Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 1: Problem 98

Acceleration due to gravity is given by $\mathrm{g}=\left(\mathrm{GM} / \mathrm{R}^{2}\right)\( what is the equation of the fractional error \)\Delta \mathrm{g} / \mathrm{g}\( in measurement of gravity \)\mathrm{g}$ ? [G \& M constant] (a) \(-(\Delta \mathrm{R} / \mathrm{R})\) (b) \(2(\Delta \mathrm{R} / \mathrm{R})\) (c) \(-2(\Delta \mathrm{R} / \mathrm{R})\) (d) \((1 / 2)(\Delta \mathrm{R} / \mathrm{R})\)

Short Answer

Expert verified
The short answer is: \( \frac{\Delta g}{g} = 2\left(\frac{\Delta R}{R}\right)\).

Step by step solution

01

Determine the function g(R)

The function g(R), representing gravity, is given by: \[g(R) = \frac{GM}{R^2}\]
02

Compute the derivative of g(R) with respect to R

To find the fractional error, we first need to compute the derivative of the function g(R) with respect to R. This will give us the sensitivity of g to changes in R: \[\frac{dg}{dR} = \frac{d}{dR} \left(\frac{GM}{R^2}\right)\] We use the power rule of differentiation on the function, \[R^{-2} \Rightarrow -2R^{-3}\] So, \[\frac{dg}{dR} = GM(-2R^{-3}) = -2\frac{GM}{R^3}\]
03

Compute the absolute error in g

Now we will find the absolute error in g, denoted as \(\Delta g\). For this, we use the following formula: \[\Delta g = \left|\frac{dg}{dR}\right|\Delta R\] Plugging in the values we computed earlier, we get \[\Delta g = \left|-2\frac{GM}{R^3}\right|\Delta R = 2\frac{GM}{R^3}\Delta R\]
04

Compute the fractional error in g

The fractional error is defined as the ratio of the absolute error to the value of the function g(R) itself. Using the expressions for \(\Delta g\) and \(g(R)\), we can find the fractional error \(\frac{\Delta g}{g}\) as follows: \[\frac{\Delta g}{g} = \frac{2\frac{GM}{R^3}\Delta R}{\frac{GM}{R^2}}\] Now we cancel out the common terms GM: \[\frac{\Delta g}{g} = \frac{2\Delta R}{R}\] Therefore, the equation of the fractional error in the measurement of gravity g is option (b) \[\boxed{\frac{\Delta g}{g} = 2\left(\frac{\Delta R}{R}\right)}\]

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

Match column - I with column - II $$ \begin{array}{|l|l|} \hline \multicolumn{1}{|c|} {\text { Column - I }} & \multicolumn{1}{|c|} {\text { Column - II }} \\ \hline \text { (1) capacitance } & \text { (a) } \mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-3} \mathrm{~A}^{-1} \\ \hline \text { (2) Electricfield } & \text { (b) } \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-1} \\ \hline \text { (3) planck's constant } & \text { (c) } \mathrm{M}^{-1} \mathrm{~L}^{-2} \mathrm{~T}^{4} \mathrm{~A}^{2} \\ \hline \text { (4) Angular momentum } & \text { (d) } \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-1} \\ \hline \end{array} $$ (a) \(a, c, b, d\) (b) \(c, a, d, b\) (c) \(c, a, b, d\) (d) \(a, b, d, c\)

Dimensional formula for conductance is ........... (a) \(\mathrm{M}^{-1} \mathrm{~L}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{2}\) (b) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2} \mathrm{~A}^{1}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{-2} \mathrm{~T}^{3} \mathrm{~A}^{2}\) (d) \(\mathrm{M}^{-1} \mathrm{~L}^{-2} \mathrm{~T}^{3} \mathrm{~A}^{2}\)

\(1 \mathrm{~g}=\ldots \ldots \ldots \ldots \ldots\) amu (a) \(6.02 \times 10^{23}\) (b) \(6.02 \times 10^{-23}\) (c) \(1.66 \times 10^{-27}\) (d) \(1.66 \times 10^{27}\)

Write the unit of angular acceleration in the SI system. (a) \(\mathrm{N} \cdot \mathrm{Kg}\) (b) \(\mathrm{rad} /(\mathrm{sec})^{2}\) (c) \(\mathrm{m} / \mathrm{sec}\) (d) \(\mathrm{N} / \mathrm{kg}\)

Which physical quantity has unit of pascal - second? (a) Force (b) Energy (c) Coefficient of viscosity (d) velocity
See all solutions

Recommended explanations on English Textbooks

Language Acquisition

Read Explanation

Creative Story

Read Explanation

Discourse

Read Explanation

Listening and Speaking

Read Explanation

English Language Study

Read Explanation

5 Paragraph Essay

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