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Found in: Page 16

### Physics For Scientists & Engineers

Book edition 9th Edition
Author(s) Raymond A. Serway, John W. Jewett
Pages 1624 pages
ISBN 9781133947271

# (a) Assume the equation ${\mathbit{x}}{\mathbf{=}}{\mathbit{A}}{\mathbit{t}}{}^{{\mathbf{3}}}{\mathbf{+}}{\mathbit{B}}{\mathbit{t}}$ 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 $\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}{\mathbf{=}}{\mathbf{3}}{\mathbit{A}}{\mathbit{t}}{}^{{\mathbf{2}}}{\mathbf{+}}{\mathbit{B}}$.

(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 $\frac{dx}{dt}=3At{}^{2}+B$ is $\frac{L}{T}$.

See the step by step solution

## 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.

$x=At{}^{3}+Bt$ …… (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).

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

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

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

And

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

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).

$\frac{dx}{dt}=3At{}^{2}+B$

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

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

Therefore, the dimension of the derivative equation $\frac{dx}{dt}=3At{}^{2}+B$ is $\frac{L}{T}$.

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