Book edition 9th Edition

Author(s) Raymond A. Serway, John W. Jewett

Pages 1624 pages

ISBN 9781133947271

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Short Answer

(a) Assume the equation $x = A t {}^{3}+ B t$ describes the motion of a particular object, with having the dimension of length and having the dimension of time. Determine the dimensions of the constants A and B .
(b) Determine the dimensions of the derivative $\frac{d x}{d t} = 3 A t {}^{2}+ B$ .

(a) The dimensions of the constants A and B are $\frac{L}{T^{3}}$ and $\frac{L}{T}$ respectively.

(b) The dimension of the derivative equation $\frac{d x}{d t} = 3 A t {}^{2}+ B$ is $\frac{L}{T}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Explanation of dimensions for the constant variables

Adimensional analysis is used to measure the dimensions of any physical quantity, which has no fixed values.Similarly, the dimensions of a constant valued parameter can be determined by using the dimensions of the variable physical quantities.

Step 2:Expression and dimensions of the motion of the object

Write the expression for the motion of the object in terms of the time physical variable.

$x = A t {}^{3}+ B t$ …… (1)

Here, x is the motion of the object, t is the time, and A, B are the constant parameters.

The dimension of the motion of the object x is the dimension of the length L.

The dimension of time t is T.

Step 3(a): Determination of dimensionsof A and B constants

Replace L for x and T for t in equation (1).

$L = A T^{3} + B T$

Here, each term on theright-hand side must be equal to left hand side individually.

$\begin{array}{rcl} A T^{3} = L \\ A & = & \frac{L}{T^{3}} \end{array}$

And

$\begin{array}{rcl} B T = L \\ B & = & \frac{L}{T} \end{array}$

Therefore, the dimensions of the constants A and B are $\frac{L}{T^{3}}$ and $\frac{L}{T}$ respectively.

Step 4(b): Determination of the dimensions for the derivative of equation (1)

Write the derivative expression of equation (1).

$\frac{d x}{d t} = 3 A t {}^{2}+ B$

Replace T for t , $\frac{L}{T^{3}}$ for A and $\frac{L}{T}$ for B in the above equation.

role="math" localid="1663588183957" $\begin{array}{rcl} \frac{d x}{d t} & = & \frac{L}{T^{3}} T^{2} + \frac{L}{T} \\ = & \frac{L}{T} \end{array}$

Therefore, the dimension of the derivative equation $\frac{d x}{d t} = 3 A t {}^{2}+ B$ is $\frac{L}{T}$ .

Hence, the dimensions for the constant parameters A and B and the derivative equation $\frac{d x}{d t} = 3 A t {}^{2}+ B$ are $\frac{L}{T^{3}}$ , $\frac{L}{T},$ and $\frac{L}{T}$ respectively.

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Physics For Scientists & Engineers

Answers without the blur.

Short Answer

(a) Assume the equation $x = A t {}^{3}+ B t$ describes the motion of a particular object, with having the dimension of length and having the dimension of time. Determine the dimensions of the constants A and B .
(b) Determine the dimensions of the derivative $\frac{d x}{d t} = 3 A t {}^{2}+ B$ .

Step by Step Solution

Step 1: Explanation of dimensions for the constant variables

Step 2:Expression and dimensions of the motion of the object

Step 3(a): Determination of dimensionsof A and B constants

Step 4(b): Determination of the dimensions for the derivative of equation (1)

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Physics For Scientists & Engineers

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(a) Assume the equation x=At+3Bt describes the motion of a particular object, with having the dimension of length and having the dimension of time. Determine the dimensions of the constants A and B .(b) Determine the dimensions of the derivative dxdt=3At+2B.

Step by Step Solution

Step 1: Explanation of dimensions for the constant variables

Step 2:Expression and dimensions of the motion of the object

Step 3(a): Determination of dimensionsof A and B constants

Step 4(b): Determination of the dimensions for the derivative of equation (1)

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(a) Assume the equation $x = A t {}^{3}+ B t$ describes the motion of a particular object, with having the dimension of length and having the dimension of time. Determine the dimensions of the constants A and B .
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