91影视

The expression for density is given by,

$\begin{array}{l}\mathbf{D}\mathbf{=}\frac{\mathbf{m}}{\frac{\mathbf{4}}{\mathbf{3}}{\mathbf{蟺搁}}^{\mathbf{3}}}\\ \mathbf{m}\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}{\mathbf{蟺搁}}^{\mathbf{3}}\mathbf{D}\end{array}$

Here, mis mass, Dis the density and Ris the radius

The expression for Heisenberg鈥檚 uncertainty principle in terms of mass and velocity is given by,

$\begin{array}{c}{\mathbf{尘螖惫}}_{\mathbf{\text{x}}}\mathbf{螖虫}\mathbf{鈮�}\frac{\mathbf{h}}{\mathbf{2}}\\ \left(\frac{4}{3}{\mathrm{蟺搁}}^{3}D\right){\mathbf{螖惫}}_{\mathbf{\text{x}}}\mathbf{螖虫}\mathbf{鈮�}\frac{\mathbf{h}}{\mathbf{2}}\\ \mathbf{8}{\mathbf{蟺顿搁}}^{\mathbf{3}}{\mathbf{螖惫}}_{\mathbf{\text{x}}}\mathbf{螖虫}\mathbf{鈮�}\mathbf{3}\mathbf{h}\\ \mathbf{螖虫}\mathbf{鈮�}\frac{\mathbf{3}\mathbf{h}}{\mathbf{8}{\mathbf{蟺顿搁}}^{\mathbf{3}}{\mathbf{螖惫}}_{\mathbf{\text{x}}}}\end{array}$

The expression for uncertainty in position as a fraction of radius is given by,

$\begin{array}{c}\frac{\mathbf{螖虫}}{\mathbf{R}}\mathbf{鈮�}\mathbf{k}\\ \frac{\left(\frac{3未}{8{\mathrm{eDR}}^3{\mathrm{螖惫}}_{\text{x}}}\right)}{\mathbf{R}}\mathbf{鈮�}\mathbf{k}\\ \frac{\mathbf{3}\mathbf{h}}{\mathbf{8}{\mathbf{蟺顿搁}}^{\mathbf{4}}{\mathbf{螖惫}}_{\mathbf{\text{x}}}}\mathbf{鈮�}\mathbf{k}\\ \mathbf{R}\mathbf{鈮�}\sqrt{\frac{\mathbf{3}\mathbf{h}}{\mathbf{8}{\mathbf{蟺kD螖惫}}_{\mathbf{\text{x}}}}}\end{array}$

Here, ${\mathbf{螖惫}}_{\mathbf{\text{x}}}$the change in velocity, Dis the density, h is Planck鈥檚 constant, $\mathbf{螖虫}$

is the uncertainty in position and Ris the radius.

Consider volume for the object is$\left(V=\frac{4}{3}蟺r^5\right)$and the uncertainty in velocity, consequently the momentum, is known.

Substitute given values we can get an estimate of the value of$r$(note$螖x=0.10\%$of $r$).

$\begin{array}{l}\mathrm{螖虫螖辫}鈮�\frac{\mathrm{鈩�}}{2}\\ \mathrm{螖虫尘螖惫}鈮�\frac{\mathrm{鈩�}}{2}\end{array}$

Here,$螖x$is the uncertainty in position, $\mathrm{螖辫}$is uncertainty in momentum, m is the mass and $螖v$is velocity change and $\mathrm{鈩�}$is Planck鈥檚 constant.

$\begin{array}{c}螖x=0.001\text{鈥�}r,\text{鈥夆赌�}\\ m=蚁V\\ m=蚁脳\frac{4}{3}蟺r^3\end{array}$

Now substitute the value in formula

$\begin{array}{c}\mathrm{螖虫尘螖惫}鈮�\frac{\mathrm{鈩�}}{2}\\ 0.001\text{鈥�}\mathrm{r}脳\mathrm{蚁}脳\frac{4}{3}{\mathrm{蟺谤}}^3\mathrm{螖惫}鈮�\frac{\mathrm{鈩�}}{2}\end{array}$

Solve further as:

$\begin{array}{l}\mathrm{r}鈮�\sqrt[4]{\frac{\mathrm{鈩�}}{2脳0.001脳\mathrm{蚁}脳\frac{4}{3}\mathrm{蟺}脳\mathrm{螖惫}}}\\ \mathrm{r}=\sqrt[4]{\frac{1.054脳{10}^{鈭�34}\text{鈥�}\frac{\text{kg}鈰�{\text{m}}^2}{\text{s}}}{2脳0.001脳2.7脳{10}^3\text{鈥�}\text{kg}/{\text{m}}^3脳\frac{4}{3}\mathrm{蟺}脳\frac{{10}^{鈭�6}\text{m}}{10脳365脳24脳60脳60\text{s}}}}\\ \mathrm{r}=\sqrt[4]{1.469脳{10}^{鈭�20}}\text{m}\\ \mathrm{r}鈮�{10}^{鈭�5}\mathbf{m}\end{array}$

Thus, the radius of the microscopic object to obtain the required uncertainty in position is$\mathrm{r}鈮�{10}^{鈭�5}\text{鈥�}\mathbf{m}$ .