We present a detailed radio morphological study and spectral analysis of the wide-angle-tail radio source 4C +24.36 associated with the dominant galaxy in the relaxed galaxy cluster AWM 4. Our study is based on new high sensitivity GMRT observations at 235 MHz, 327 MHz and 610 MHz, and on literature and archival data at other frequencies. We find that the source major axis is likely oriented at a small angle with respect to the plane of the sky. The wide-angle-tail morphology can be reasonably explained by adopting a simple hydrodynamical model in which both ram pressure (driven by the motion of the host galaxy) and buoyancy forces contribute to bend the radio structure. The spectral index progressively steepens along the source major axis from $\alpha \sim$0.3 in the region close to the radio nucleus to beyond 1.5 in the lobes. The results of the analysis of the spectral index image allow us to derive an estimate of the radiative age of the source of $\sim$ 160 Myr. The cluster X-ray emitting gas has a relaxed morphology and short cooling time, but its temperature profile is isothermal out to at least 160 kpc from the centre. Therefore we seek evidence of energy ejection from the central AGN to prevent catastrophic cooling. We find that the energy injected by 4C +24.36 in the form of synchrotron luminosity during its lifetime is far less than the energy required to maintain the high gas temperature in the core. We also find that it is not possible for the central source to eject the requisite energy in the intracluster gas in terms of the enthalpy of buoyant bubbles of relativistic fluid, without creating discernible large cavities in the existing X-ray XMM-Newton observations.