Is it possible for a body to have inertia but no weight?
Yes, a body can have inertia but no weight. Inertia of a body is related to mass. Whereas weight of body is defined as
M = mass of body
g = acceleration due to gravity
therefore, weight of the object can be zero when the acceleration due to gravity becomes zero.
Since we know that at the centre of earth, the value of acceleration due to gravity is zero and hence the weight of the object is zero at the centre, but it remains mass(inertia) remains constant.
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