Is the normal force exerted on an object always equal to the object's weight?
No. The normal force exerted on an object need not be equal to its weight all the time. It depends on the position of the object with respect to the reference frame.
For example consider the two situations given below
1) Object of mass M at rest on a horizontal surface
2) The same object on an incline at an angle of with respect to the horizontal.
In the first case Normal force is equal to the weight of the object in magnitude. But in the second case the normal force is not equal to weight.
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