In the above case, the normal force exerted by the elevator on the box would be $= 10\mathrm{kg}\ g + 10\mathrm{kg}\ 5\ \mathrm{m/s^2}$, assuming $g = 10\ \mathrm{m/s^2}$, the elevator would exert a normal force of $150\ \mathrm N$ on the box. But wouldn't the box exert an equal and opposite force on the elevator resulting in no motion? I know this is a really stupid question but I've not been able to understand where I'm going wrong.
Another related question: the box exert a force of $mg = 10\ \mathrm{kg}\ g = 100\ \mathrm N$ on the elevator, or, since the elevator is pushing on the box with $150\ \mathrm N$, it will react by pushing down on the elevator with $150\ \mathrm N$
Best Answer
When you do a force balance on a body, you include only the forces that are acting on that body, not forces that the body exerts on other bodies. The force that the box exerts on the elevator should not included in the force balance on the box. Similarly, the force that the elevator exerts on the box should not be included in the force balance on the elevator.