The Internet of Things (IoT) is empowering various sectors and aspects of daily life. Green IoT systems typically involve Low-Power and Lossy Networks (LLNs) with resource-constrained nodes. Lightweight routing protocols, such as the Routing Protocol for Low-Power and Lossy Networks (RPL), are increasingly being applied for efficient communication in LLNs. However, RPL is susceptible to various attacks, such as the black hole attack, which compromises network security. The existing black hole attack detection methods in Green IoT rely on static thresholds and unreliable metrics to compute trust scores. This results in increasing false positive rates, especially in resource-constrained IoT environments. To overcome these limitations, we propose a delta-threshold-based trust model called the Optimized Reporting Module (ORM) to mitigate black hole attacks in Green IoT systems. The proposed scheme comprises both direct trust and indirect trust and utilizes a forgetting curve. Direct trust is derived from performance metrics, including honesty, dishonesty, energy, and unselfishness. Indirect trust requires the use of similarity. The forgetting curve provides a mechanism to consider the most significant and recent feedback from direct and indirect trust. To assess the efficacy of the proposed scheme, we compare it with the well-known trust-based attack detection scheme. Simulation results demonstrate that the proposed scheme has a higher detection rate and low false positive alarms compared to the existing scheme, confirming the applicability of the proposed scheme in green IoT systems.