Determining the power of frictional force

[Gourab_chill]

Homework Statement

The question was:
As a ball rolls down the inclined plane, the power of gravitational force Pg and frictional force Pf thereafter :-
(A) Pg increases with time and Pf remains constant with time
(B)Pg as well as Pf remain constant with time.
(C) Pg as well as Pf increase with time.
(D)Pg remains constant and Pf decreases with time.

The correct answer is (A). But my question is how come?

Relevant Equations

P=F.v

I can say that the frictional force always against the rolling sphere and the velocity is increasing for the ball. So The dot product F.v keeps on getting more and more negative, so how can the Pf remain constant? Well the velocity increases along the incline and the force of gravity is down the incline too so the Fg keeps on increasing, right? Or am I making a mistake somewhere?

 

[jbriggs444]

(A) Pg increases with time and Pf remains constant with time
[...]
The correct answer is (A). But my question is how come?
[...]
I can say that the frictional force always against the rolling sphere and the velocity is increasing for the ball. So The dot product F.v keeps on getting more and more negative, so how can the Pf remain constant?

My best guess is that it is something of a trick question. Since the ball is rolling down the plane, the frictional force is from static friction. The force of static friction dissipates no mechanical energy. So the power associated with it is constant and zero.

For this to be the intended explanation, one must count the work done by static friction in terms of the motion of the contacted surface rather than in terms of the motion of the center of mass. The equation:$$P=\vec{F} \cdot \vec{v}$$ does not tell you what $\vec{v}$ is the velocity of. It matters.

[It also does not tell you what reference frame to use, but we can assume the lab frame in which the inclined plane is at rest. That choice eliminates some complications that might otherwise arise]

 

[Gourab_chill]

For this to be the intended explanation, one must count the work done by static friction in terms of the motion of the contacted surface rather than in terms of the motion of the center of mass. The equation:$$P=\vec{F} \cdot \vec{v}$$ does not tell you what $\vec{v}$ is the velocity of. It matters.

Yes, i agree that's the reason behind the power being constant; it is actually zero.