We present the first spatially resolved (~0.5 kpc) measurements of the molecular gas depletion time $tau_{depl}$ across the disk of the interacting spiral galaxy NGC,2276, a system with an asymmetric morphology in various SFR tracers. To estimate $tau_{depl}$, we use new NOEMA observations of the $^{12}$CO(1-0) emission tracing the bulk molecular gas reservoir in NGC 2276, and extinction-corrected H$alpha$ measurements obtained with the PMAS/PPaK integral field unit for robust estimates of the SFR. We find a systematic decrease in $tau_{depl}$ of 1-1.5 dex across the disk of NGC 2276, with a further, abrupt drop in $tau_{depl}$ of ~1 dex along the galaxys western edge. The global $tau_{depl}$ in NGC 2776 is $tau_{depl}=0.55$ Gyr, insistent with literature measurements for the nearby galaxy population. Such a large range in $tau_{depl}$ on sub-kpc scales has never previously been observed within an individual isolated or pre-merger system. When using a metallicity-dependent molecular gas conversion factor X$rm_{CO}$ the variation decreases by 0.5 dex. We attribute the variation in $tau_{depl}$ to the influence of galactic-scale tidal forces and ram pressure on NGC 2276s molecular interstellar medium (ISM). Our observations add to the growing body of numerical and observational evidence that galaxy-galaxy interactions significantly modify the molecular gas properties and star-forming activity within galactic disks throughout the interaction, and not just during the final merger phase.
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