We describe the dynamical instability of spherical polytropes of index 5 that are truncated and confined by a constant pressure matching the pressure at the truncation radius. The instability is realized at the maximum mass along the sequence of different truncation radii and is analogous to the Bonnor - Ebert mass in the isothermal case. We have confirmed through direct numerical simulation that the polytropic star oscillates as expected when truncated at a normalized coordinate less than three . When truncated at normalized radii three or greater, oscillations seeded by errors in equilibrium grow in amplitude and the star rapidly collapses. Truncated polytropic configurations for a range of polytropic indices including n = 5 are expected to be dynamically unstable in a similar manner. Finally, a pressure-truncated n = 5 polytrope would be an interesting test case for verifying self-gravitating hydrodynamics codes as the initial data are analytic and an analytic expectation for the subsequent evolution can be checked quickly.