Chapter 1, Problem 91P

(a)

To determine

Combination of fundamental constants to get the dimension of time.

Answer to Problem 91P

Combination of fundamental constants to get the dimension of time is $\underset{\_}{\sqrt{\frac{\text{Gh}}{{\text{c}}^{\text{5}}}}}$.

Explanation of Solution

Consider the following combination of dimension of gravitational constant, Plank’s constant, and speed of light.

  $\begin{array}{c}{G}^x{h}^y{c}^z=t\\ {\left(\frac{{\text{m}}^3}{\text{kg}\cdot {\text{s}}^2}\right)}^x{\left(\frac{\text{kg}\cdot {\text{m}}^2}{\text{s}}\right)}^y{\left(\frac{\text{m}}{\text{s}}\right)}^z={\text{s}}^1\end{array}$        (I)

Consider coefficient of dimension of length.

  $3x+2y+z=0$        (II)

Consider the coefficient of dimension of mass.

  $\begin{array}{c}-x+y=0\\ x=y\end{array}$        (III)

Consider the coefficient of dimension of time.

  $-2x-y-z=1$        (IV)

Substitute, equation (III) in (IV).

  $\begin{array}{c}-2y-y-z=1\\ -3y-z=1\\ -3y-1=z\end{array}$        (V)

Substitute equation (V) and (III) in equation (II).

  $\begin{array}{c}3y+2y-3y-1=0\\ y=\frac{1}{2}=x\end{array}$        (VI)

Substitute, $\frac{1}{2}$ for $y$ in equation (V).

  $\begin{array}{c}-3\left(\frac{1}{2}\right)-1=z\\ z=-\frac{5}{2}\end{array}$

Conclusion:

Substitute, $\frac{1}{2}$ for $y$, and $x$, and $-\frac{5}{2}$ for $z$ in equation (I).

  $\begin{array}{c}{G}^{1/2}{h}^{1/2}{c}^{-5/2}=t\\ t=\sqrt{\frac{Gh}{c^2}}\end{array}$        (VII)

Therefore, the Combination of fundamental constants to get the dimension of time is $\underset{\_}{\sqrt{\frac{\text{Gh}}{{\text{c}}^{\text{5}}}}}$.

(b)

To determine

The value of time in seconds.

Answer to Problem 91P

The value of time in seconds is $\underset{\_}{12.141\times {10}^{-22}\text{s}}$.

Explanation of Solution

Use equation (VII) to obtain the answer.

Conclusion:

Substitute, $6.7\times {10}^{-11}{\text{m}}^3\cdot {\text{kg}}^{-1}\cdot {\text{s}}^{-2}$ for $G$, $6.6\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$ for $h$, and $3\times {10}^8\text{m}/\text{s}$ for $c$ in equation (VII).

  $\begin{array}{c}t=\sqrt{\frac{\left(6.7\times {10}^{-11}{\text{m}}^3\cdot {\text{kg}}^{-1}\cdot {\text{s}}^{-2}\right)\left(6.6\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}\right)}{{\left(3\times {10}^8\text{m}/\text{s}\right)}^2}}\\ =12.141\times {10}^{-22}\text{s}\end{array}$

Therefore, the value of time in seconds is $\underset{\_}{12.141\times {10}^{-22}\text{s}}$.