Given that the normal acceleration of the object along the circle is proportional to \(t^\alpha\), where \(t\) is time and \(\alpha\) is a positive constant, let's analyze the power developed by the forces acting on the object.
Power (\(P\)) is the rate of doing work or transferring energy. In the context of circular motion, the power developed by forces can be calculated using the formula:
\[P = \vec{F} \cdot \vec{v}\]
Where:
- \(\vec{F}\) is the net force acting on the object.
- \(\vec{v}\) is the velocity of the object.
For circular motion, velocity (\(\vec{v}\)) is perpendicular to the net force (\(\vec{F}\)) at every point along the circle. Therefore, the dot product of \(\vec{F}\) and \(\vec{v}\) is zero, and the power is zero at each instant.
This means that no power is being developed by the forces acting on the object as it moves along the circle with normal acceleration proportional to \(t^\alpha\).
In conclusion, the time dependence of the power developed by all the forces acting on the object is proportional to zero, which means the power remains constant and independent of time.
