It has been suggested that gravitational potential can have a significant role in suppressing star formation in nearby galaxies. To establish observational constraints on this scenario, we investigated the connection between the dynamics – taking the circular velocity curves (CVCs) as a proxy for the inner gravitational potential – and star formation quenching in 215 non-active galaxies across the Hubble sequence from the Calar Alto Legacy Integral Field Area (CALIFA) survey. Our results show that galaxies with similar CVCs tend to have a certain star-formation quenching pattern. To explore these findings in more details, we constructed kiloparsec(kpc)-resolved relations of the equivalent width of the H α ( W H α ) versus the amplitude ( V c ) and shape ( β  = dln V c /dln R ) of the circular velocity at given radii. We find that the W H α  −  V c is a declining relationship, where the retired regions of the galaxies (the ones with W H α values of below 3 Å) tend to have higher V c . Concurrently, W H α  −  β is a bimodal relationship, which is characterised by two peaks: concentration of the star forming regions at a positive β (rising CVC) and a second concentration of the retired regions with a negative β (declining CVC). Our results show that both the amplitude of the CVC – driven by the mass of the galaxies – and its shape – which reflects the internal structure of the galaxies – play an important role in the quenching history of a galaxy.