Protons in the LHC gain relativistic mass as they approach the speed of light, making them harder to accelerate, but the concept of "weight" becomes complex due to relativistic effects. While two objects moving at relativistic speeds will experience a stronger gravitational attraction toward each other, this cannot be simply calculated using Newton's gravitational equation. The increase in perceived mass is relative to the observer's frame of reference, meaning different observers will measure different masses based on their relative speeds. Time dilation affects interactions, such as the electromagnetic force between particles, but the underlying physics remains consistent across different frames. Understanding these concepts requires a solid grasp of Special Relativity, as intuitive explanations may not suffice for deeper inquiries.