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In an upward accelerating elevator with acceleration a, from the inertial reference frame, a normal force equal to N= m(a+g) will be acting on the object inside the elevator of mass m. Now, for normal force to be felt, the object must be exerting a force on the elevator floor. I understand the mg part of the normal force, but I don't understand why the object feels another downward force ma if it is moving upwards.

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I don't understand why the object feels another downward force ma if it is moving upwards

It doesn't. The forces on the object are (1) its weight $mg$ acting downwards and (2) a normal force from the floor on the elevator $m(g+a)$ acting upwards. The net force on the object is therefore $ma$ upwards, which is what makes it accelerate upwards with acceleration $a$.

If you stand in a lift you only feel the upwards force from the elevator $m(g+a)$ - since gravity affects all parts of your body, you do not feel your weight directly. However, you are used to being in a state of equilibrium in which your weight is equal to the upwards force that you feel from the ground. So when you feel a greater upwards force $m(g+a)$ from the elevator you assume that this must mean a greater downwards force also. This mistaken assumption/feeling arises from not realising that your body is no longer in equilibrium.

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If you are standing on a scale in the elevator, the reading on the scale (the apparent weight) increases because the scale must exert an extra force to accelerate you upward.

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