Estimate the range of the force mediated by an \({\omega ^0}\)g meson that has mass \(783\;MeV/{c^2}\).

The term force may be defined as the product of mass and acceleration.

The mass of the\({\omega ^0}\)particle is\(783\;{\rm{MeV/}}{{\rm{c}}^{\rm{2}}}\), and the minimum uncertainty energy is:

\(\begin{array}{l}\Delta E = 783\;{\rm{MeV}}\left( {1.60 \times {{10}^{ - 13}}\;{\rm{J/MeV}}} \right)\\\Delta E = 1.25 \times {10^{ - 10}}\;{\rm{J}}\end{array}\)

The lifetime\(\delta t\)of the particle, according to the Heisenberg uncertainty principle:\(\begin{array}{l}\delta t = \frac{h}{{2\Delta E}}\\\delta t = \frac{{1.055 \times {{10}^{ - 34}}\;{\rm{J}} \cdot {\rm{s}}}}{{2\left( {1.25 \times {{10}^{ - 10}}\;{\rm{J}}} \right)}}\\\delta t = 4.21 \times {10^{ - 25}}\;{\rm{s}}\end{array}\)

If we consider the speed of light is the average speed of the particle than

\(\begin{array}{l}r = c\delta t\\r = \left( {3 \times {{10}^8}\;{\rm{m/s}}} \right)\left( {4.21 \times {{10}^{ - 25}}\;{\rm{s}}} \right)\\r = 1.26 \times {10^{ - 16}}\;{\rm{m}}\end{array}\)

Hence, the range of the force is \(1.26 \times 1{0^{ - 16}}\;m\).