Question

Use the potential-energy diagram in Figure $42.8$to estimate the ratio of the gravitational potential energy to the nuclear potential energy for two neutrons separated by $1.0$$fm$.

Short answer

Ratio of gravitational potential energy to the nuclear potential energy is

$-0.822\times {10}^{-27}.$

Step-by-step solution

Given information.

We have been given that,

Two neutrons are separated by $1.0fm$,

Mass of neutron=$1.67493\times {10}^{-27}Kg$.

We need to find the ratio of gravitational potential energy to the nuclear potential energy for the two neutrons.

Simplify.

The gravitational potential energy is given by,

$U_g$$=$$-\frac{GMm}{r}$

Since,

$M=m=1.67493\times {10}^{-27}Kg$$=$mass of neutrons

Therefore, $G=6.67\times {10}^{-11}Nk{g}^{-2}{m}^2$, $r=1.0fm$

Putting the values in formula,

$$\begin{gathered} U_g=-\frac{6.67\times {10}^{-11}\times {\left(1.67493\times {10}^{-27}\right)}^2}{1.0\times {10}^{-15}}J/kg \\ =-\frac{18.711\times {10}^{-65}}{{10}^{-15}}J/kg \\ =-18.711\times {10}^{-50}J/kg \end{gathered}$$

Now,

The nuclear potential energy is given by,

$U_e=\frac{{\rm K}{q}^2}{r}J/kg$

$$\begin{gathered} =\frac{8.89\times {(1.6\times {10}^{-19})}^2}{{10}^{-15}}J/kg \\ =22.758\times {10}^{-23}J/kg \end{gathered}$$

Now,

$$\begin{gathered} \frac{U_g}{Ue}=\frac{-18.711\times {10}^{-50}}{22.758\times {10}^{-23}} \\ =-0.822\times {10}^{-27} \end{gathered}$$