Q: Car slightly shutters when coming to an abrupt stop applying brakes hard. Failing master cylinder?

This may be a sign of warped brake rotors. As you may know, when you press on your brake pedal, this activates a hydraulic pressure which forces small pistons inside your brake calipers to push the brake pads against the brake rotors which is what causes the stopping power you are familiar with. When this happens, an enormous amount of heat is generated due to the friction between the brake pads and rotors. As the brake rotors eventually cool down and then heat up again the next time you press on the brakes, over time this can cause them to warp slightly causing very small low spots on the surface of the brake rotor. When this happens and the brake pads are applied again to the rotors, this causes an uneven surface (on the rotor) to be applied to the flat surface brake pad, which can cause a vibration, shaking or very loud grinding sound. I would recommend having an expert from YourMechanic come to your location to diagnose and replace your brake rotors.