Explain why you can never ignore the smaller number, \(0.0030,\) for case (c) and case (d)? (f) What rule can you make about ignoring small values?
You have just performed an experiment in which you measured many values of two quantities, \(A\) and \(B\). According to theory, \(A=c B^{3}+A_{0} .\) You want to verify that the values of \(c\) and \(A_{0}\) are correct by making a graph of your data that enables you to determine their values from a slope and a vertical axis intercept. What quantities do you put on the vertical and horizontal axes of the plot?
Estimate the number of hairs on the average human head. [Hint: Consider the number of hairs in an area of 1 in. and then consider the area covered by hair on the head.]
See all solutions

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Read Explanation

Physical Quantities And Units

Read Explanation

Radiation

Read Explanation

Atoms and Radioactivity

Read Explanation

Waves Physics

Read Explanation

Physics of Motion

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free